GRE 2025 Model Paper Set 3 Question Paper with Solutions PDF

Step 1: Understanding the Concept:

This question requires us to evaluate the arithmetic expressions in both columns and then compare their final values. The order of operations (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is crucial.


Step 2: Key Formula or Approach:

First, solve the expression inside the parentheses. Then, multiply the result by the number outside the parentheses.


Step 3: Detailed Explanation:

For Column A:

The expression is \(2(8-7)\).

First, evaluate the subtraction inside the parentheses:
\[ 8 - 7 = 1 \]
Now, multiply this result by 2:
\[ 2 \times 1 = 2 \]
So, the value of Column A is 2.


For Column B:

The expression is \(2(7-8)\).

First, evaluate the subtraction inside the parentheses:
\[ 7 - 8 = -1 \]
Now, multiply this result by 2:
\[ 2 \times (-1) = -2 \]
So, the value of Column B is -2.


Step 4: Final Answer:

Comparing the two quantities:

Column A = 2

Column B = -2

Since 2 is greater than -2, the quantity in Column A is greater.
Quick Tip: Be very careful with signs when performing arithmetic operations. A simple sign error is a common mistake in these types of questions.

Step 1: Understanding the Concept:

We are given a linear equation with two variables, \(x\) and \(y\). We need to determine if there is a fixed relationship between \(x\) and \(y\).


Step 2: Key Formula or Approach:

The strategy is to test different possible values for \(x\) and \(y\) that satisfy the given equation, \(x+y=2\). If the relationship between \(x\) and \(y\) changes for different valid pairs of values, then the relationship cannot be determined.


Step 3: Detailed Explanation:

The given equation is \(x+y=2\). There are infinitely many pairs of \((x, y)\) that satisfy this equation. Let's test a few cases:


Case 1: Let \(x = 1\).

Substituting into the equation: \(1 + y = 2\), which gives \(y = 1\).

In this case, Column A (\(x\)) = 1 and Column B (\(y\)) = 1. The quantities are equal.


Case 2: Let \(x = 2\).

Substituting into the equation: \(2 + y = 2\), which gives \(y = 0\).

In this case, Column A (\(x\)) = 2 and Column B (\(y\)) = 0. Column A is greater than Column B.


Case 3: Let \(x = 0\).

Substituting into the equation: \(0 + y = 2\), which gives \(y = 2\).

In this case, Column A (\(x\)) = 0 and Column B (\(y\)) = 2. Column B is greater than Column A.


Step 4: Final Answer:

Since we have found cases where the two quantities are equal, where A is greater than B, and where B is greater than A, we can conclude that the relationship between \(x\) and \(y\) cannot be determined from the given information.
Quick Tip: For quantitative comparison questions with variables, always try to plug in different types of numbers (positive, negative, zero, fractions) to see if the relationship holds true for all cases. If you find conflicting results, the answer is always (D).

Step 1: Understanding the Concept:

This problem requires the addition of two fractions with a common denominator and then comparing the result to the integer 1.


Step 2: Key Formula or Approach:

To add fractions with the same denominator, you add the numerators and keep the denominator the same: \(\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}\).


Step 3: Detailed Explanation:

For Column A:

The expression is \(\frac{3}{7} + \frac{2}{7}\).

Since the denominators are the same, we add the numerators:
\[ \frac{3 + 2}{7} = \frac{5}{7} \]
So, the value of Column A is \(\frac{5}{7}\).


For Column B:

The value is 1.


Step 4: Final Answer:

Now we compare Column A and Column B.

Column A = \(\frac{5}{7}\)

Column B = 1

A fraction is less than 1 if its numerator is less than its denominator. Here, \(5 < 7\), so \(\frac{5}{7} < 1\).

Therefore, the quantity in Column B is greater.
Quick Tip: When comparing a proper fraction (numerator \(<\) denominator) to 1, the fraction is always smaller. This can save you calculation time.

Step 1: Understanding the Concept:

The problem involves finding a missing side in a right-angled triangle inscribed in a circle. The key concept is that the hypotenuse of both triangles is the radius of the circle, and we can use the Pythagorean theorem.


Step 2: Key Formula or Approach:

The Pythagorean theorem states that for a right-angled triangle with legs \(a\) and \(b\) and hypotenuse \(c\), we have \(a^2 + b^2 = c^2\).


Step 3: Detailed Explanation:

Find the radius of the circle:

Look at the bottom right-angled triangle. Its legs are of length 3 and 4. The hypotenuse is the line segment from the center O to the circle's edge, which is the radius (let's call it \(r\)).

Using the Pythagorean theorem:
\[ r^2 = 3^2 + 4^2 \] \[ r^2 = 9 + 16 \] \[ r^2 = 25 \] \[ r = \sqrt{25} = 5 \]
So, the radius of the circle is 5.


Find the value of x:

Now look at the other right-angled triangle. Its legs are of length 3 and \(x\). Its hypotenuse is also the radius of the circle, which we found to be 5.

Using the Pythagorean theorem again:
\[ 5^2 = x^2 + 3^2 \] \[ 25 = x^2 + 9 \]
Subtract 9 from both sides:
\[ 25 - 9 = x^2 \] \[ 16 = x^2 \] \[ x = \sqrt{16} = 4 \]
Since \(x\) represents a length, we take the positive root. So, \(x=4\).


Step 4: Final Answer:

We are comparing Column A (\(x\)) and Column B (5).

Column A = 4

Column B = 5

Since \(4 < 5\), the quantity in Column B is greater.
Quick Tip: Recognizing common Pythagorean triples like (3, 4, 5) can save time. When you see a right triangle with legs 3 and 4, you should immediately know the hypotenuse is 5.

Step 1: Understanding the Concept:

This is a counting problem based on the fundamental counting principle. We need to find the total number of connections (lines) between two distinct groups of items (houses).


Step 2: Key Formula or Approach:

If there are \(m\) items in one group and \(n\) items in another group, and each item from the first group is connected to each item from the second group, the total number of connections is \(m \times n\).


Step 3: Detailed Explanation:

For Column A:

We have two groups of houses:

Group 1: 6 houses on one side of the street.

Group 2: 4 houses on the other side.

Each house from Group 1 must be connected to each house in Group 2.

Consider the first house in Group 1. It is connected to all 4 houses in Group 2. That's 4 lines.

The second house in Group 1 is also connected to all 4 houses in Group 2. That's another 4 lines.

This pattern continues for all 6 houses in Group 1.

So, the total number of lines is the number of houses in Group 1 multiplied by the number of houses in Group 2.
\[ Total lines = 6 \times 4 = 24 \]
The value of Column A is 24.


For Column B:

The value is 12.


Step 4: Final Answer:

Comparing the two quantities:

Column A = 24

Column B = 12

Since \(24 > 12\), the quantity in Column A is greater.
Quick Tip: This type of problem is a classic application of the multiplication principle in combinatorics. Be careful to read the question to ensure you are connecting items between groups, not within the same group.

Step 1: Understanding the Concept:

This problem requires calculating the areas of two different triangles defined by coordinates on a Cartesian plane.


Step 2: Key Formula or Approach:

The area of a triangle is given by the formula: Area \( = \frac{1}{2} \times base \times height\). We need to identify a base and the corresponding perpendicular height for each triangle.


Step 3: Detailed Explanation:

For Column A: Area of triangular region OPQ

The vertices are O(0,0), P(2,2), and Q(2,0).

Let's choose the segment OQ as the base. O is at \(x=0\) and Q is at \(x=2\), so the length of the base OQ is \(2 - 0 = 2\).

The height corresponding to this base is the perpendicular distance from point P to the x-axis, which is the y-coordinate of P. The height is 2.

The triangle OPQ is a right-angled triangle with the right angle at Q.
\[ Area_{OPQ} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 2 \times 2 = 2 \]
The area of triangle OPQ is 2.


For Column B: Area of triangular region ORS

The vertices are O(0,0), R(4,0), and S(4,-1).

Let's choose the segment OR as the base. O is at \(x=0\) and R is at \(x=4\), so the length of the base OR is \(4 - 0 = 4\).

The height corresponding to this base is the perpendicular distance from point S to the x-axis, which is the absolute value of the y-coordinate of S. The height is \(|-1| = 1\).

The triangle ORS is a right-angled triangle with the right angle at R.
\[ Area_{ORS} = \frac{1}{2} \times base \times height = \frac{1}{2} \times 4 \times 1 = 2 \]
The area of triangle ORS is 2.


Step 4: Final Answer:

Comparing the two areas:

Column A = 2

Column B = 2

The two quantities are equal.
Quick Tip: When a triangle has a horizontal or vertical side, choosing that side as the base simplifies finding the height, which will be the difference in the y-coordinates or x-coordinates, respectively.

Step 1: Understanding the Concept:

The problem involves multiplying decimals and a whole number. It tests understanding of decimal place values.


Step 2: Key Formula or Approach:

To multiply decimals, one can convert them to fractions or multiply them as whole numbers and then place the decimal point correctly in the result.


Step 3: Detailed Explanation:

For Column A:

The expression is \((0.01)(0.07)(70)\).

Let's multiply the decimals first: \(0.01 \times 0.07\).
\(1 \times 7 = 7\). The first number has 2 decimal places and the second has 2, so the product will have \(2+2=4\) decimal places.

So, \(0.01 \times 0.07 = 0.0007\).

Now, multiply this by 70:
\[ 0.0007 \times 70 \]
This is the same as \(0.0007 \times 7 \times 10\).
\(0.0007 \times 7 = 0.0049\).
\(0.0049 \times 10 = 0.049\).

Alternatively, using fractions:
\[ \left(\frac{1}{100}\right) \times \left(\frac{7}{100}\right) \times 70 = \frac{1 \times 7 \times 70}{100 \times 100} = \frac{490}{10000} = \frac{49}{1000} = 0.049 \]
The value of Column A is 0.049.


For Column B:

The value is 0.49.


Step 4: Final Answer:

Comparing the two values:

Column A = 0.049

Column B = 0.49

To compare, we can write them with the same number of decimal places: 0.049 and 0.490. It is clear that \(49 < 490\), so \(0.049 < 0.49\).

Therefore, the quantity in Column B is greater.
Quick Tip: When multiplying by powers of 10, move the decimal point. For example, multiplying \( (0.01)(70) \) first gives \(0.7\). Then \(0.7 \times 0.07 = 0.049\). Rearranging the multiplication order can sometimes simplify the calculation.

Step 1: Understanding the Concept:

Column A represents the arithmetic mean (average) of three numbers \(x, y,\) and \(z\). Column B is the largest of these three numbers. The question compares the mean of a set of distinct numbers to its maximum value.


Step 2: Key Formula or Approach:

The arithmetic mean of a set of numbers is always less than the maximum value in the set, and greater than the minimum value, provided the numbers are not all equal.


Step 3: Detailed Explanation:

We are given the inequality \(x < y < z\).

This tells us that \(x, y,\) and \(z\) are distinct numbers, and \(z\) is the largest.

Let's analyze the expression for the mean in Column A. Since \(x < z\) and \(y < z\), we can write the following inequalities:
\[ x < z \] \[ y < z \]
We also know that \(z = z\).

Adding these three inequalities:
\[ x + y + z < z + z + z \] \[ x + y + z < 3z \]
Now, divide both sides by 3 (since 3 is a positive number, the inequality direction does not change):
\[ \frac{x+y+z}{3} < z \]
This shows that the quantity in Column A is strictly less than the quantity in Column B.


Example with numbers:

Let \(x=1, y=2, z=3\). They satisfy \(x < y < z\).

Column A: \(\frac{1+2+3}{3} = \frac{6}{3} = 2\).

Column B: \(z = 3\).

Here, \(2 < 3\), so Column B is greater.


Step 4: Final Answer:

The arithmetic mean of three distinct numbers is always less than the largest number.

Column A = \(\frac{x+y+z}{3}\)

Column B = \(z\)

Therefore, Column A \(<\) Column B. The quantity in Column B is greater.
Quick Tip: Remember the property of averages: the average of a set of numbers must lie between the smallest and largest numbers in the set. If the numbers are not all identical, the average will be strictly between the minimum and maximum.

Step 1: Understanding the Concept:

The problem involves determining the ratio of the side lengths of a large rectangle that is composed of three identical smaller rectangles. The key is to express the sides of the large rectangle in terms of the dimensions of the smaller ones.


Step 2: Key Formula or Approach:

Let the dimensions of the small identical rectangles be length \(l\) and width \(w\), with \(l > w\). We will use the geometric constraints of the figure to find a relationship between \(l\) and \(w\).


Step 3: Detailed Explanation:

Let's analyze the figure. The large rectangle QRSP is divided into one rectangle on top and two side-by-side rectangles on the bottom. Since all three small rectangles are identical, they all have dimensions \(l \times w\).


The two bottom rectangles are standing vertically. Their width is \(w\) and their height is \(l\).

The bottom side of the large rectangle, PS, is the sum of the widths of these two rectangles.
\[ PS = w + w = 2w \]
The height of the large rectangle, RS, is the same as the height of the vertically oriented bottom-right rectangle.
\[ RS = l \]

The top rectangle is lying horizontally. Its width must match the total width of the two rectangles below it. Therefore, its width is \(2w\). Its height is \(w\).

Since the dimensions of this top rectangle must also be \(l\) and \(w\), we can see that its longer side is \(l\) and its shorter side is \(w\). From the figure, the width of the top rectangle is its longer side.
\[ l = 2w \]
This establishes the relationship between the length and width of the small rectangles.


Now, we can calculate the ratio in Column A.
\[ \frac{PS}{RS} = \frac{2w}{l} \]
Substitute the relationship \(l = 2w\) into the expression:
\[ \frac{PS}{RS} = \frac{2w}{2w} = 1 \]
The value of Column A is 1.


Step 4: Final Answer:

We are comparing Column A and Column B.

Column A = 1

Column B = \(\frac{1}{2}\)

Since \(1 > \frac{1}{2}\), the quantity in Column A is greater.
Quick Tip: In geometry problems with composite shapes, assign variables to the dimensions of the basic components. Then, use the properties of the larger shape (e.g., opposite sides of a rectangle are equal) to form equations and solve for the relationships between the variables.

Step 1: Understanding the Concept:

The problem requires finding a missing value from a set, given the average of the set.


Step 2: Key Formula or Approach:

The formula for the arithmetic mean (average) is:
\[ Average = \frac{Sum of values}{Number of values} \]
We can rearrange this to find the sum: Sum of values = Average \( \times \) Number of values.


Step 3: Detailed Explanation:

We are given three temperatures: \(x\), 25, and 37.

The number of values is 3.

The average of these temperatures is 20.

Using the average formula:
\[ 20 = \frac{x + 25 + 37}{3} \]
To solve for \(x\), first multiply both sides by 3:
\[ 20 \times 3 = x + 25 + 37 \] \[ 60 = x + 62 \]
Now, subtract 62 from both sides to isolate \(x\):
\[ 60 - 62 = x \] \[ x = -2 \]
So, the value for Column A is -2.


Step 4: Final Answer:

We are comparing Column A (\(x\)) and Column B (0).

Column A = -2

Column B = 0

Since \(-2 < 0\), the quantity in Column B is greater.
Quick Tip: A useful trick for average problems is to think about the "sum deficit/surplus". The sum should be \(20 \times 3 = 60\). The known values are 25 (which is +5 from the average) and 37 (which is +17 from the average). The total surplus is \(5+17=22\). To balance this, \(x\) must be 22 below the average. \(20 - 22 = -2\).

Step 1: Understanding the Concept:

This is an algebraic manipulation problem. We need to substitute the given expression for \(m\) into the expression in Column A and simplify it.


Step 2: Key Formula or Approach:

The key steps are substitution and factorization. We will factor a common term from the expression for \(m\) to allow for simplification.


Step 3: Detailed Explanation:

For Column A:

The expression is \(\frac{2m}{x+y}\).

We are given \(m = 4x+4y\). Substitute this into the expression in Column A:
\[ \frac{2(4x+4y)}{x+y} \]
Now, look at the term in the parentheses, \(4x+4y\). We can factor out a common factor of 4:
\[ 4x+4y = 4(x+y) \]
Substitute this factored form back into the expression:
\[ \frac{2 \times 4(x+y)}{x+y} \] \[ \frac{8(x+y)}{x+y} \]
We are given the condition that \(x \neq -y\), which means \(x+y \neq 0\). Since the denominator is not zero, we can safely cancel the \((x+y)\) term from the numerator and the denominator.
\[ \frac{8\cancel{(x+y)}}{\cancel{(x+y)}} = 8 \]
The value of Column A is 8.


Step 4: Final Answer:

Comparing the two quantities:

Column A = 8

Column B = 8

The two quantities are equal.
Quick Tip: Whenever you see an algebraic expression to be simplified, always look for common factors that can be canceled out. The condition \(x \neq -y\) is a strong hint that the term \((x+y)\) will be in the denominator and needs to be canceled.

Step 1: Understanding the Concept:

This problem tests knowledge of the properties of a rectangle, specifically its angles and diagonals. We need to compare two sums of angles.


Step 2: Key Formula or Approach:

Properties of a rectangle:

1. Opposite sides are parallel (AB \(||\) DC, AD \(||\) BC).

2. All corner angles are 90\(^\circ\).

3. Diagonals are equal in length and bisect each other.

Alternate interior angles formed by a transversal intersecting two parallel lines are equal.


Step 3: Detailed Explanation:

The comparison between Column A (\(r+u+v\)) and Column B (\(r+u+w\)) can be simplified by subtracting the common terms \(r+u\) from both columns. The problem then reduces to comparing \(v\) and \(w\).


Let's identify the angles from the diagram:
\(v\) is the measure of angle ABD.
\(w\) is the measure of angle CBD.


In rectangle ABCD, the angle at vertex B, \(\angle ABC\), is 90\(^\circ\). From the diagram, we can see that \(\angle ABC = \angle ABD + \angle CBD = v + w\). So, \(v+w = 90^\circ\).


Now, let's consider the relationship between \(v\) and \(w\).

Consider the right-angled triangle \(\triangle ABC\). We have \(\tan(u) = \frac{BC}{AB}\).

Consider the right-angled triangle \(\triangle ABD\). We have \(\tan(v) = \frac{AD}{AB}\).

Since ABCD is a rectangle, \(AD = BC\). Therefore, \(\tan(v) = \frac{BC}{AB}\). This means \(\tan(u) = \tan(v)\), so \(u=v\).


Now, let's relate \(v\) and \(w\). The diagonal DB divides the corner angle \(\angle ABC\) into two angles, \(v\) and \(w\). The diagonal bisects the corner angle only if the rectangle is a square (i.e., if adjacent sides are equal, \(AB=BC\)).


Case 1: The rectangle is a square.

If \(AB = BC\), then in \(\triangle ABC\), it is an isosceles right triangle, and the diagonal AC would bisect \(\angle B\). Similarly, diagonal DB would bisect \(\angle B\). Thus, \(v = w = 45^\circ\). In this case, Column A and Column B are equal.


Case 2: The rectangle is not a square.

Suppose \(AB > BC\).
Consider the right-angled triangle \(\triangle BCD\). We have \(\tan(w) = \frac{DC}{BC}\). Since \(DC=AB\), \(\tan(w) = \frac{AB}{BC}\).

We are comparing \(v\) and \(w\). Let's compare \(\tan(v)\) and \(\tan(w)\).
\(\tan(v) = \frac{BC}{AB}\) and \(\tan(w) = \frac{AB}{BC}\).

Since we assumed \(AB > BC\), then \(\frac{AB}{BC} > 1\) and \(\frac{BC}{AB} < 1\).

So, \(\tan(w) > \tan(v)\). Since tangent is an increasing function for acute angles, this implies \(w > v\). In this case, Column B is greater.

If we assumed \(BC > AB\), we would find \(v > w\), making Column A greater.


Step 4: Final Answer:

Since the relationship between \(v\) and \(w\) depends on the side lengths of the rectangle (whether it is a square or an elongated rectangle), and this information is not given, we cannot determine the relationship between the columns.
Quick Tip: Do not assume a geometric figure has special properties unless stated. A rectangle is not necessarily a square. If the relationship changes when you consider a special case (like a square) versus a general case (like a non-square rectangle), the answer is (D).

Step 1: Understanding the Concept:

This question compares two exponential expressions where the base and the exponent are interchanged. We need to determine if one expression is consistently larger than the other for all positive integers \(n\).


Step 2: Key Formula or Approach:

The best approach is to test several different values for the positive integer \(n\) and observe how the relationship between the two columns changes.


Step 3: Detailed Explanation:

Let's test various values of \(n\), where \(n\) is a positive integer (\(n=1, 2, 3, \dots\)).


Case 1: Let \(n = 1\).

Column A: \(n^{100} = 1^{100} = 1\).

Column B: \(100^n = 100^1 = 100\).

In this case, Column B > Column A.


Case 2: Let \(n = 2\).

Column A: \(n^{100} = 2^{100}\).

Column B: \(100^n = 100^2 = 10000 = 10^4\).

To compare these, we can approximate \(2^{100}\). We know \(2^{10} = 1024 \approx 10^3\).

So, \(2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}\).

Clearly, \(10^{30}\) is vastly larger than \(10^4\).

In this case, Column A > Column B.


Case 3: Let \(n = 100\).

Column A: \(n^{100} = 100^{100}\).

Column B: \(100^n = 100^{100}\).

In this case, Column A = Column B.


Step 4: Final Answer:

We have found a case where Column B is greater (\(n=1\)), a case where Column A is greater (\(n=2\)), and a case where they are equal (\(n=100\)). Since the relationship between the two quantities is not constant and depends on the value of \(n\), the relationship cannot be determined from the information given.
Quick Tip: When comparing exponential functions like \(x^a\) and \(a^x\), remember that their relationship is not fixed. Testing small integer values, and sometimes the value where the bases and exponents are equal (like \(n=100\) here), is a good strategy to see if the relationship changes.

Step 1: Understanding the Concept:

This problem deals with a simple linear function, representing direct proportionality (\(f(t)\) is proportional to \(t\)). We are given a specific value of the function and asked to compare the constant of proportionality \(k\) with another value of the function.


Step 2: Key Formula or Approach:

First, use the given information \(f(3) = \frac{1}{2}\) to find the value of the constant \(k\). Then, use this value of \(k\) to evaluate the expression in Column B.


Step 3: Detailed Explanation:

The function is defined as \(f(t) = kt\).

We are \(f(3) = \frac{1}{2}\).

Using the function definition, we can write:
\[ f(3) = k \times 3 = 3k \]
Since we know \(f(3) = \frac{1}{2}\), we can set up the equation:
\[ 3k = \frac{1}{2} \]
To solve for \(k\), divide both sides by 3:
\[ k = \frac{1/2}{3} = \frac{1}{6} \]
So, the value of Column A is \(\frac{1}{6}\).


Now for Column B, we need to find \(f(1)\).

Using the function definition again:
\[ f(1) = k \times 1 = k \]
Since we have already found that \(k = \frac{1}{6}\), the value of Column B is also \(\frac{1}{6}\).


Step 4: Final Answer:

Comparing the two quantities:

Column A: \(k = \frac{1}{6}\)

Column B: \(f(1) = \frac{1}{6}\)

The two quantities are equal.
Quick Tip: For any function of the form \(f(t) = kt\), the value of \(f(1)\) is always equal to \(k\), because \(f(1) = k \times 1 = k\). Recognizing this directly makes the comparison trivial without needing to calculate the specific value of k.

Step 1: Understanding the Concept:

This problem involves a system of two linear inequalities with two variables. We need to use these inequalities to determine the relationship between the expressions in Column A and Column B. Since there are no constraints on \(x\) and \(y\) (e.g., being positive), we must consider all possibilities.


Step 2: Key Formula or Approach:

First, simplify the given inequalities to get the clearest possible bound for \(y\) in terms of \(x\). Then, test different values for \(x\) (positive, negative, and zero) to see if the relationship between Column A and Column B remains constant.


Step 3: Detailed Explanation:

We are given two inequalities:

1) \(100x < y\), which can be written as \(y > 100x\).

2) \(1000x < 2y\). Dividing by 2 (a positive number), we get \(500x < y\), which can be written as \(y > 500x\).


For both inequalities to hold true, \(y\) must be greater than the larger of the two lower bounds.

If \(x > 0\), then \(500x > 100x\). So, the controlling inequality is \(y > 500x\).

If \(x < 0\), then \(500x < 100x\). So, the controlling inequality is \(y > 100x\).

If \(x = 0\), both inequalities become \(y > 0\).


Let's test cases based on the sign of \(x\).


Case 1: Let \(x\) be positive (e.g., \(x=1\)).

The condition becomes \(y > 500(1)\), so \(y > 500\).

Column A: \(1,100x = 1,100(1) = 1,100\).

Column B: \(y\).

We are comparing 1,100 with \(y\). We know \(y\) must be greater than 500.

- If we choose \(y = 600\) (which is > 500), then Column B (600) < Column A (1,100).

- If we choose \(y = 1,200\) (which is > 500), then Column B (1,200) > Column A (1,100).

Since we get a different relationship depending on the value of \(y\) we choose, the relationship cannot be determined for positive \(x\).


Because we have already found a scenario where the relationship is indeterminate, we can conclude the answer is (D). We do not need to test \(x \le 0\).


Step 4: Final Answer:

The given inequalities only provide a lower bound for \(y\) in terms of \(x\). They do not provide an upper bound. This means \(y\) can be arbitrarily large. When \(x\) is positive, we need to compare \(1,100x\) with a value \(y\) that is only known to be greater than \(500x\). Since \(y\) could be \(600x\) or \(1,200x\), we cannot establish a fixed relationship. Therefore, the relationship cannot be determined.
Quick Tip: In inequality problems, pay close attention to whether variables can be positive, negative, or zero. If the problem doesn't specify, you must consider all cases. Finding just one scenario with conflicting outcomes is enough to prove the answer is (D).

Step 1: Understanding the Concept:

This is a word problem that requires careful reading to identify the correct numbers to use for the calculation. The problem involves division to determine how many cartons are used and then subtraction to find out how many are left empty. There appears to be a typo in the question, as two different numbers of eggs (372 and 327) are mentioned. Using 372 eggs leads to one of the answer choices, while using 327 does not. We will proceed assuming 372 is the correct number of eggs.


Step 2: Key Formula or Approach:

1. Calculate the number of cartons filled by dividing the total number of eggs by the number of eggs per carton.

2. Calculate the number of remaining empty cartons by subtracting the number of filled cartons from the total number of available cartons.


Step 3: Detailed Explanation:

Let's assume the total number of eggs to be packed is 372.

The capacity of each carton is 12 eggs.

First, we find the number of cartons that can be completely filled:
\[ Number of cartons filled = \frac{Total eggs}{Eggs per carton} = \frac{372}{12} \] \[ \frac{372}{12} = 31 \]
So, 31 cartons are completely filled.


Next, we find how many cartons remain empty.

Total number of available cartons = 50.

Number of cartons used = 31.
\[ Number of empty cartons = Total cartons - Cartons filled \] \[ Number of empty cartons = 50 - 31 = 19 \]
19 cartons will remain empty. This matches option (C).


(Note: If we were to use the number 327, the number of filled cartons would be \(\lfloor \frac{327}{12} \rfloor = \lfloor 27.25 \rfloor = 27\). The number of empty cartons would be \(50 - 27 = 23\), which is not among the options.)


Step 4: Final Answer:

Based on the calculation using 372 eggs, 19 cartons will remain empty.
Quick Tip: In word problems, always watch out for extraneous information or possible typos. If your initial calculation leads to an answer that isn't an option, re-read the problem carefully and see if another interpretation of the numbers makes sense and matches an answer choice.

Step 1: Understanding the Concept:

The question asks to identify the greatest number from a list of negative decimal numbers. For negative numbers, the "greatest" number is the one that is least negative, meaning it is closest to 0 on the number line.


Step 2: Key Formula or Approach:

To find the greatest negative number, we can compare the absolute values of the numbers. The negative number with the smallest absolute value is the greatest.


Step 3: Detailed Explanation:

We have the following numbers: -0.225, -0.0225, -0.323, -0.0325, and -0.3205.

Let's find their absolute values:

(A) \(|-0.225| = 0.225\)

(B) \(|-0.0225| = 0.0225\)

(C) \(|-0.323| = 0.323\)

(D) \(|-0.0325| = 0.0325\)

(E) \(|-0.3205| = 0.3205\)

Now, we compare these positive decimal values to find the smallest one.

- Comparing 0.0225 and 0.0325, we see that 0.0225 is smaller.

- Comparing 0.0225 with the rest (0.225, 0.323, 0.3205), it is clearly the smallest because its first non-zero digit is in the hundredths place, while for the others it's in the tenths place (or in the case of 0.0325, the value in the hundredths place is larger).

The smallest absolute value is 0.0225.


Step 4: Final Answer:

The number with the smallest absolute value is -0.0225. Therefore, -0.0225 is the greatest number in the list.
Quick Tip: Visualize a number line. Numbers increase as you move from left to right. For negative numbers, numbers closer to zero (e.g., -1) are greater than numbers further from zero (e.g., -10).

Step 1: Understanding the Concept:

The amount of gasoline consumed is inversely proportional to the car's fuel efficiency (miles per gallon). To find the maximum amount of gasoline consumed, we must use the minimum fuel efficiency.


Step 2: Key Formula or Approach:

The formula to calculate gasoline consumption is:
\[ Gasoline Consumed = \frac{Total Distance}{Miles Per Gallon} \]
To maximize the "Gasoline Consumed", we need to minimize the denominator, "Miles Per Gallon".


Step 3: Detailed Explanation:

The total distance of the trip is 360 miles.

The fuel efficiency range is between 20 and 24 miles per gallon (mpg), inclusive.

The minimum fuel efficiency is 20 mpg.

Now, we calculate the maximum gasoline consumption using the minimum fuel efficiency:
\[ Maximum Gasoline Consumed = \frac{360 miles}{20 mpg} \] \[ Maximum Gasoline Consumed = \frac{36}{2} gallons = 18 gallons \]

Step 4: Final Answer:

The maximum amount of gasoline the automobile would consume is 18.0 gallons.
Quick Tip: Be careful with keywords like "maximum" and "minimum". In problems involving rates like this, maximizing one quantity (consumption) often requires minimizing a related quantity (efficiency).

Step 1: Understanding the Concept:

The given equations describe the distances and relative ordering between three points on a number line. We need to combine these pieces of information to determine the overall order of \(x, y,\) and \(z\).


Step 2: Key Formula or Approach:

Analyze each equation to determine the relationship between the two variables involved. An equation of the form \(a - b = k\) where \(k\) is positive implies \(a > b\).


Step 3: Detailed Explanation:

Let's analyze the first equation:
\[ y - x = 2 \]
Since the result is positive, this means \(y > x\). We can rewrite this as \(y = x + 2\), which tells us that \(y\) is located 2 units to the right of \(x\) on the number line.


Now let's analyze the second equation:
\[ y - z = 3 \]
Since the result is positive, this means \(y > z\). We can rewrite this as \(y = z + 3\), which tells us that \(y\) is located 3 units to the right of \(z\) on the number line.


So far, we know that \(y\) is the greatest of the three numbers. Now we need to compare \(x\) and \(z\). We can express both in terms of \(y\):

From the first equation: \(x = y - 2\)

From the second equation: \(z = y - 3\)

To find \(x\), we subtract 2 from \(y\). To find \(z\), we subtract 3 from \(y\). Since we are subtracting a larger number from \(y\) to get \(z\), \(z\) must be smaller than \(x\).

Thus, \(z < x\).


Combining all the information, we get the final order: \(z < x < y\).


Step 4: Final Answer:

The correct order of the points on the number line from left to right is \(z\), then \(x\), then \(y\). This corresponds to the arrangement shown in option (C).
Quick Tip: A quick way to solve this is to pick a value for one variable and solve for the others. For example, let \(y=5\). Then \(5 - x = 2 \Rightarrow x = 3\), and \(5 - z = 3 \Rightarrow z = 2\). The numbers are \(z=2, x=3, y=5\), confirming the order \(z < x < y\).

Step 1: Understanding the Concept:

This problem involves calculating the probability of two independent events both occurring. The events are "independent" because the selection of a bead from one bag does not influence the selection from the other bag.


Step 2: Key Formula or Approach:

The probability of two independent events, A and B, both happening is given by the product of their individual probabilities:
\[ P(A and B) = P(A) \times P(B) \]

Step 3: Detailed Explanation:

Let event A be the selection of a yellow bead from the first bag. The probability of this event is given as:
\[ P(A) = \frac{2}{7} \]
Let event B be the selection of a yellow bead from the second bag. The probability of this event is given as:
\[ P(B) = \frac{3}{7} \]
We want to find the probability that both beads are yellow, which is \(P(A and B)\). Since the events are independent, we multiply their probabilities:
\[ P(both yellow) = P(A) \times P(B) = \frac{2}{7} \times \frac{3}{7} \] \[ P(both yellow) = \frac{2 \times 3}{7 \times 7} = \frac{6}{49} \]

Step 4: Final Answer:

The probability that both beads selected will be yellow is \(\frac{6}{49}\).
Quick Tip: Remember the keyword "and" in probability usually implies multiplication (for independent events), while "or" usually implies addition (for mutually exclusive events).

Step 1: Understanding the Concept:

This question asks for the percent increase in sales for a specific company between two years. We need to read the values from the bar graph and apply the percent change formula.


Step 2: Key Formula or Approach:

The formula for percent increase is:
\[ Percent Increase = \left( \frac{New Value - Old Value}{Old Value} \right) \times 100% \]

Step 3: Detailed Explanation:

From the graph, we find the sales for Compaq:

- Old Value (1992 sales) = 0.7 million.

- New Value (1993 sales) = 1.4 million.

First, calculate the amount of increase:
\[ Increase = New Value - Old Value = 1.4 - 0.7 = 0.7 million \]
Now, use the percent increase formula:
\[ Percent Increase = \left( \frac{0.7}{0.7} \right) \times 100% \] \[ Percent Increase = 1 \times 100% = 100% \]
The number of computers sold by Compaq doubled, which is a 100% increase.


Step 4: Final Answer:

The number of personal computers sold by Compaq increased by 100% from 1992 to 1993.
Quick Tip: When a quantity doubles, the percent increase is 100%. If it triples, the increase is 200%, and so on. Recognizing this pattern can save calculation time.

Step 1: Understanding the Concept:

This question asks for a part-to-whole percentage. We need to find the total number of computers sold by all four companies in 1992 and then determine what percentage of that total was sold by Packard Bell.


Step 2: Key Formula or Approach:

The formula for percentage is:
\[ Percentage = \left( \frac{Part}{Whole} \right) \times 100% \]

Step 3: Detailed Explanation:

First, we read the sales data for 1992 (the lighter bars) for all four companies from the graph:

- IBM = 1.4 million

- Apple = 1.6 million

- Compaq = 0.7 million

- Packard Bell = 0.6 million

Next, we calculate the total sales (the "Whole") in 1992:
\[ Total Sales = 1.4 + 1.6 + 0.7 + 0.6 = 4.3 million \]
The "Part" is the sales by Packard Bell, which is 0.6 million.

Now, we calculate the percentage:
\[ Percentage = \left( \frac{0.6}{4.3} \right) \times 100% \] \[ Percentage = \frac{6}{43} \times 100% \approx 0.1395 \times 100% = 13.95% \]
This value is closest to 14%.


Step 4: Final Answer:

In 1992, Packard Bell accounted for approximately 14% of the computers sold by the four companies.
Quick Tip: For quick estimation on multiple-choice questions, you can approximate the denominator. \(\frac{6}{43}\) is slightly less than \(\frac{6}{42} = \frac{1}{7}\). Since \(\frac{1}{7} \approx 14.3%\), the answer must be very close to 14%.

Step 1: Understanding the Concept:

This is a ratio problem that requires us to use data from the graph. We are given a ratio connecting the sales of three companies, two of which have sales data on the graph. We can use this known data to find the value of one "part" of the ratio, and then use that to find the sales for the unknown company (Tandy).


Step 2: Key Formula or Approach:

Set up a proportion based on the given ratio and the known sales figures.

Ratio: IBM : Compaq : Tandy = 6 : 4 : 1.


Step 3: Detailed Explanation:

From the graph, we find the sales data for 1993 (the darker bars):

- IBM sales = 2.1 million = 2,100,000.

- Compaq sales = 1.4 million = 1,400,000.

We can use either IBM's or Compaq's data. Let's use IBM.

According to the ratio, IBM's sales correspond to 6 parts.
\[ 6 parts = 2,100,000 \]
To find the value of 1 part, we divide by 6:
\[ 1 part = \frac{2,100,000}{6} = 350,000 \]
Tandy's sales correspond to 1 part of the ratio. Therefore, Tandy sold 350,000 computers.


To verify, let's use Compaq's data. According to the ratio, Compaq's sales correspond to 4 parts.
\[ 4 parts = 1,400,000 \] \[ 1 part = \frac{1,400,000}{4} = 350,000 \]
Both calculations yield the same result.


Step 4: Final Answer:

Approximately 350,000 personal computers were sold by Tandy in 1993.
Quick Tip: When solving ratio problems with real-world data, check if your value for "1 part" is consistent with all the given information. Here, using both IBM and Compaq data to find the value of 1 part confirms our reading of the graph and our calculation.

Step 1: Understanding the Concept:

The question asks to identify which degree category consistently represents about one-quarter (25%) of the total number of graduates across the given years. We need to calculate the total for at least one year and compare each category to that total.


Step 2: Key Formula or Approach:

1. For a sample year, sum the number of graduates across all categories to find the total.

2. Calculate \(\frac{1}{4}\) of this total.

3. Compare this value with the number of graduates in each category to find the best match.


Step 3: Detailed Explanation:

Let's perform the calculation for the year 1995. The numbers are in thousands.

- High School: 2,329

- Associate: 458

- Bachelor's: 1,047

- Master's: 324

- Doctoral: 36

Total number of graduates in 1995:
\[ Total = 2329 + 458 + 1047 + 324 + 36 = 4194 (thousands) \]
Now, let's find \(\frac{1}{4}\) of this total:
\[ \frac{1}{4} \times 4194 = 1048.5 (thousands) \]
Now we compare this value to the numbers for each category in 1995:

- High School (2,329) is much larger.

- Associate (458) is much smaller.

- Bachelor's (1,047) is extremely close to 1048.5.

- Master's (324) is much smaller.

- Doctoral (36) is much smaller.

The Bachelor's degree category is the only one that is approximately \(\frac{1}{4}\) of the total for 1995. A quick check of other years confirms this trend. For example, in 2001, the total is \(2865 + 489 + 1037 + 327 + 36 = 4754\). One quarter of this is \(4754/4 = 1188.5\). The Bachelor's degree number is 1037, which is the closest value among the categories.


Step 4: Final Answer:

The Bachelor's degree category consistently accounts for approximately \(\frac{1}{4}\) of the total number of graduates.
Quick Tip: When a question asks for an approximation across multiple data points (like years), you can often find the correct answer by doing a detailed calculation for just one of those points. The result is usually clear enough that you don't need to repeat the full calculation for every single data point.

Step 1: Understanding the Concept:

This question asks for the percent increase in the number of associate degrees from 1995 to 2001. The phrase "percent greater than" is equivalent to percent increase.


Step 2: Key Formula or Approach:

The formula for percent increase is:
\[ Percent Increase = \left( \frac{New Value - Old Value}{Old Value} \right) \times 100% \]

Step 3: Detailed Explanation:

From the table, we find the number of associate degrees (in thousands):

- Old Value (1995) = 458.

- New Value (2001) = 489.

First, calculate the amount of increase:
\[ Increase = 489 - 458 = 31 \]
Now, we use the percent increase formula. The base for the percentage is the original number from 1995.
\[ Percent Increase = \left( \frac{31}{458} \right) \times 100% \]
To calculate this, we can divide 3100 by 458:
\[ \frac{3100}{458} \approx 6.768...% \]
The question asks for the nearest percent. 6.768% is most nearly 7%.


Step 4: Final Answer:

The number of associate degrees in 2001 is most nearly 7% greater than the number in 1995.
Quick Tip: Estimation can get you close to the answer quickly. \(\frac{31}{458}\) is close to \(\frac{30}{450} = \frac{3}{45} = \frac{1}{15}\). Knowing that \(\frac{1}{15}\) is about 6.7% helps you quickly identify 7% as the likely correct answer.

Step 1: Understanding the Concept:

This problem requires finding the perimeter of a square given the area of a triangular region within it. The key is to correctly interpret the geometry of the triangle from the diagram to relate its area to the side length of the square. While the diagram is not perfectly clear, we can deduce the intended geometry by finding an interpretation that leads to one of the answer choices. A likely interpretation is that the triangle's vertices are one corner of the square and the midpoints of the two opposite sides.


Step 2: Key Formula or Approach:

1. Let the side length of the square be \(s\).

2. Express the area of the shaded triangle in terms of \(s\). The area of a triangle is \(\frac{1}{2} \times base \times height\).

3. Set this area expression equal to 20 and solve for \(s\).

4. Calculate the perimeter using the formula \(P = 4s\).


Step 3: Detailed Explanation:

Let the side length of the square be \(s\). Let's assume the vertices of the shaded triangle are one corner (e.g., the bottom-left) and the midpoints of the two opposite sides (top and right sides).

Let's place the square on a coordinate plane with vertices at (0,0), (s,0), (s,s), and (0,s).

The triangle's vertices would be at (0,0), the midpoint of the top side \((s/2, s)\), and the midpoint of the right side \((s, s/2)\).

The area of this triangle can be found by taking the area of the square and subtracting the three unshaded right triangles in the corners.

- Triangle 1 (bottom right): base \(s/2\), height \(s\). Area = \(\frac{1}{2}(s/2)(s) = s^2/4\). No, base s, height s/2. Area = \(\frac{1}{2}(s)(s/2) = s^2/4\).

- Triangle 2 (top left): base \(s/2\), height \(s\). Area = \(\frac{1}{2}(s/2)(s) = s^2/4\).

- Triangle 3 (top right): base \(s/2\), height \(s/2\). Area = \(\frac{1}{2}(s/2)(s/2) = s^2/8\).

This is too complicated and depends on a specific interpretation.


Let's try a simpler interpretation that fits the options. Let the vertices of the triangle be the midpoint of the top side, the midpoint of the bottom side, and one of the other corners (e.g., the bottom-left).

- Base of the triangle: The line connecting the midpoints \((s/2, 0)\) and \((s/2, s)\). This is a vertical line of length \(s\).

- Height of the triangle: The perpendicular distance from the corner vertex (0,0) to the base line \(x=s/2\). This distance is \(s/2\).

- Area = \(\frac{1}{2} \times base \times height = \frac{1}{2} \times s \times \frac{s}{2} = \frac{s^2}{4}\).


This interpretation gives a simple formula. Let's assume it is the correct one.

We are given that the area of the shaded region is 20.
\[ \frac{s^2}{4} = 20 \]
Multiply both sides by 4 to solve for \(s^2\):
\[ s^2 = 80 \]
Take the square root of both sides to find the side length \(s\):
\[ s = \sqrt{80} \]
To simplify the square root, find the largest perfect square factor of 80. \(80 = 16 \times 5\).
\[ s = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5} \]
Now, calculate the perimeter of the square:
\[ P = 4s = 4 \times (4\sqrt{5}) = 16\sqrt{5} \]

Step 4: Final Answer:

The perimeter of the square is \(16\sqrt{5}\).
Quick Tip: If a diagram in a geometry problem is ambiguous, consider different plausible interpretations. The correct one is usually the one that leads to one of the given answer choices. Here, interpreting the triangle's area as \(\frac{s^2}{2}\) (base=side, height=side) leads to an answer not in the options, while interpreting it as \(\frac{s^2}{4}\) (base=side, height=half side) leads to a correct option.

Step 1: Understanding the Concept:

This problem presents a system of two equations with two variables, \(x\) and \(y\). To find the value of \(y\), we need to combine the equations to eliminate one variable, in this case, \(x\), and solve for the remaining variable, \(y\).


Step 2: Key Formula or Approach:

The method of substitution is ideal here. We will substitute the expression for \(x\) from the first equation into the second equation. This will result in an equation solely in terms of \(y\), which we can then solve.


Step 3: Detailed Explanation:

We are given two equations:

1) \(x = \frac{1}{y}\)

2) \(y = \frac{1}{1-x}\)

Substitute the expression for \(x\) from equation (1) into equation (2):
\[ y = \frac{1}{1 - \left(\frac{1}{y}\right)} \]
To simplify the denominator, find a common denominator:
\[ y = \frac{1}{\frac{y}{y} - \frac{1}{y}} \] \[ y = \frac{1}{\frac{y-1}{y}} \]
Now, invert the fraction in the denominator and multiply:
\[ y = 1 \times \frac{y}{y-1} \] \[ y = \frac{y}{y-1} \]
To solve for \(y\), multiply both sides by \((y-1)\). Note that this assumes \(y \neq 1\).
\[ y(y-1) = y \]
Distribute \(y\) on the left side:
\[ y^2 - y = y \]
Move all terms to one side to form a quadratic equation:
\[ y^2 - y - y = 0 \] \[ y^2 - 2y = 0 \]
Factor out a common factor of \(y\):
\[ y(y-2) = 0 \]
This gives two possible solutions: \(y = 0\) or \(y - 2 = 0 \Rightarrow y = 2\).

We must check these solutions in the original equations. If \(y=0\), the first equation becomes \(x = \frac{1}{0}\), which is undefined. Therefore, \(y=0\) is not a valid solution.

The only valid solution is \(y=2\).


Step 4: Final Answer:

After substituting and solving the resulting equation, we find that the only valid value for \(y\) is 2.
Quick Tip: When solving equations that involve variables in the denominator, always check your final answers against the original equations. Solutions that lead to division by zero are extraneous and must be discarded.

Step 1: Understanding the Concept:

The problem describes a product of consecutive integers starting from 2. This is similar to a factorial product. We need to find the last integer in this sequence, \(n\), and then calculate \(n-1\).


Step 2: Key Formula or Approach:

We are given the equation \(2 \times 3 \times 4 \times \dots \times n = 720\). The most direct approach is to start multiplying the integers from 2 upwards and see when the product reaches 720.


Step 3: Detailed Explanation:

Let's calculate the product of consecutive integers starting with 2:

- Start with 2.

- Multiply by the next integer, 3: \(2 \times 3 = 6\).

- Multiply the result by the next integer, 4: \(6 \times 4 = 24\).

- Multiply the result by the next integer, 5: \(24 \times 5 = 120\).

- Multiply the result by the next integer, 6: \(120 \times 6 = 720\).

We have reached the target product, 720. The sequence of integers is 2, 3, 4, 5, 6. The last integer in this sequence is \(n\).

Therefore, \(n=6\).

The question asks for the value of \(n-1\).
\[ n - 1 = 6 - 1 = 5 \]

Step 4: Final Answer:

The value of \(n\) is 6, so the value of \(n-1\) is 5.
Quick Tip: Recognizing common factorial values can be very helpful. \(720\) is \(6!\) (6 factorial), which is \(1 \times 2 \times 3 \times 4 \times 5 \times 6\). Since the product in the problem starts from 2, it is \(2 \times 3 \times 4 \times 5 \times 6\), which is also 720. This immediately tells you that the last number, \(n\), must be 6.

Step 1: Understanding the Concept:

This is a word problem that requires translating the given information into a system of algebraic equations to solve for an unknown variable, the seating capacity.


Step 2: Key Formula or Approach:

1. Define a variable for the seating capacity (e.g., \(C\)).

2. Create an expression for the total number of tickets sold in terms of \(C\).

3. Create an expression for the total attendance across both performances in terms of \(C\).

4. Equate these two expressions and solve for \(C\).


Step 3: Detailed Explanation:

Let \(C\) be the seating capacity of the auditorium.

Let \(T\) be the total number of tickets sold.


From the problem statement, "150 more tickets... were sold than the seating capacity":
\[ T = C + 150 \]

There were two performances, and for each one, the auditorium was \(\frac{2}{3}\) full. So, the number of people who attended each performance is:
\[ Attendance per performance = \frac{2}{3} \times C \]
The total number of people who attended the two performances is:
\[ Total Attendance = 2 \times \left(\frac{2}{3} C\right) = \frac{4}{3} C \]

The problem states that "the total number of tickets sold was equal to the total number who attended". Therefore:
\[ T = Total Attendance \]
We can now set our two expressions for \(T\) and Total Attendance equal to each other:
\[ C + 150 = \frac{4}{3} C \]
To solve for \(C\), first get all the \(C\) terms on one side. Subtract \(C\) from both sides:
\[ 150 = \frac{4}{3} C - C \] \[ 150 = \frac{4}{3} C - \frac{3}{3} C \] \[ 150 = \frac{1}{3} C \]
Multiply both sides by 3 to find \(C\):
\[ C = 150 \times 3 = 450 \]

Step 4: Final Answer:

The seating capacity of the auditorium is 450.
Quick Tip: Break down complex word problems sentence by sentence. Assign variables to unknown quantities and translate each piece of information into a mathematical equation or expression. This systematic approach helps prevent confusion and errors.

Step 1: Understanding the Concept:

This question is about number theory, specifically finding the total number of positive divisors (or factors) of a given number. The number is expressed as a product of three distinct prime numbers.


Step 2: Key Formula or Approach:

There are two common methods to solve this:

1. Listing Method: Systematically list all possible combinations of the prime factors.

2. Formula Method: If a number's prime factorization is \(n = p_1^{a_1} p_2^{a_2} \dots p_k^{a_k}\), then the total number of divisors is given by the product \((a_1 + 1)(a_2 + 1) \dots (a_k + 1)\).


Step 3: Detailed Explanation:

Method 1: Listing the Divisors

The number is \(n = pqr\). The divisors are formed by taking combinations of its prime factors \(p, q,\) and \(r\).

- Divisors with zero prime factors: 1. (1 divisor)

- Divisors with one prime factor: \(p, q, r\). (3 divisors)

- Divisors with two prime factors: \(pq, pr, qr\). (3 divisors)

- Divisors with three prime factors: \(pqr\) (which is \(n\)). (1 divisor)

The total number of divisors is the sum of these counts: \(1 + 3 + 3 + 1 = 8\).


Method 2: Using the Formula

The given number is \(n = pqr\). The prime factorization can be written with exponents as:
\[ n = p^1 q^1 r^1 \]
The exponents of the prime factors are \(a_1 = 1\), \(a_2 = 1\), and \(a_3 = 1\).

Using the formula for the number of divisors:
\[ Number of divisors = (a_1 + 1)(a_2 + 1)(a_3 + 1) \] \[ Number of divisors = (1 + 1)(1 + 1)(1 + 1) \] \[ Number of divisors = 2 \times 2 \times 2 = 8 \]
Both methods yield the same result. The number of divisors includes 1 and \(n\) itself.


Step 4: Final Answer:

The number \(n\) has 8 different positive divisors.
Quick Tip: The formula method is very powerful and much faster than listing, especially for numbers with higher powers in their prime factorization. For any number \(n = p^a q^b\), the number of divisors is \((a+1)(b+1)\). This is a fundamental concept in number theory and very useful for competitive exams.

Step 1: Understanding the Concept:

This is a sentence completion question that tests vocabulary and logical structure. The sentence starts with "Though...", which signals a concession or contrast. The structure is: "Although X is true, Y is also true." The two blanks must create a logical and meaningful sentence based on this structure.


Step 2: Key Formula or Approach:

1. Analyze the first part of the sentence: "Though ---- to some degree...". This blank should describe a quality of "telling a small lie." The word "Though" suggests this quality will be somewhat negative.

2. Analyze the second part: "...enables one to avoid --- another's feelings." This blank should describe a negative action that the lie helps to prevent.

3. Evaluate the options to see which pair of words best fits this logical structure.


Step 3: Detailed Explanation:

The sentence contrasts a negative aspect of lying with a positive outcome. A small lie, while perhaps not ideal, can serve a social purpose.

- Let's look at option (B): "Though regrettable... avoid harming another's feelings." This fits perfectly. A lie is generally seen as regrettable (a negative quality), but in this context, its positive outcome is that it avoids harming someone's feelings.

- Option (A) is incorrect because "mollifying" means to soothe or appease. The sentence says the lie helps one avoid an action, not perform one. You don't "avoid mollifying" feelings in this context.

- Option (C) is incorrect because one does not "avoid exaggerating" another's feelings by telling a lie.

- Option (D) is incorrect because a lie is not typically described as "attractive," and one uses a lie to spare feelings, not to "avoid considering" them.

- Option (E) is incorrect because "avoid resisting" feelings does not make logical sense in this context.


Step 4: Final Answer:

The words "regrettable" and "harming" create the most logical and coherent sentence, acknowledging the negative nature of a lie while stating its potential to prevent hurt.
Quick Tip: Pay close attention to structural keywords like "Though," "Although," "Despite," "However," etc. They provide crucial clues about the logical relationship (e.g., contrast, cause-and-effect) between different parts of the sentence.

Step 1: Understanding the Concept:

This sentence completion question describes a cause-and-effect relationship. The cause is scientists' fascination with dogs' smell and hearing. The effect is how they have treated the study of dogs' eyesight. The blank must describe this effect.


Step 2: Key Formula or Approach:

1. Identify the cause: Scientists were "intrigued by dogs' superior senses of smell and hearing."

2. Identify the consequence or assumption: Researchers assumed dogs live in a "drab, black-and-white world."

3. The blank must be a verb that logically connects the cause to this assumption. If scientists focused on other senses and made a negative assumption about eyesight, they likely neglected or undervalued it.


Step 3: Detailed Explanation:

The sentence suggests that the intense focus on smell and hearing led to a neglect or misjudgment of the sense of sight.

- Option (E), "underestimated," fits perfectly. Because researchers were so impressed with other senses, they paid less attention to eyesight and wrongly assumed it was poor.

- Option (A), "studied," is too neutral. While they did study it, the context implies they did so with a flawed perspective, which "underestimated" captures more precisely.

- Options (B), "coveted" (to desire), and (D), "resented" (to feel bitterness at), are emotions that do not fit the scientific context described.

- Option (C), "appreciated," is the opposite of what the sentence implies. The assumption of a "drab, black-and-white world" shows a lack of appreciation.


Step 4: Final Answer:

The word "underestimated" correctly describes how the researchers' focus on other senses led them to undervalue and misjudge dogs' eyesight.
Quick Tip: Look for clues in the latter part of the sentence that define or explain the missing word. Here, the assumption of a "drab, black-and-white world" is a direct result of the action in the blank, pointing clearly to "underestimated."

Step 1: Understanding the Concept:

This is a sentence completion question where the logic is based on contrast. The word "Despite" at the beginning of the sentence is a key indicator of this contrast.


Step 2: Key Formula or Approach:

1. Identify the first part of the contrast: "a string of dismal earnings reports." This is a negative situation.

2. The word "Despite" means that the second part of the sentence will present an opposing or surprising outcome.

3. Therefore, the blank must be filled with a word that indicates a positive development for the company's strategy.


Step 3: Detailed Explanation:

The sentence sets up a contrast between past negative results and a current positive trend.

- Option (D), "work," means to be effective or successful. If the strategy is beginning to "work," it creates the perfect contrast with the "dismal earnings reports."

- Options (A), "falter" (to lose strength), and (B), "disappoint," are negative words. They would be consistent with the dismal reports, but they contradict the logical structure set up by "Despite."

- Options (C), "compete," and (E), "circulate," do not fit the context. A strategy itself doesn't "compete" or "circulate"; it either succeeds (works) or fails.


Step 4: Final Answer:

The word "work" provides the necessary positive contrast to the preceding negative clause, indicating that the strategy is starting to show success.
Quick Tip: In sentences with contrast keywords ("Despite," "Although," "However"), look for an answer choice that presents an opposite idea to the one already stated in the sentence.

Step 1: Understanding the Concept:

This sentence completion question has two blanks that must be filled to create a logically coherent statement. The two words must be consistent with each other and with the context of a president making a difficult decision.


Step 2: Key Formula or Approach:

1. Analyze the first blank. It follows "lengthy," suggesting a long process of careful consideration before making a decision.

2. Analyze the second blank. It describes the opinions that were being "painstakingly" weighed. The word "painstakingly" implies difficulty and complexity, suggesting the opinions were not simple or all the same.

3. Evaluate the pairs of words to find the best fit.


Step 3: Detailed Explanation:

- Let's examine option (A): "deliberation.. divergent." A "deliberation" is a long and careful discussion, which fits perfectly with "lengthy." "Divergent" opinions are ones that differ or conflict, which would require "painstakingly weighing." This pair is an excellent fit.

- Option (B) is incorrect because if the opinions were "unanimous" (all in agreement), there would be no need for lengthy or painstaking weighing.

- Option (C) is incorrect for a similar reason. If opinions were "consistent," the decision would be much easier.

- Option (D) is weak. While there might be "speculation," "deliberation" is a more formal and fitting term for a presidential process. "Conciliatory" (intended to pacify) opinions might not require such difficult weighing.

- Option (E) is incorrect. "Canvassing" (soliciting votes/opinions) is part of the process, but "deliberation" is a better word for the overall consideration. "Arbitrary" (random or baseless) is not a fitting description for opinions from cabinet members.


Step 4: Final Answer:

The pair "deliberation" and "divergent" accurately describes a president engaging in a long process of careful thought to resolve differing viewpoints among advisors.
Quick Tip: Look for descriptive adverbs and adjectives in the sentence. Words like "lengthy" and "painstakingly" provide strong clues about the nature of the words that should fill the blanks.

Step 1: Understanding the Concept:

This sentence uses the word "Although" to set up a contrast between Jones's ability as a writer and his ability as an editor. The two blanks must contain words that reflect this contrast.


Step 2: Key Formula or Approach:

1. The first clause, "Although just barely ---- as a writer," suggests that his writing skill was minimal or only just acceptable. The word "lucid prose" implies he could write clearly, so the blank should reflect a low level of proficiency, not complete incompetence.

2. The second clause, "...was an extremely ---- editor who worked superbly," indicates that his editing skill was very high.

3. The correct pair of words will show a contrast between "minimally skilled" and "highly skilled."


Step 3: Detailed Explanation:

- Let's analyze option (C): "adequate.. capable." "Just barely adequate" means he met the minimum standard as a writer, which fits the sentence structure. "Extremely capable" means he was very skilled as an editor, which fits with the description "worked superbly." This pair provides the perfect contrast.

- Option (A) is incorrect. "Deficient" contradicts the idea that he wrote "lucid prose," and a "muddling" editor is a bad one, which contradicts "worked superbly."

- Option (B) is incorrect because "contentious" (argumentative) is not necessarily a positive trait for an editor.

- Option (D) is incorrect because the phrase "just barely appalling" does not make sense. "Appalling" is a strong negative, not something one can be "just barely."

- Option (E) reverses the logic. It suggests he was a good writer ("engaging") but a bad editor ("inept"), which contradicts the structure set up by "Although."


Step 4: Final Answer:

The pair "adequate" and "capable" correctly establishes the contrast between Jones's minimal skill as a writer and his excellent skill as an editor.
Quick Tip: The phrase "just barely" usually modifies a word that indicates a minimum level of competence or acceptability. It's used to say someone just meets a standard, but no more.

Step 1: Understanding the Concept:

This sentence completion question presents a philosophical or moral statement. The two parts of the sentence, separated by a semicolon, are related and explain each other. The blanks must be filled with words that create a meaningful and logical piece of advice.


Step 2: Key Formula or Approach:

1. Analyze the first clause: "The accusations we bring against others should be ---- ourselves". This suggests that when we criticize others, that criticism should have a reflective purpose for ourselves.

2. Analyze the second clause: "...they should not ---- complacency and easy judgments on our part...". This states what the accusations should not do. They should not encourage us to be self-satisfied or lazy in our own moral conduct.

3. Find a pair of words that fits this structure of self-reflection versus self-satisfaction.


Step 3: Detailed Explanation:

- Let's examine option (D): "warnings to.. justify". The sentence would read: "The accusations we bring against others should be warnings to ourselves; they should not justify complacency...". This makes perfect sense. Seeing a flaw in someone else should serve as a warning to avoid that flaw ourselves. It should not be used as an excuse to feel superior and become complacent about our own behavior.

- Option (A) is illogical. Accusations are not "definitions of" ourselves.

- Option (B) is weak. "Instructions to" is plausible, but "warnings to" is a much better fit for the moral context. Furthermore, "equate complacency" does not make sense.

- Option (C) is incorrect. "Denigrations of" (criticisms of) ourselves is redundant and "exclude complacency" is the opposite of the intended meaning.

- Option (E) is not a good fit. "Parodies of" ourselves is too specific, and "satirize complacency" doesn't fit the logical flow.


Step 4: Final Answer:

The words "warnings to" and "justify" correctly convey the intended moral lesson: use criticism of others as a tool for self-improvement, not as a reason for self-satisfaction.
Quick Tip: When a sentence has two clauses separated by a semicolon, the second clause often restates, explains, or elaborates on the first. Use this relationship to ensure the words you choose for the blanks are consistent across both parts of the sentence.

Step 1: Understanding the Concept:

This sentence completion question is built on a contrast, indicated by the word "Although." It contrasts the nature of word meanings with the nature of spelling.


Step 2: Key Formula or Approach:

1. Identify the first part of the contrast: "the meanings of words may necessarily be liable to change." This establishes that word meanings are not fixed; they are flexible.

2. The phrase "it does not follow that" signals a break in logic. Just because meanings change, it doesn't mean the same is true for spelling.

3. The blank describes what a lexicographer (a dictionary writer) can make spelling. The logic requires a word that is the opposite of "liable to change."


Step 3: Detailed Explanation:

The sentence argues that the flexibility of word meaning does not necessitate the flexibility of spelling. A lexicographer can still establish a fixed standard for spelling.

- Option (E), "constant," means unchanging or fixed. This is the direct opposite of "liable to change" and fits the logic of the sentence perfectly. A lexicographer can render spelling constant (standardized).

- Options (A), "arbitrary" (based on random choice), and (D), "flexible," are similar in meaning to "liable to change." Choosing them would destroy the contrast that the sentence is built on.

- Options (B), "superfluous" (unnecessary), and (C), "interesting," are irrelevant to the contrast between change and stability.


Step 4: Final Answer:

The word "constant" correctly completes the contrast, arguing that even if meanings are fluid, spelling can be made stable and unchanging.
Quick Tip: In sentences with the structure "Although A, not necessarily B," the word that fills the blank in B is often an antonym or a word with an opposing concept to a key word in A. Here, "constant" is the opposite of "change."

Step 1: Understanding the Concept:

This is an analogy question. We need to identify the relationship between the first pair of words (ELEGIAC: SORROW) and then find another pair with the same relationship.


Step 2: Key Formula or Approach:

First, define the relationship in the stem pair. Something that is "elegiac" is mournful or expresses sorrow. An elegy is a poem of serious reflection, typically a lament for the dead. Therefore, the relationship is that the first word is characterized by or expresses the second word. (X is characterized by Y).


Step 3: Detailed Explanation:

Let's apply the "X is characterized by Y" relationship to the answer choices:

- (A) Is something polemical characterized by resolution? No, a polemic is a strong verbal or written attack, characterized by controversy or argument, not resolution.

- (B) Is something fictional characterized by humor? No, fiction can be of any genre (tragedy, horror, etc.), not necessarily humor.

- (C) Is something devotional characterized by reverence? Yes, a devotional work or act is one that shows deep religious feeling or reverence. This relationship matches the stem pair.

- (D) Is something didactic characterized by inspiration? No, something didactic is intended to teach, particularly a moral lesson. It is not necessarily inspirational.

- (E) Is something literary characterized by emotion? This is too broad. While literature often involves emotion, it's not its sole defining characteristic. The relationship in the stem pair is much more specific.


Step 4: Final Answer:

The relationship between ELEGIAC and SORROW (characterized by) is best mirrored by the relationship between devotional and reverence.
Quick Tip: For analogy questions, try to form a precise sentence that connects the two words. For example, "Something elegiac expresses sorrow." Then, plug the answer choices into the same sentence structure: "Something devotional expresses reverence." This helps to test the relationship accurately.

Step 1: Understanding the Concept:

This is an analogy question. We need to find the relationship between ROSTRUM and ORATOR and find an answer pair with the same relationship.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair. A "rostrum" is a raised platform where an "orator" (a public speaker) stands to perform their function. The relationship is "X is the specific place/station from which Y performs their professional function."


Step 3: Detailed Explanation:

Let's test this relationship with the answer choices:

- (A) Is a stage the place from which an audience performs its function? No, the audience watches from their seats; actors perform on the stage. This is an incorrect relationship.

- (B) Is a bench the place from which a judge performs their professional function? Yes, a judge presides over a courtroom from the bench. This is a very strong match.

- (C) Is a shelf the place from which a clerk performs their function? No, a clerk may use shelves, but it is not their designated station in the same way a rostrum or bench is. A clerk works at a counter or desk.

- (D) Is a municipality the place from which a citizen performs their function? No, a municipality is a city or town where a citizen lives.

- (E) Is a crosswalk the place from which a pedestrian performs their function? A pedestrian uses a crosswalk to cross the street, but it's not a professional station. The relationship in the stem pair is more specific to a profession or formal role.


Step 4: Final Answer:

The relationship of "a specific station for a professional role" is best represented by BENCH: JUDGE.
Quick Tip: Pay attention to the specificity of the relationship. While a pedestrian uses a crosswalk, the relationship between a judge and the bench is much more specific and role-defined, just like an orator and a rostrum. Always look for the closest possible parallel.

Step 1: Understanding the Concept:

This analogy question asks us to identify a relationship between two words and find a pair that shares that same relationship.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: MISUNDERSTOOD: CLARIFY. To "clarify" something is the action one takes to correct something that has been "misunderstood." The relationship is "Y is the action taken to correct X," where X is a negative state or error.


Step 3: Detailed Explanation:

Let's apply this "action to correct an error" relationship to the options:

- (A) To "espouse" a cause is to adopt or support it, often strongly. This does not correct being "fanatical."

- (B) To "renounce" something is to formally give it up. One might renounce something popular, but it's not an action to correct popularity.

- (C) To "shock" is a reaction to something "fantastic," not a correction of it.

- (D) "Erroneous" means incorrect or wrong. To "retract" a statement is to withdraw it because it was erroneous. This is a perfect match: retracting is the action taken to correct an erroneous statement.

- (E) To "flaunt" something is to display it ostentatiously, which makes it more "conspicuous," not less. This action intensifies the state, rather than correcting it.


Step 4: Final Answer:

The relationship of taking an action (retract) to correct a problem (erroneous) perfectly mirrors the relationship in the stem pair (clarify: misunderstood).
Quick Tip: Focus on the function of the words. CLARIFY is a corrective action. Look for another corrective action in the second position of the answer pairs. Retract is a corrective action, while espouse, renounce, shock, and flaunt are not (in the same way).

Step 1: Understanding the Concept:

This is an analogy question that relates a place to the raw material that is processed there.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: REFINERY: PETROLEUM. A "refinery" is a factory or facility where a raw material, "petroleum," is processed and made into useful products. The relationship is "X is a place where Y is processed."


Step 3: Detailed Explanation:

Let's test the "X is a place where Y is processed" relationship with the options:

- (A) Is a mill a place where grain is processed? Yes, a mill is a place where grain is ground into flour. This is a strong match.

- (B) Is a mine a place where ore is processed? No, a mine is where ore is extracted or dug from the ground. It is then sent elsewhere for processing.

- (C) Is a warehouse a place where merchandise is processed? No, a warehouse is where merchandise is stored.

- (D) Is a generator a place where electricity is processed? No, a generator is a machine that produces electricity.

- (E) Is a forest a place where lumber is processed? No, a forest is where trees grow. Trees are harvested and then taken to a mill to be processed into lumber.


Step 4: Final Answer:

The relationship of a place of processing to its raw material is best mirrored in the pair MILL: GRAIN.
Quick Tip: Distinguish between different stages of production: extraction (mine: ore), growth (forest: lumber), production (generator: electricity), storage (warehouse: merchandise), and processing (refinery: petroleum). The question asks for the processing relationship.

Step 1: Understanding the Concept:

This analogy question relates a quality or situation to a state or resource that it diminishes.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: TEDIOUS: ENERGY. Something that is "tedious" (long, slow, or dull; tiresome) causes a loss of or depletes one's "energy." The relationship is "X causes a loss of Y."


Step 3: Detailed Explanation:

Let's apply the "X causes a loss of Y" relationship to the answer choices:

- (A) Does being avaricious (greedy) cause a loss of satisfaction? Avarice is a state defined by a lack of satisfaction, but it doesn't necessarily cause a loss of it in the same way tedium causes a loss of energy.

- (B) Does being fractious (irritable) cause a loss of irritation? No, being fractious is a state of being irritated.

- (C) Does something disturbing cause a loss of composure? Yes, a disturbing event or piece of news can make a person lose their composure (calmness). This relationship is a very strong match.

- (D) Does something improbable cause a loss of ambition? There is no clear, direct relationship here.

- (E) Does being informed cause a loss of intelligence? No, being informed generally enhances one's use of intelligence.


Step 4: Final Answer:

The cause-and-effect relationship where a quality or event leads to the loss of a state of being is best represented by DISTURBING: COMPOSURE.
Quick Tip: Frame the relationship as a cause-and-effect sentence. "A tedious task drains your energy." Then test the options: "A disturbing sight can make you lose your composure." This works well. "An avaricious person lacks satisfaction" - this is a definition, not cause-and-effect in the same way.

Step 1: Understanding the Concept:

This is an analogy question where the first word is a descriptive adjective that characterizes the second word, which is a noun. The adjective describes a pleasing aesthetic quality.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: GRACEFUL: MOVEMENT. "Graceful" is a positive quality that describes "movement." It refers to a pleasing elegance or beauty of motion. The relationship is "X is a pleasing aesthetic quality of Y."


Step 3: Detailed Explanation:

Let's test this relationship with the options:

- (A) Is euphonious a pleasing aesthetic quality of sound? Yes, "euphonious" literally means pleasing to the ear. This is a perfect match.

- (B) Is forbidding a pleasing quality of a countenance (facial expression)? No, "forbidding" means unfriendly or threatening, which is a negative quality.

- (C) Is ephemeral a pleasing quality of duration? "Ephemeral" means lasting for a very short time. It is a neutral descriptor of duration, not an aesthetic quality.

- (D) Is melodramatic a pleasing quality of emotion? No, "melodramatic" means exaggerated and overemotional, which is generally considered a negative quality.

- (E) Is vibrant a pleasing quality of color? "Vibrant" means bright and striking. It is a quality of color and often positive, but "euphonious" for sound is a more precise parallel to "graceful" for movement, as both describe a specific type of aesthetic harmony.


Step 4: Final Answer:

Both "graceful" and "euphonious" describe a harmonious and aesthetically pleasing quality of a sensory experience (movement and sound, respectively), making EUPHONIOUS: SOUND the best match.
Quick Tip: Look for nuances in the relationship. Both (A) and (E) seem possible at first. However, 'graceful' and 'euphonious' both relate to a sense of harmony and beauty, making their relationship more specific and parallel than 'vibrant' and 'color', which relates more to intensity.

Step 1: Understanding the Concept:

This is an analogy question. We need to determine the relationship between the given pair of words. Note that "bravura" is a noun, while the first words in the options are adjectives. This suggests we should think of "bravura" as a quality. A "bravura performance" is one characterized by great skill and brilliance.


Step 2: Key Formula or Approach:

Define the relationship: "Bravura" is a quality of great brilliance and flair in a "performance." The relationship is "X is a quality of dazzling brilliance or skill in Y." A performance characterized by bravura is a "bravura performance."


Step 3: Detailed Explanation:

We are looking for a pair where the first word describes a dazzling or brilliant quality of the second. Let's form sentences like "a Y characterized by X."

- A performance characterized by bravura is a dazzling display.

- (A) An expenditure characterized by being extravagant is excessive. This is often negative.

- (B) An oration characterized by being elaborate is detailed. "Elaborate" lacks the specific meaning of brilliance or flair.

- (C) An outcome characterized by being foreseeable is predictable, the opposite of dazzling.

- (D) An analysis characterized by being thorough is complete. This describes diligence, not brilliance in the same way as bravura.

- (E) An appearance characterized by being resplendent is dazzling. "Resplendent" means attractive and impressive through being richly colorful or sumptuous. This is a very strong parallel to bravura. A resplendent appearance and a bravura performance are both impressive, brilliant displays.


Step 4: Final Answer:

The quality of being "resplendent" relates to an "appearance" in the same way the quality of "bravura" relates to a "performance"; both describe a brilliant, dazzling display.
Quick Tip: Don't be thrown off by slight differences in parts of speech if the underlying conceptual relationship is strong. The core idea here is a quality of impressive, brilliant display. "Bravura" and "resplendent" both capture this idea perfectly for their respective nouns.

Step 1: Understanding the Concept:

This analogy question compares two verbs. The first verb is a more specific or intense version of the second verb.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: BADGER: BOTHER. To "badger" someone is to repeatedly and annoyingly ask them for something; it is to "bother" them persistently. The relationship is "To X is to Y persistently and excessively."


Step 3: Detailed Explanation:

Let's apply this "To X is to Y persistently" relationship to the options:

- (A) To "persecute" is to subject someone to hostility and ill-treatment, especially because of their race or political or religious beliefs. It may "injure" them, but it is not defined as "injuring persistently."

- (B) To "haunt" (of a ghost or memory) is to be persistently present. It is related to "remember," but the subject is different. A person remembers; a memory haunts. The structure does not match.

- (C) To "belabor" a point is to argue or elaborate on it in excessive detail. It is to "mention" or discuss it persistently and to an annoying degree. This is a perfect match.

- (D) To "quibble" is to argue about a trivial matter. It is a specific type of arguing, but the definition isn't "to argue persistently."

- (E) To "censure" is to express severe disapproval. This is an action that might follow an "evaluation," not a persistent form of it.


Step 4: Final Answer:

The relationship of doing an action persistently or excessively is best mirrored in the pair BELABOR: MENTION.
Quick Tip: For verb analogies, think about the manner in which the action is performed. The word 'badger' adds the idea of persistence and annoyance to the word 'bother'. Look for a similar modification of meaning in the answer choices. 'Belabor' adds persistence and excess to 'mention'.

Step 1: Understanding the Concept:

This is an analogy question relating an adjective to a noun. The adjective describes a state of sameness or identity with respect to the noun.


Step 2: Key Formula or Approach:

Define the relationship in the stem pair: CONGRUENT: DIMENSIONS. In geometry, figures that are "congruent" have the exact same "dimensions" (size and shape). The relationship is "Things that are X have identical Y."


Step 3: Detailed Explanation:

Let's test the "Things that are X have identical Y" relationship with the options:

- (A) Do things that are convenient have identical time? No, something "convenient" happens at a suitable or opportune time, not an identical time.

- (B) This pair does not fit the structure. An axis is used to define a coordinate.

- (C) Is a conglomerate made of identical parts? No, a "conglomerate" is a thing consisting of a number of different and distinct parts. This is the opposite relationship.

- (D) Do things that are coincident have identical chance? No, "coincident" means occurring at the same time. It is often attributed to chance, but this doesn't fit the relationship structure.

- (E) Are things that are coeval of identical age? Yes, "coeval" means having the same age or date of origin. This is a perfect match.


Step 4: Final Answer:

Just as congruent things have identical dimensions, coeval things have an identical age. The relationship is perfectly parallel.
Quick Tip: Many challenging vocabulary words in analogies begin with common prefixes. Here, "con-" and "co-" both mean "with" or "together." Recognizing this can help you see the shared idea of sameness in both CONgruent and COeval.

Step 1: Understanding the Concept:

This question asks for the primary purpose or main idea of the entire passage. We need to identify the overall goal of the author in writing this text.


Step 2: Key Formula or Approach:

Read the passage to identify the central theme. The author starts by identifying a problem ("artificial aspects of the conventional course"), then describes a solution ("an experimental version of the traditional scholarly methods course"), and explains how and why it was implemented. This structure is key to finding the primary concern.


Step 3: Detailed Explanation:

- The passage begins by outlining a "shortcoming" in traditional scholarly methods courses: they are often artificial and don't provide an "authentic experience."

- The bulk of the passage is then dedicated to "describing an attempt to correct" this shortcoming—the experimental course focused on Elizabeth Griffith. The author details the course's design, rationale, and the benefits it offers.

- (A) is too narrow. The author does more than just reveal a deficiency; they describe a solution.

- (B) is incorrect. The author is describing a new, non-traditional approach, not a return to tradition.

- (D) is also too narrow. While the author's tone is positive, the passage is more descriptive than evaluative. It focuses on explaining what the new approach is, rather than formally assessing its overall success with data.

- (E) is incorrect. The author describes a current experiment, not a future prediction.

Therefore, the passage is best summarized as a description of an experimental course designed to fix a problem in traditional teaching.


Step 4: Final Answer:

The author's primary concern is to describe a new course that was created as an attempt to correct the shortcomings of the traditional approach.
Quick Tip: For "primary purpose" questions, look at the overall structure of the passage. A "problem-solution" structure, like the one here, often points to an answer choice that includes words like "describe," "explain," or "propose" in relation to a problem or solution.

Step 1: Understanding the Concept:

This is an inference question focused on a specific example in the passage. We need to determine the intended lesson or effect of the student's discovery of an error in a reference book.


Step 2: Key Formula or Approach:

Locate the relevant part of the passage. The last paragraph describes the student's experience and explicitly states the author's hope for the outcome: "Such experiences... serve to vaccinate the student... against credulous use of reference sources." We need to interpret what "vaccinate against credulous use" means.


Step 3: Detailed Explanation:

- "Credulous use" means believing things too easily or without proper evidence. To be "vaccinated" against this means to be protected from this tendency. In other words, the student learns to be skeptical and critical.

- Option (B) perfectly captures this idea. The experience of finding a clear error in a published reference work teaches the student to "question the accuracy" of such sources, especially when dealing with neglected authors where errors are more common.

- Option (A) is too strong. The goal is to question, not to completely "disregard" all information. Critical use is different from non-use.

- Option (C) is also too extreme. The author does not suggest avoiding reference sources, but rather using them critically.

- Option (D) is related, as a first edition was used, but the main point of the anecdote is the error in the secondary source (\textit{Bibliotheca Britannica), not the importance of the primary source.

- Option (E) is a broader goal of the course, but it is not the specific lesson learned from this particular incident of finding an error.


Step 4: Final Answer:

The author expects the student's shocking discovery to instill a healthy skepticism, teaching them to question the reliability of reference materials.
Quick Tip: When a passage uses a strong metaphor, like "vaccinate the student," take a moment to decode it. A vaccine protects you from a disease. Here, the "disease" is credulity (gullibility), and the "protection" is skepticism.

Step 1: Understanding the Concept:

This question asks us to identify a disadvantage of the new, experimental course as acknowledged by the author.


Step 2: Key Formula or Approach:

The passage is structured as a comparison. The author contrasts the new course with the "conventional course." Often, in such comparisons, the author will concede a point or mention a trade-off. We need to find where the author admits a benefit of the old course that was lost in the new one.


Step 3: Detailed Explanation:

- In the second paragraph, the author describes abandoning the traditional method: "...the usual procedure of assigning a large number of small problems drawn from the entire range of historical periods was abandoned, though this procedure has the obvious advantage of at least superficially familiarizing students with a wide range of reference sources."

- This sentence explicitly states the advantage of the old system that was given up. The disadvantage of the new, focused system is therefore the loss of this broad, albeit superficial, exposure.

- Option (E) rephrases this idea perfectly. By focusing deeply on one author, "Students were not given an opportunity to encounter certain sources of information [the 'wide range'] that could prove useful in their future studies."

- (A) is incorrect; the course was specifically about a woman writer outside the canon.

- (B) is not mentioned in the passage.

- (C) is mentioned as an advantage of the new course (it provided a "clear field"), not a disadvantage.

- (D) is not mentioned or implied.


Step 4: Final Answer:

The author concedes that by abandoning the broad survey method of the traditional course, the experimental course forgoes the benefit of introducing students to a wide variety of reference sources.
Quick Tip: Look for concession words like "though," "although," or "while it is true that." These words often signal that the author is about to acknowledge a downside of their preferred approach or an upside of the approach they are criticizing.

Step 1: Understanding the Concept:

This question asks us to define the specific "pedagogical purpose" (teaching goal) for which the writer Elizabeth Griffith was chosen as the subject of the course.


Step 2: Key Formula or Approach:

We must look at the context surrounding line 28. The sentences immediately preceding line 28 define the goal of the course. The "pedagogical purpose" is the implementation of this goal.


Step 3: Detailed Explanation:

- The passage states that the experimental course was designed "to give them [students] an authentic experience of literary scholarship and to inspire them to take responsibility for the quality of their own work."

- This is the core teaching goal. Choosing Griffith served this purpose because her obscurity and the state of her texts created a perfect environment for students to do real, original research, make their own discoveries, and learn the scholarly process firsthand.

- Option (D), "To encourage scholarly rigor in students' own research," is a perfect summary of this goal. The "authentic experience" is one that requires rigor, and making students "take responsibility" is a way of encouraging it.

- (A) and (E) are potential outcomes or larger academic goals, but they are not the immediate pedagogical (teaching) purpose of the course, which is focused on student development.

- (B) describes a reason for creating the course, not the purpose of choosing Griffith specifically.

- (C) is too simplistic. The goal was not just to transmit information about Griffith, but to use her as a subject for students to learn research skills.


Step 4: Final Answer:

The choice of Elizabeth Griffith served the specific teaching goal of creating an environment where students could engage in genuine research, thereby fostering scholarly rigor and responsibility.
Quick Tip: Pay attention to the specific wording of the question. "Pedagogical purpose" means "teaching purpose." This directs you to look for the answer that focuses on what the students are meant to learn or how they are meant to develop, rather than on broader outcomes like changing the literary canon.

Step 1: Understanding the Concept:

This question asks about the role of a specific paragraph (the last one) within the context of the entire passage. We need to understand what this paragraph contributes to the author's overall argument.


Step 2: Key Formula or Approach:

The passage first makes a general argument: the experimental course provides an "authentic experience" that teaches valuable lessons. A common rhetorical strategy is to follow a general claim with a specific example to make the claim more concrete and understandable. We should check if the last paragraph serves this function.


Step 3: Detailed Explanation:

- The main body of the passage explains the theory and design of the experimental course.

- The last paragraph shifts from theory to practice by telling the story of one student's specific discovery: finding an error in a major reference work.

- This story is not a summary of all benefits (A), nor is its primary purpose to discuss the canon (B, E) or contrast with the traditional course (D).

- Its function is to be an "illustration" or a concrete example of the "immediate" and "authentic" experiences the course provides. The student's shock and outrage is a powerful example of the engagement the author values.

- Therefore, option (C) is the most accurate description of the paragraph's function. It shows, rather than just tells, what students can get out of the course.


Step 4: Final Answer:

The final paragraph serves as a specific, illustrative anecdote that demonstrates the kind of powerful, hands-on learning experiences the experimental course offers.
Quick Tip: When a question asks for the "function" of a paragraph, think about its relationship to the paragraphs before it. Is it providing an example, a counter-argument, a summary, or a transition? Identifying this structural role is key.

Step 1: Understanding the Concept:

This is an inference question that asks us to identify a specific question or "issue" that the course is designed to raise, based on context clues.


Step 2: Key Formula or Approach:

We must read the sentence containing line 38 carefully. The line is part of the clause: "...her exclusion from the canon and virtual disappearance from literary history also helped raise issues concerning the current canon." The issue must be related to this statement.


Step 3: Detailed Explanation:

- The passage establishes a key fact about Griffith: she was "successful in the eighteenth century" with "continued productivity and favorable reviews."

- It then contrasts this success with her modern status: "her exclusion from the canon and virtual disappearance."

- The natural "issue" or question that arises from this contrast is: Why did this happen? Why would a successful and popular writer, who was also a woman, be forgotten and left out of the list of great authors?

- Option (A) frames this question perfectly.

- Options (B), (C), and (E) are questions about Griffith's work and reception, but they don't directly address the central mystery of her exclusion from the modern canon.

- Option (D) refers to the anecdote in the last paragraph, which is separate from the "issues concerning the current canon" mentioned in the third paragraph.


Step 4: Final Answer:

The most prominent issue raised by Griffith's career is the question of why a writer who was successful in her own time was subsequently excluded from the literary canon.
Quick Tip: For inference questions tied to a specific line, the answer is almost always found in the sentences immediately surrounding that line. The context provides the clues needed to make a logical leap.

Step 1: Understanding the Concept:

This question asks for the author's inferred opinion about the traditional courses that the experimental course was designed to replace.


Step 2: Key Formula or Approach:

The author's opinion is revealed through the criticisms leveled against the traditional course and the praise given to the new one. We must find the key descriptive words used for each.


Step 3: Detailed Explanation:

- The author criticizes the "artificial aspects of the conventional course" and its "usual procedure of assigning a large number of small problems drawn from the entire range of historical periods." This establishes that the course is "wide-ranging."

- The author states this traditional procedure only "superficially familiarizing students" with sources.

- This is contrasted with the new course, which provides an "authentic experience of literary scholarship," also described as "genuine scholarly activity."

- Putting these pieces together, the author's view is that the traditional course, because it is too broad and wide-ranging, offers only a superficial experience that doesn't feel like real, genuine scholarship.

- Option (E) captures this criticism perfectly. The wide range is the cause, and the failure to approximate genuine activity is the effect.

- (B) is incorrect; the author criticizes their focus for being too broad, not too narrow.

- (D) is a possible consequence, but (E) better describes the fundamental problem according to the author: the lack of authenticity caused by the broad-survey design.

- (A) and (C) are not directly supported by the text.


Step 4: Final Answer:

The author implies that the traditional course's attempt to cover a vast range of topics makes the experience superficial and prevents students from engaging in what feels like genuine, focused scholarly work.
Quick Tip: Often, an author's opinion is best understood by looking at the contrast they create. What they praise in one thing reveals what they dislike about its opposite. Here, praising the "authentic" nature of the new course implies the old one is "inauthentic."

Step 1: Understanding the Concept:

This is a detail-oriented question that asks for a specific cause-and-effect relationship mentioned in the passage.


Step 2: Key Formula or Approach:

We need to scan the passage for the keyword "size" in relation to male cycad cones and identify what the passage says is a direct consequence of that size.


Step 3: Detailed Explanation:

- In the middle of the passage, the author presents a counterargument to the wind pollination theory. After mentioning large cones that produce a lot of pollen, the author states: "Still, many male cycad cones are comparatively small and thus produce far less pollen."

- This sentence explicitly and directly links the size of the cone to the amount of pollen it produces.

- Option (E) is a direct paraphrase of this statement.

- The other options are not supported by the text. The passage does not link cone size to structural arrangement (A), release mechanism (B), female accessibility (C), or attractiveness to insects (D).


Step 4: Final Answer:

The passage explicitly states that smaller male cycad cones produce less pollen, directly linking size to the amount of pollen produced.
Quick Tip: For "According to the passage" questions, you should be able to point to a specific sentence or phrase in the text that directly supports your answer. Avoid making inferences and look for the stated fact.

Step 1: Understanding the Concept:

This question asks us to summarize what the passage says about the evidence provided by the physical structure of male and female cones.


Step 2: Key Formula or Approach:

We need to synthesize the information about male and female cones. The passage discusses them separately and points out a conflict or inconsistency in the evidence they provide.


Step 3: Detailed Explanation:

- First, the passage states: "The structure of cycads male cones is quite consistent with the wind dispersal of pollen..." This means male cones support the wind pollination theory.

- Then, the passage states: "Furthermore, the structure of most female cycad cones seems inconsistent with direct pollination by wind." This means female cones contradict the wind pollination theory.

- Option (E) accurately summarizes this conflict: Male cone structure is consistent with a certain means (wind pollination), but that same means is inconsistent with the structure of most female cones.

- (A) is incorrect; the evidence is conflicting, not conclusive.

- (C) is close, but uses the strong word "irresolvable." The passage presents a discrepancy but doesn't claim it can never be resolved. (E) is a more direct and neutral summary of the facts presented.

- (D) misstates the relationship; the male and female cones present conflicting, not mutually exclusive, evidence regarding the same mechanism (wind).


Step 4: Final Answer:

The passage highlights a conflict where the structure of male cones supports the idea of wind pollination, while the structure of most female cones suggests that wind pollination would not work.
Quick Tip: In questions that ask you to synthesize information, be wary of answer choices with extreme words like "conclusive," "irresolvable," or "rules out." Scientific passages often present conflicting evidence without making such absolute claims. A more moderately worded answer is often correct.

Step 1: Understanding the Concept:

This is a logical reasoning question. It asks us to identify a piece of additional information that would strengthen a specific argument made in the passage.


Step 2: Key Formula or Approach:

The argument in lines 2-4 is: Cycads removed from their native habitats (and native insects) are usually infertile. The implied conclusion is that the native insects are necessary for pollination. A good way to strengthen a causal argument is to eliminate alternative explanations for the observed effect.


Step 3: Detailed Explanation:

- The observed effect is that transplanted cycads become infertile. The proposed cause is the absence of native insects.

- An alternative cause could be that the environment in the new habitat is different in some crucial way. For example, perhaps the wind conditions are different, and the cycads are actually wind-pollinated but cannot be in the new location.

- Option (D) addresses this alternative explanation. If the wind conditions in the new habitat are the same as in the old one, but the plant is still infertile, it makes it much less likely that wind is the key factor.

- By eliminating wind as a likely cause, the argument that the absence of insects is the true cause becomes much stronger and more convincing.

- (A) and (C) are irrelevant to the pollination argument.

- (B) would weaken the argument by introducing other possibilities.

- (E) is ambiguously worded and could weaken the argument if it implies that insects (either native or new) are present after transplantation.


Step 4: Final Answer:

By showing that a major environmental variable (wind) is unchanged between the two locations, we can more confidently attribute the plant's infertility to the variable that did change: the absence of native insects. This strengthens the argument.
Quick Tip: To strengthen a causal argument (X causes Y), you can do one of two things: 1) provide more evidence showing that when X happens, Y also happens, or 2) eliminate other possible causes for Y. This question is a classic example of the second strategy.

Step 1: Understanding the Concept:

This question asks us to infer the current state of scientific knowledge about cycad pollination, based on the evidence presented in the passage.


Step 2: Key Formula or Approach:

Review all the evidence the passage presents for each theory. Pay close attention to the specific words used to describe that evidence (e.g., "experiments," "anecdotal," "consistent with").


Step 3: Detailed Explanation:

- For insect pollination, the passage mentions "Experiments show that insects can function as pollinators" and evidence from transplanted cycads. This sounds like systematic evidence.

- For wind pollination, the passage mentions that "anecdotal reports of wind pollination in cycads cannot be ignored." The word "anecdotal" means the evidence is based on personal accounts rather than formal scientific study. It is by definition not systematic.

- While the passage also mentions cone structure as being "consistent with" wind pollination, this is presented as conflicting and inconclusive.

- Therefore, the passage suggests that while there are hints and anecdotes, there is a lack of hard, systematic proof for wind pollination.

- Option (A) captures this state of affairs perfectly.

- (B) is incorrect; "cannot be ignored" is very different from "confirmed."

- (C) is incorrect; the passage opens with "Experiments show," suggesting evidence for insect pollination exists and is not necessarily recent or scant.

- (D) and (E) are too strong; the passage mentions these as parts of the investigation, but does not state they are the primary or usual focus.


Step 4: Final Answer:

Based on the author's description of the evidence for wind pollination as "anecdotal," it is reasonable to infer that systematic evidence for it has not yet been produced.
Quick Tip: The specific vocabulary scientists and authors use to describe evidence is crucial. "Anecdotal evidence" is a key term that contrasts with "systematic evidence," "experimental proof," or "clinical trials." Recognizing this distinction is key to answering this question.

Step 1: Understanding the Concept:

This question asks for the antonym of the given word.


Step 2: Key Formula or Approach:

First, define the given word. "Procrastination" is the act of delaying or postponing tasks, especially out of habitual carelessness or laziness. We need to find the word that means the opposite.


Step 3: Detailed Explanation:

- The opposite of delaying and being lazy is working hard, promptly, and conscientiously.

- Let's look at the options. "Diligence" means careful and persistent work or effort. This is a direct opposite of procrastination.

- (B) Complacence is self-satisfaction.

- (C) Reasonableness is the quality of being fair and sensible.

- (D) Allegiance is loyalty.

- (E) Rehabilitation is the action of restoring something to a former condition.


Step 4: Final Answer:

The direct antonym of PROCRASTINATION is diligence.
Quick Tip: For antonym questions, first define the root word in your own terms. Then, think of what its opposite would be before you even look at the options. This can help you avoid being confused by attractive but incorrect choices.

Step 1: Understanding the Concept:

This question asks for the antonym of the given word.


Step 2: Key Formula or Approach:

Define the word "circuity." It comes from the word "circuitous," which means longer than the most direct way; roundabout, indirect. Circuity is the quality of being indirect.


Step 3: Detailed Explanation:

- We are looking for the word that means the quality of being direct.

- Option (A), "straightforwardness," means the quality of being direct and uncomplicated. This is a perfect antonym.

- The other options are unrelated: inventiveness (creativity), authenticity (genuineness), insightfulness (deep understanding), practicality (usefulness).


Step 4: Final Answer:

The opposite of circuity (indirectness) is straightforwardness (directness).
Quick Tip: Look for root words you recognize. "Circuity" contains the root "circ-", as in circle, which suggests going around rather than going straight. This can help you deduce the meaning even if the exact word is unfamiliar.

Step 1: Understanding the Concept:

This question asks for the antonym of the given verb.


Step 2: Key Formula or Approach:

Define the word "conclude." It means to bring something to an end; to finish. We need to find a word that means to begin.


Step 3: Detailed Explanation:

- We are looking for a word that means to start or begin.

- Option (E), "embark on," means to begin a course of action, especially one that is important or challenging. This is a direct antonym for conclude.

- The other options are unrelated: foster (encourage), frequent (visit often), emanate from (originate from), empower to (give authority to).


Step 4: Final Answer:

The opposite of concluding (ending) something is to embark on (begin) it.
Quick Tip: Think of how the words are used in phrases. You can "conclude a journey" or you can "embark on a journey." Setting up a parallel phrase like this can help confirm the antonym relationship.

Step 1: Understanding the Concept:

This question asks for the antonym of the given word.


Step 2: Key Formula or Approach:

Define the word "rite." A rite is a social custom, practice, or conventional act; a formal ceremony or procedure prescribed or customary in a religious or other solemn use. Key characteristics are that it is formal, planned, and follows a set structure.


Step 3: Detailed Explanation:

- The opposite would be an act that is informal, unplanned, and spontaneous.

- Option (B), "improvised act," is an act that is created spontaneously or without preparation. This is a direct antonym to the planned, structured nature of a rite.

- The other options are not direct opposites. They describe concepts from different domains (interpretation, logic, observation, etc.).


Step 4: Final Answer:

The opposite of a rite (a planned, formal act) is an improvised act (an unplanned, spontaneous act).
Quick Tip: Focus on the core concept of the word. The essence of a "rite" or "ritual" is its structure and pre-planned nature. The opposite must be something that lacks structure and is not planned in advance.

Step 1: Understanding the Concept:

This question asks for the antonym of the given adjective.


Step 2: Key Formula or Approach:

Define the word "blatant." It means (of bad behavior) done openly and unashamedly; completely lacking in subtlety; very obvious.


Step 3: Detailed Explanation:

- We are looking for a word that means subtle, not obvious, or not attracting attention.

- Option (C), "unobtrusive," means not conspicuous or attracting attention. This is a direct antonym of blatant.

- The other options are unrelated: indecisive (not able to make decisions), perceptive (insightful), involuntary (done without will), spontaneous (unplanned).


Step 4: Final Answer:

The opposite of blatant (obvious and conspicuous) is unobtrusive (not conspicuous).
Quick Tip: Think of common phrases. A "blatant lie" is an obvious one. The opposite would be a subtle, hard-to-notice action. "Unobtrusive" captures this sense of not being noticeable.

Step 1: Understanding the Concept:

This question asks for the antonym of the given verb.


Step 2: Key Formula or Approach:

Define the word "pontificate." It means to express one's opinions in a way considered annoyingly pompous and dogmatic. It implies speaking with an air of great authority, as if one's opinion is infallible.


Step 3: Detailed Explanation:

- The opposite of speaking in a pompous, overly authoritative way is to speak in a humble or reserved manner.

- Option (E), "speak modestly," perfectly captures this opposite meaning. Modesty is the quality of having a moderate or humble view of one's importance.

- The other options describe actions that are unrelated to the specific manner of speaking described by "pontificate."


Step 4: Final Answer:

The opposite of pontificating (speaking pompously) is to speak modestly.
Quick Tip: The word "pontificate" is related to "pontiff," another name for the Pope. This origin can help you remember that the word implies speaking with supreme, unquestionable authority. The opposite would naturally be to speak with humility.

Step 1: Understanding the Concept:

This question asks for the antonym of the given verb.


Step 2: Key Formula or Approach:

Define the word "posit." It means to put forward as a fact or as a basis for argument; to postulate or assume. It is an act of affirmation or proposing something as true.


Step 3: Detailed Explanation:

- The opposite of proposing or affirming something as true is to state that it is not true.

- Option (C), "deny," means to state that something is not true. This is a direct antonym.

- (A) deceive is to mislead.

- (B) begrudge is to envy or resent.

- (D) consent is to agree. While related, "deny" is a more direct opposite to "posit" (propose as true vs. state as untrue).

- (E) reinforce is to strengthen, which is a synonym, not an antonym.


Step 4: Final Answer:

The opposite of positing (affirming) a claim is to deny it.
Quick Tip: Think of a simple sentence: "Scientists posit that the universe is expanding." The opposing action would be to "deny that the universe is expanding."

Step 1: Understanding the Concept:

This question asks for the antonym of the given word, which can be used as both a noun and a verb.


Step 2: Key Formula or Approach:

First, define the word "fetter." As a verb, it means to restrain, chain up, or confine. As a noun, a fetter is a chain or shackle used to restrain a prisoner, typically placed around the ankles. The core meaning is one of restriction and lack of freedom. The antonym must mean to set free.


Step 3: Detailed Explanation:

We are looking for a word that means the opposite of to restrain or confine.

- (A) justify means to show or prove to be right or reasonable. This is unrelated.

- (B) comfort means to ease the distress of someone. This is unrelated.

- (C) intrude means to enter without invitation. This is unrelated.

- (D) liberate means to set someone free from a situation, such as imprisonment or slavery, where their liberty is severely restricted. This is a direct opposite of fetter.

- (E) optimize means to make the best or most effective use of something. This is unrelated.


Step 4: Final Answer:

The direct antonym of FETTER (to restrain) is to liberate (to set free).
Quick Tip: Think of the literal meaning of "fetter," which is a physical chain. The opposite action is to remove the chains and release someone. This image directly leads to the word "liberate."

Step 1: Understanding the Concept:

This question asks for the antonym of the adjective "synergic."


Step 2: Key Formula or Approach:

Define the word "synergic" (or the more common form, "synergistic"). It describes a state where different elements work together to produce a result that is greater than the sum of their individual effects. The core meaning is about cooperation, working together, and mutual enhancement. The opposite would describe elements that work against each other.


Step 3: Detailed Explanation:

We are looking for a word that means working against each other or being in opposition.

- (A) natural in origin is unrelated to cooperation.

- (B) fragile in structure is unrelated.

- (C) untainted means pure or clean, which is unrelated.

- (D) inessential means not necessary, which is unrelated.

- (E) antagonistic means showing or feeling active opposition or hostility toward someone or something. Antagonistic forces work against each other, creating a negative or destructive effect. This is the direct opposite of synergic forces working together.


Step 4: Final Answer:

The opposite of SYNERGIC (working together) is antagonistic (working against).
Quick Tip: Look at the prefixes. The prefix "syn-" means "together" (as in synthesis, synchronize). The prefix "anti-" means "against." Recognizing these prefixes can often lead you directly to the correct antonym pair.

Step 1: Understanding the Concept:

This question asks for the antonym of the noun "deprivation."


Step 2: Key Formula or Approach:

Define the word "deprivation." It means the lack or denial of something considered to be a necessity. It is a state of having too little of something. The opposite must be a state of having too much of something.


Step 3: Detailed Explanation:

We are looking for a word that means an excess or an overabundance.

- (A) sanity is the state of having a healthy mind. It is not the opposite of deprivation.

- (B) awareness is knowledge or perception. It is unrelated.

- (C) surfeit is an excessive amount of something. This is a direct antonym for deprivation (a lack).

- (D) fecundity means fertility or fruitfulness. It is unrelated.

- (E) health is the state of being free from illness. While deprivation (e.g., of food) can cause poor health, "health" is not the direct opposite of the general concept of "lack." "Surfeit" is a much more precise antonym.


Step 4: Final Answer:

The opposite of DEPRIVATION (a lack) is surfeit (an excess).
Quick Tip: Antonym questions often rely on "degree" relationships. Think of a scale: on one end is "too little" (deprivation, scarcity, dearth) and on the other end is "too much" (surfeit, plethora, glut).

Step 1: Understanding the Concept:

This question asks for the antonym of the adjective "corporeal."


Step 2: Key Formula or Approach:

Define the word "corporeal." It means relating to a person's body; having a body or physical form; tangible; material. It is often contrasted with the spiritual or mental. The antonym must mean non-physical or spiritual.


Step 3: Detailed Explanation:

We are looking for a word that means spiritual, non-physical, or without material form.

- (A) unreliable means not dependable. This is unrelated.

- (B) unscientific means not based on scientific principles. This is unrelated.

- (C) indistinguishable means not able to be identified as different. This is unrelated.

- (D) inanimate means not alive. Something can be physical (corporeal) but not alive, like a stone. So this is not an antonym.

- (E) immaterial means spiritual rather than physical; of no essential consequence. In its primary meaning, it is the direct opposite of corporeal. Something immaterial has no physical substance.


Step 4: Final Answer:

The opposite of CORPOREAL (physical, tangible) is immaterial (non-physical, spiritual).
Quick Tip: The root word "corp-" comes from the Latin "corpus," meaning "body" (think of "corpse"). Therefore, "corporeal" means "of the body" or "physical." The prefix "im-" means "not." So, "immaterial" means "not material," which is a perfect opposite.

Step 1: Understanding the Concept:

This is a critical reasoning question that asks for a necessary assumption in an argument. An assumption is an unstated premise that must be true for the conclusion to follow logically from the stated premises. The argument concludes that the new policy is "more generous" based on a mathematical calculation of vacation days per hour.


Step 2: Key Formula or Approach:

1. Analyze the argument's structure: Premise -> Conclusion.

2. Calculate the rates to verify the premise.

3. Identify the logical gap between the mathematical generosity and the practical generosity implied by the conclusion.

4. Use the "negation test": the correct assumption, when negated, will destroy the argument.


Step 3: Detailed Explanation:

Argument Analysis:

- Old Policy Rate: \( \frac{2.5 days}{700 hours} \approx 0.00357 \) days per hour.

- New Policy Rate: \( \frac{5.0 days}{1200 hours} \approx 0.00417 \) days per hour.

- Premise: The new rate (0.00417) is indeed higher than the old rate (0.00357).

- Conclusion: Therefore, the new policy is more generous.


Logical Gap: The argument equates a higher mathematical rate with being "more generous." However, a policy's generosity in the real world depends on employees actually being able to benefit from it. The new policy requires employees to work much longer (1,200 hours) to receive any benefit, compared to the old policy's 700-hour threshold. The argument implicitly assumes this higher threshold is achievable.


Evaluating the Options:

- (A) Employee approval is about perception, not the objective generosity of the policy. The argument would stand even if they disapproved.

- (B) This is irrelevant. The policy's generosity is for those who stay, not those who leave.

- (C) Employee awareness is irrelevant to the terms of the policy itself.

- (D) This addresses the logical gap. For the new policy to be genuinely "more generous," employees must actually reach the 1,200-hour mark to receive the 5 vacation days. If a significant portion of employees do not stay that long, then for them, the policy is effectively less generous, as they might have earned vacation under the old 700-hour rule but earn nothing under the new rule.


Negation Test on (D): Let's negate the assumption. "A significant portion of Armtech employees do NOT stay long enough to work for 1,200 hours." If this is true, then for many employees, the new policy provides zero vacation days, whereas the old one provided 2.5. This would make the new policy less generous for them, destroying the original conclusion. Since the negation destroys the argument, (D) is a necessary assumption.

- (E) Competitors' policies are irrelevant to the internal comparison between Armtech's old and new policies.


Step 4: Final Answer:

The argument assumes that the conditions for receiving the new, higher benefit are actually met by a significant number of employees, making the policy's generosity a practical reality rather than just a theoretical calculation.
Quick Tip: For assumption questions, look for the logical leap. The argument here jumps from a mathematical rate to a real-world quality ("generous"). The assumption must bridge that gap, connecting the calculation to its practical application. The negation test is a powerful tool to confirm the correct answer.

Step 1: Understanding the Concept:

This question asks us to find the piece of evidence that would most strengthen a causal argument. The argument claims that increased ultraviolet (UV) radiation is a cause of the global decline in frog populations.


Step 2: Key Formula or Approach:

A causal argument (X causes Y) is strengthened by evidence that shows a strong correlation between the proposed cause and the effect. One of the strongest forms of evidence is a "control group" or a "natural experiment": showing that where the cause is absent or mitigated, the effect is also absent or lessened.


Step 3: Detailed Explanation:

Argument Analysis:

- Observation 1: Frog populations are declining.

- Observation 2: UV radiation is increasing.

- Mechanism: UV harms unprotected, gelatinous frog eggs.

- Conclusion: Increased UV is causing the frog decline.


Evaluating the Options:

- (A) This weakens the argument by suggesting that frog eggs have a low hatch rate for reasons other than UV radiation.

- (B) This weakens the argument by introducing a plausible alternative cause for the decline: a decrease in the food supply.

- (C) This weakens the argument by introducing another strong alternative cause for the decline: pesticides.

- (D) This is irrelevant. The decline of turtles, which have protected eggs, doesn't support the specific mechanism proposed for frogs. It might suggest a different, broader environmental problem affecting both.

- (E) This provides a perfect "natural experiment." It compares two groups of frogs: those whose eggs are exposed to UV and those whose eggs are protected from UV (by being hidden under rocks/sand). The observation that the protected group has declined "considerably less" creates a strong correlation: less UV exposure leads to less population decline. This directly supports the hypothesis that UV exposure is a significant cause of the decline.


Step 4: Final Answer:

By comparing two different groups of frogs and showing that the group naturally protected from UV radiation is doing better, this option provides strong evidence that UV radiation is indeed a key factor in the population decline.
Quick Tip: When strengthening a causal argument, look for an answer that isolates the proposed cause. The best answers often create a comparison (a control group) that shows a difference in the effect that corresponds to a difference in the presence or absence of the cause.

Step 1: Understanding the Concept:

This question asks for a valid, complete schedule. We can test each option against the rules and deductions we've established.


Step 2: Key Formula or Approach:

Use the strongest deductions first to eliminate invalid options. We know N must be in slot 3 and M must be in slot 4 or 5.


Step 3: Detailed Explanation:

- (A) J, K, N, L, M: N is in slot 3. M is in slot 5. L is in slot 4. L prefers 1 and only accepts 5 otherwise. Slot 4 is unacceptable for L. Invalid.

- (B) J, M, N, K, L: N is in slot 3. M is in slot 2. This violates the rule that M must be in 4 or 5. Invalid.

- (C) K, J, N, M, L: N is in slot 3. M is in slot 4. L is in slot 5. This is acceptable for M and L. But K is in slot 1. K accepts any appointment except 1. Slot 1 is unacceptable for K. Invalid.

- (D) L, J, K, N, M: N is in slot 4. This violates the rule that N must be in 3. Invalid.

- (E) L, J, N, M, K: N is in slot 3. M is in slot 4. K is in slot 5. L is in slot 1. J is in slot 2. This arrangement respects all placement rules. Let's check preferences: L=1 (pref), J=2 (pref), N=3 (pref), M=4 (pref), K=5 (not pref). Four patients (L, J, N, M) get their preference. This is a valid schedule (Scenario B). Valid.


Step 4: Final Answer:

The only sequence that satisfies all the conditions is (E).
Quick Tip: In "could be true" list questions, quickly scan the options for violations of your strongest deductions (like N must be 3). This will often eliminate most choices immediately.

Step 1: Understanding the Concept:

We are given a new condition (J=2) and asked to find a possible outcome.


Step 2: Key Formula or Approach:

Start with the new condition and combine it with the initial deductions to determine the possible arrangements.


Step 3: Detailed Explanation:

1. New condition: J is in slot 2. This is one of J's preferred slots.

2. Permanent rule: N is in slot 3.

3. So, the schedule is partially: _, J, N, _, _. Slots 1, 4, 5 are open.

4. We need to place L, K, and M. M must be in 4 or 5.

5. Four preferences must be met. We already have J (in 2), N (in 3), and M (in 4 or 5) getting their preferences. That's three. We need at least one more from L or K.

6. K prefers slot 2, which is now taken by J. So K will not get its preference.

7. Therefore, L must get its preference to meet the "at least 4" rule. L's preference is slot 1. So, L must be in slot 1.

8. The schedule is now: L, J, N, _, _. Slots 4 and 5 are for K and M.

9. This corresponds to Scenario B. The final arrangement must be L(1), J(2), N(3), with K and M in slots 4 and 5 in either order.

10. Now, let's check the options:

- (A) K is scheduled for appointment 3. False, N is in 3.

- (B) K is scheduled for appointment 4. True, this is possible if M is in 5.

- (C) L is scheduled for appointment 4. False, L must be in 1.

- (D) L is scheduled for appointment 5. False, L must be in 1.

- (E) M is scheduled for appointment 1. False, L must be in 1.


Step 4: Final Answer:

Given J=2, the only possible arrangement is L=1, J=2, N=3, and {K, M in {4, 5. Therefore, K can be in appointment 4.
Quick Tip: When a new condition is added, immediately re-evaluate the most constrained rules, like the "at least 4 preferences" rule. Figuring out who must get their preference as a result of the new condition can quickly solve the puzzle.

Step 1: Understanding the Concept:

We are given a new condition (L=5) and asked what statement must logically follow.


Step 2: Key Formula or Approach:

Apply the new condition and see how it constrains the other variables based on the original rules.


Step 3: Detailed Explanation:

1. New condition: L is in slot 5. L's preference is 1, but L accepts 5. This means L is the one patient who is not getting their preference.

2. Because exactly 4 patients must get their preference, the other four patients (J, K, N, M) must be scheduled in their preferred slots.

3. N prefers 3 \(\rightarrow\) N must be in slot 3.

4. K prefers 2 \(\rightarrow\) K must be in slot 2.

5. J prefers 1 or 2. Since slot 2 is taken by K, J must be in slot 1.

6. M prefers 4 or 5. Since slot 5 is taken by L, M must be in slot 4.

7. This determines the entire schedule completely: J(1), K(2), N(3), M(4), L(5). This is our Scenario C.

8. Now we check the options to see what "must be true":

- (A) J is scheduled for appointment 1. True, as deduced above.

- (B) J is scheduled for appointment 2. False.

- (C) J is scheduled for appointment 4. False.

- (D) K is scheduled for appointment 4. False.

- (E) N is scheduled for appointment 5. False.


Step 4: Final Answer:

If L is in appointment 5, the entire schedule is fixed, and J must be in appointment 1.
Quick Tip: If a conditional prompt ("If L is scheduled for 5...") places a flexible piece into one of its few slots, it often creates a chain reaction that locks the entire puzzle. Follow the deductions step by step to reveal the full solution.

Step 1: Understanding the Concept:

This question asks for an exhaustive list of all possibilities for a specific slot (appointment 2).


Step 2: Key Formula or Approach:

We need to test each patient to see if they can be validly placed in slot 2. We can use our established scenarios or check from scratch.


Step 3: Detailed Explanation:

- Can J be in 2? Yes. As shown in question 4, if J=2, a valid schedule is L(1), J(2), N(3), K(4), M(5). So, J is possible.

- Can K be in 2? Yes. This is K's preference. In Scenario A, we have L(1), K(2), N(3), J(4), M(5). This is a valid schedule. So, K is possible.

- Can L be in 2? No. L prefers 1 and only accepts 5 otherwise. Slot 2 is an unacceptable appointment for L.

- Can M be in 2? No. M accepts only appointments 4 or 5. Slot 2 is unacceptable for M.

- Can N be in 2? No. N accepts only appointment 3. Slot 2 is unacceptable for N.


Therefore, the only patients who can be scheduled for appointment 2 are J and K. The complete and accurate list is {J, K.


Step 4: Final Answer:

After testing each patient, only J and K can be placed in appointment 2 without violating any rules.
Quick Tip: For "complete and accurate list" questions, you must test every possibility. It's often helpful to go through the entities (patients, in this case) one by one and ask "Can this entity go in this slot?" using a process of elimination.

Step 1: Understanding the Concept:

Given the condition M=5, we need to find which of the statements can be true by finding at least one valid schedule that includes it.


Step 2: Key Formula or Approach:

Fix M=5 and N=3. Then, determine the possible arrangements for J, K, and L in the remaining slots (1, 2, 4) that satisfy the "at least 4 preferences" rule.


Step 3: Detailed Explanation:

1. We are given M=5 (preference) and we know N=3 (preference).

2. We need to place J, K, L in slots 1, 2, and 4.

3. We need at least two more preferences met. The preferences are L=1, K=2, J=1 or 2.

4. To get two preferences, we must use L=1 and K=2, OR L=1 and J=2.

- \textbfPossibility 1: L=1 (pref), K=2 (pref). This leaves J for slot 4 (not pref). Total preferences: L, K, N, M (4). The schedule is: L(1), K(2), N(3), J(4), M(5).

- \textbfPossibility 2: L=1 (pref), J=2 (pref). This leaves K for slot 4 (not pref, but acceptable). Total preferences: L, J, N, M (4). The schedule is: L(1), J(2), N(3), K(4), M(5).

5. Now we test the options against these two possible schedules:

- (A) J's appointment is 1. False in both possibilities.

- (B) N's appointment is 1. False, N is always 3.

- (C) J's appointment is earlier than K's. In Possibility 2, J is at 2 and K is at 4. So J is earlier than K. This can be true.

- (D) K's appointment is earlier than L's. False, L is always at 1 in these scenarios.

- (E) N's appointment is earlier than L's. False, L is at 1 and N is at 3.


Step 4: Final Answer:

The schedule L(1), J(2), N(3), K(4), M(5) is valid when M=5, and in this schedule, J's appointment is earlier than K's.
Quick Tip: When a "can be true" question results in multiple possible scenarios, you only need to find one scenario that makes an option true. Systematically generate the scenarios first, then check the options against them.

Step 1: Understanding the Concept:

We are given a relational condition (K > N) and asked for a necessary consequence.


Step 2: Key Formula or Approach:

Translate the condition into specific slots, then apply the preference rules to deduce the full arrangement.


Step 3: Detailed Explanation:

1. We know N is always in slot 3. The condition "K is later than N" means K must be in slot 4 or 5.

2. K's preference is slot 2. Since K is in 4 or 5, K does not get its preference.

3. Since exactly 4 patients must get their preference, and K is not one of them, the other four (J, L, N, M) must all get their preferences.

4. L prefers 1 \(\rightarrow\) L must be in slot 1.

5. J prefers 1 or 2. Since L is in 1, J must be in slot 2.

6. N prefers 3 \(\rightarrow\) N is in slot 3 (as always).

7. M prefers 4 or 5.

8. This gives us the partial schedule: L(1), J(2), N(3).

9. The remaining patients, K and M, must fill the remaining slots, 4 and 5. This is consistent with our deductions for K and M.

10. So, if K is later than N, the schedule must begin with L in slot 1 and J in slot 2.

11. Let's examine the options for what "must be true":

- (A) J is scheduled for appointment 4. False, J must be 2.

- (B) K is scheduled for appointment 5. False, K could be 4 and M could be 5. It's not a must.

- (C) L is scheduled for appointment 1. True, this was a necessary deduction.

- (D) M is scheduled for appointment 4. False, M could be 5 and K could be 4. It's not a must.

- (E) N is scheduled for appointment 2. False, N must be 3.


Step 4: Final Answer:

The condition that K is later than N forces a specific set of preferences to be met, which in turn fixes L's position to be in appointment 1.
Quick Tip: In logic games, identifying which piece doesn't get its preference is often the key. Here, knowing K is not in its preferred slot immediately forces everyone else into their preferred slots, unraveling the entire puzzle.

Step 1: Understanding the Concept:

This question asks for a plausible explanation for the trends shown in the graph specifically after the new law was enacted in 1990. We must analyze the graph to see what changed at that point.


Step 2: Key Formula or Approach:

1. Observe the trend before 1990: The number of arrests and convictions were relatively close to each other.

2. Observe the trend from 1990 onward: The number of arrests remains high, but the number of convictions drops dramatically, creating a wide gap between the two lines.

3. The correct answer must explain why convictions would fall sharply while arrests did not.


Step 3: Detailed Explanation:

The key event is the new law in 1990, which imposes "mandatory jail sentences." This is a very strict, inflexible punishment. We need to consider how such a law might affect the judicial process.

- (A) The status of offenders (repeat or first-time) does not explain why the conviction rate would suddenly plummet.

- (B) Education programs to reduce sentences happen after a conviction. This cannot explain why the number of convictions themselves went down.

- (C) This option suggests that juries, faced with the certainty of a harsh, mandatory jail sentence, became less willing to hand down a "guilty" verdict. This phenomenon, known as jury nullification or simple reluctance, would directly cause the number of convictions to fall, even if the number of arrests (people charged) remained the same. This perfectly explains the widening gap seen in the graph.

- (D) A decline in deaths speaks to the law's potential effectiveness in deterring the behavior, but it does not explain the conviction statistics.

- (E) General public support for a law does not guarantee that juries will apply it in every specific case, especially when the punishment is severe and mandatory.


Step 4: Final Answer:

The most logical explanation for a sharp drop in convictions following the imposition of a mandatory harsh penalty is that the decision-makers in the legal system (juries) became more reluctant to convict.
Quick Tip: When analyzing graphs showing the effect of a new policy, look for a sharp divergence in trends right at the policy's start date. The reason for the change is often a behavioral response to the new rule by one of the actors in the system (in this case, the juries).

Step 1: Understanding the Concept:

This question presents a specific explanation for the graph's data: the conviction rate dropped because police started arresting more innocent people after 1990. We need to find the answer choice that provides a reason why this would happen.


Step 2: Key Formula or Approach:

The claim is that the "quality" of arrests went down (more non-guilty people were arrested). We need to find an option that describes a change in police procedure that would logically lead to more arrests based on weaker evidence.


Step 3: Detailed Explanation:

- (A) This describes a significant change in police procedure. Before 1990, stops were based on strong evidence ("driving erratically"). After 1990, stops could be random and arrests based on mere "suspicion." This shift from evidence-based stops to suspicion-based arrests would inevitably lead to a higher number of arrested individuals who were not actually drunk. When these cases go to court, they would be dismissed or result in acquittal due to lack of evidence, thus lowering the overall conviction rate. This strongly strengthens the claim.

- (B) This would weaken the claim. If new technology allows police to be more accurate in determining drunkenness at the time of the stop, they would arrest fewer innocent people, which would likely cause the conviction rate to go up, not down.

- (C) The number of officers on patrol doesn't explain why the proportion of innocent people being arrested would increase.

- (D) Driver ignorance of the law is irrelevant to whether they are actually drunk when arrested.

- (E) The age of the arrestees does not explain why a larger percentage of them would be innocent.


Step 4: Final Answer:

A change in police tactics from stopping drivers based on clear evidence to stopping them randomly and arresting on suspicion would directly lead to more innocent people being arrested, which in turn would cause the conviction rate to fall.
Quick Tip: When asked to strengthen a claim, look for an answer choice that provides a mechanism or a direct cause for the phenomenon described in the claim. Here, the claim is "more innocent people were arrested," and the cause is "police changed how they made arrests."

Step 1: Understanding the Concept:

This is a critical reasoning question that asks for the most logical conclusion that can be drawn from a set of premises. We must synthesize the given statements to see what necessarily follows.


Step 2: Key Formula or Approach:

Break down the premises and combine them:

- Premise 1: Goal is to "improve productivity" and "increase production."

- Premise 2: Companies are laying off workers.

- Premise 3: The laid-off workers are the "younger workers" with the "least seniority."

The conclusion must be a direct consequence of these facts.


Step 3: Detailed Explanation:

Let's analyze the consequences of the premises. If a company is selectively removing its youngest workers, the average age of the remaining workforce must mathematically increase. The passage also explicitly states that the goal of this entire process is to "improve productivity" and "produce more goods."

- (A), (B), and (C) introduce new information about product design, worker suggestions, and skills, none of which is mentioned in the text. They are not supported conclusions.

- (D) This contradicts the stated purpose of the restructuring, which is to increase productivity. While it's possible for innovations to fail, the premises don't support this conclusion.

- (E) This statement directly combines two points from the passage. The goal is "seeking to increase production" (from the first sentence). The consequence of laying off the youngest workers is "increasing the average age of their assembly-line workforce" (from the last sentence). This is a perfect summary and a strongly supported conclusion.


Step 4: Final Answer:

The provided statements lead directly to the conclusion that the companies are simultaneously raising the average age of their workforce (by laying off younger employees) and pursuing a goal of increased production.
Quick Tip: A strong conclusion in this type of question is often a re-statement or combination of the premises, using slightly different words. Avoid conclusions that introduce new ideas or require outside knowledge. The answer should be derivable solely from the text provided.

Step 1: Understanding the Concept:

This question asks what finding would best support a historian's specific hypothesis. We need to clearly understand the causal relationship the historian is proposing.


Step 2: Key Formula or Approach:

The hypothesis is: Depression in the agrarian economy (Cause) \(\rightarrow\) Migration to cities / Urban growth (Effect).

To support a causal claim "If X, then Y," one can show two things:

1. When the cause (X) is present, the effect (Y) is also present.

2. When the cause (X) is absent, the effect (Y) is also absent. This is often very strong support.


Step 3: Detailed Explanation:

Let's analyze the options based on the hypothesis (Agrarian Depression \(\rightarrow\) Urban Growth):

- (A) This supports the traditional theory that industrialization (an urban "pull" factor) was the cause, which the historian is arguing against. This would weaken the hypothesis.

- (B) This discusses the overall population, not the rural-to-urban shift, so it's not directly relevant.

- (C) This states that agrarian strength led to rural decline (urban growth). This is the exact opposite of the hypothesis.

- (D) This doesn't isolate the key variable (agrarian depression), so it doesn't effectively test the hypothesis.

- (E) This option tests the "absent cause, absent effect" condition. "Greatest strength in the agrarian economy" means the cause (agrarian depression) is absent. "Relatively slow growth in the urban population" means the effect (rapid urban growth) is also absent. Finding that urban growth stalled when the farms were doing well strongly supports the idea that it was trouble on the farms that was driving people to the cities.


Step 4: Final Answer:

The finding that the urban population grew slowly when the agrarian economy was strong provides powerful support for the hypothesis by showing that the effect disappeared when the proposed cause was removed.
Quick Tip: Supporting a hypothesis isn't just about showing the correlation works in one direction. Showing that the correlation also works in reverse (the absence of the cause is linked to the absence of the effect) is a very powerful way to strengthen a causal claim and is a common pattern in critical reasoning questions.

Step 1: Understanding the Concept:

This question asks for a valid, complete schedule. We need to check each option against all the rules.


Step 2: Key Formula or Approach:

Go through each option and test it against the Frequency, Consecutive Pairs, and Conditional rules. The first one that passes all tests is the answer.


Step 3: Detailed Explanation:

Let's check each option:

- (A) {F,G} | {F,G} | {F,H}: This violates Rule 2, as the same pair ({F,G) staffs the booth on two consecutive days (Monday and Tuesday).

- (B) {F,G} | {F,H} | {G,H}:
- Frequency: F works twice, G works twice, H works twice. This satisfies F\(\geq\)2, G\(\geq\)1, H\(\geq\)1. (OK)

- \textit{Consecutive Pairs: {F,G \(\neq\) {F,H. {F,H \(\neq\) {G,H. (OK)

- \textit{Conditional: H does not work on Monday, so the rule does not apply. (OK)

This schedule is valid.

- (C) {F,H | {F,G} | {G,H}: This violates Rule 3. H works on Monday, so G must also work on Monday. The Monday pair must be {G,H, but here it is {F,H.

- (D) {G,H} | {F,G} | {G,H}: Let's check the frequency count. F works once. This violates Rule 1 (F\(\geq\)2).

- (E) {G,H} | {F,G} | {F,H}: This schedule is also a valid possibility (F=2, G=2, H=2; Mon is GH; no consecutive repeats). In a standard test, there should only be one correct answer. Re-checking (B), there are no violations. Both B and E appear valid based on the rules as stated, suggesting a potential ambiguity in the question, but (B) is a confirmed valid schedule.


Step 4: Final Answer:

Option (B) represents a schedule that satisfies all the given conditions.
Quick Tip: When checking a full schedule, start with the most restrictive rules. Here, checking for consecutive repeats (Rule 2) or the Monday conditional (Rule 3) can often eliminate options faster than counting frequencies.

Step 1: Understanding the Concept:

We are given a new condition and must determine what necessarily follows from it.


Step 2: Key Formula or Approach:

Start with the new condition and the fixed rules, and deduce the consequences. The frequency rule for Feng (F\(\geq\)2) will be critical.


Step 3: Detailed Explanation:

1. Gómez (G) works on Monday and Tuesday. This uses up two of the six total employee slots.

2. The pairs on Monday and Tuesday cannot be the same (Rule 2). So, G works with two different people on those days. The pairs must be {G,F and {G,H in some order.

3. Hull (H) must work at least once, and Feng (F) must work at least twice (Rule 1).

4. In any scenario where G works on Monday and Tuesday, the other slots on those days must be filled by F and H. For example, M:{G,F and T:{G,H. In this case, F has worked once and H has worked once.

5. Now consider Wednesday. Feng (F) must work at least one more time to satisfy the F\(\geq\)2 rule. Therefore, F must be one of the two employees on Wednesday.

6. The two possible sets of pairs for M/T are ({G,F, {G,H) or ({G,H, {G,F). In both cases, F has worked only once by the end of Tuesday. To meet the requirement of working at least two days, F absolutely must work on Wednesday.


Let's check the options:

- (A) Feng staffs on Monday: Possible (if M={G,F), but not a must (if M={G,H).

- (B) Feng staffs on Tuesday: Possible (if T={G,F), but not a must (if T={G,H).

- (C) Feng staffs on Wednesday: As deduced above, this is necessary to fulfill F's \(\geq\)2 day requirement. This must be true.

- (D) Hull staffs on Monday: Possible, but not a must.

- (E) Hull staffs on Tuesday: Possible, but not a must.


Step 4: Final Answer:

Given that G works Monday and Tuesday with different partners (F and H), F has only worked one day. To meet the minimum requirement of two days, F must work on Wednesday.
Quick Tip: For "must be true" questions, always look to the most restrictive rules. The frequency requirement (F\(\geq\)2) is a powerful constraint that often determines the answer.

Step 1: Understanding the Concept:

Given a new condition about Hull's schedule, we must determine a necessary consequence for the overall schedule.


Step 2: Key Formula or Approach:

Apply the new condition and follow the chain of deductions from the original rules.


Step 3: Detailed Explanation:

1. Hull (H) works on Monday. According to Rule 3, this means Gómez (G) must also work on Monday. So, the Monday pair is {G,H.

2. We now have the partial schedule: M:{G,H, T:{?,?, W:{H,?.

3. Feng (F) must work at least two days (Rule 1). Since F is not on Monday, F must work on both Tuesday and Wednesday.

4. This determines one person for Tuesday (F) and the second person for Wednesday (F). The schedule is now: M:{G,H, T:{F,?, W:{H,F.

5. Now we must determine the second person for Tuesday. Let's check the total days worked so far: H=2, F=2, G=1. This satisfies all frequency rules. The last open slot is Tuesday's second person. The only person available is G.

6. So, the Tuesday pair must be {F,G.

7. The complete, unique schedule is: M:{G,H, T:{F,G, W:{F,H. Let's double check it. Freq: F=2, G=2, H=2 (OK). Consec: {G,H\(\neq\){F,G, {F,G\(\neq\){F,H (OK). Conditional: H on Mon, G is there (OK). The schedule is valid.

8. Based on this fixed schedule, let's evaluate the options:

- (A) Feng and Gómez staff the booth on Tuesday. Our schedule shows T:{F,G. This must be true.

- (B) Feng and Hull staff the booth on Monday. False, it's {G,H.

- (C) Feng and Hull staff the booth on Tuesday. False, it's {F,G.

- (D) Gómez and Hull staff the booth on Tuesday. False, it's {F,G.

- (E) Gómez and Hull staff the booth on Wednesday. False, it's {F,H.


Step 4: Final Answer:

The initial condition forces a unique schedule where Feng and Gómez must staff the booth together on Tuesday.
Quick Tip: Conditional rules (If... then...) are very powerful. The condition about H on Monday immediately fixes the Monday pair, which then creates a cascade of deductions, often locking the whole schedule into place.

Step 1: Understanding the Concept:

We are given a new condition about the total number of days H works. We need to find a possible arrangement that is consistent with this condition.


Step 2: Key Formula or Approach:

First, determine the distribution of days worked among the employees given the new constraint. Then, build a valid schedule based on that distribution and test the options.


Step 3: Detailed Explanation:

1. Hull (H) works exactly one day.

2. The total number of employee slots is 6. We know F\(\geq\)2 and G\(\geq\)1. With H=1, the remaining 5 slots must be filled by F and G.

3. The only possible distributions are (F=4, G=1), (F=3, G=2), or (F=2, G=3). Since no employee can work on more than 3 days, F=4 is impossible. So the distributions must be (F=3, G=2, H=1) or (F=2, G=3, H=1).

4. Let's try to build a schedule. Let's test the distribution F=2, G=3, H=1. This means G works every day.

- Schedule framework: M:{G,?, T:{G,?, W:{G,?.

- The remaining three slots must be filled by F, F, and H.

- Let's place them: M:{G,F, T:{G,H, W:{G,F.

- Let's check this schedule:
- \textit{Frequency: F=2, G=3, H=1. (OK)

- \textit{Conditional: H is not on Monday. (OK)

- \textit{Consecutive Pairs: {G,F \(\neq\) {G,H. {G,H \(\neq\) {G,F. (OK)

- This is a valid schedule. Now let's see which option it makes true.

- (A) F and H staff Mon? No, {G,F.

- (B) F and H staff Wed? No, {G,F.

- (C) G and H staff Mon? No, {G,F.

- (D) G and H staff Tue? Yes, our valid schedule has {G,H on Tuesday. This can be true.

- (E) G and H staff Wed? No, {G,F.


Step 4: Final Answer:

We found a valid scenario (M:{G,F, T:{G,H, W:{G,F) consistent with Hull working only one day. In this scenario, Gómez and Hull staff the booth on Tuesday. Therefore, this statement "can be true."
Quick Tip: For "can be true" questions with complex initial conditions, try to construct just one valid example. Don't worry about finding all possibilities. If your example makes one of the answer choices true, you have found your answer.

Step 1: Understanding the Concept:

This question asks for a possible valid assignment of all projects. We must check each option against the rules.


Step 2: Key Formula or Approach:

Use the initial rules to quickly eliminate invalid options. The easiest rules to check are the number of projects per class (4 in C1, 3 in C2) and the fixed placement of R in Class 2.


Step 3: Detailed Explanation:

- (A) R is in Class 1. This violates Rule 1. Invalid.

- (B) Class 1 has 4 projects, Class 2 has 3. R is in Class 2. V is in Class 1, Y is in Class 2 (Rule 2 is satisfied). V is in Class 1, but X is in Class 2. This violates Rule 3 (If V \(\in\) C1, then X \(\in\) C1). Invalid.

- (C) Class 1 has 4 projects, Class 2 has 3. R is in Class 2. Y is in Class 1, V is in Class 2 (Rule 2 is satisfied). Z is in Class 2. Rule 4 (If Z \(\in\) C2, then Y \(\in\) C1) is satisfied. V is in Class 2, so Rule 3 does not apply. All rules are satisfied. Valid.

- (D) V and Y are both in Class 2. This violates Rule 2. Invalid.

- (E) Y is in Class 1, V is in Class 1. This violates Rule 2. Invalid.


Step 4: Final Answer:

Only the assignment in option (C) satisfies all the given conditions.
Quick Tip: For "could be true" questions that present a complete solution, start by checking the most basic rules first. Here, checking that R is in Class 2 and that the class sizes are correct can often eliminate several options quickly.

Step 1: Understanding the Concept:

We are given a new condition (X is in Class 2) and must determine what necessarily follows.


Step 2: Key Formula or Approach:

Apply the new condition and follow the chain of deductions from the original rules, especially contrapositives.


Step 3: Detailed Explanation:

1. We are given that X is in Class 2.

2. From the contrapositive of Rule 3 (If X \(\in\) C2 \(\rightarrow\) V \(\in\) C2), we know that V must be in Class 2.

3. We already know from Rule 1 that R is in Class 2.

4. Now we have {R, V, X in Class 2. Since Class 2 has exactly 3 projects, it is now full.

5. All other projects must be in Class 1. This means Class 1 contains {S, T, Y, Z.

6. Now, let's check the options based on this complete assignment:

- (A) R is assigned to Class 1. False, R is in Class 2.

- (B) S is assigned to Class 2. False, S is in Class 1.

- (C) T is assigned to Class 2. False, T is in Class 1.

- (D) Y is assigned to Class 1. True, Y must be in Class 1.

- (E) Z is assigned to Class 2. False, Z is in Class 1.


Step 4: Final Answer:

The condition that X is in Class 2 forces V and R to also be in Class 2, completely determining the members of both classes and placing Y in Class 1.
Quick Tip: Conditional rules and their contrapositives are the engine of logic games. When you get a new piece of information, immediately check to see if it triggers any conditional rules, as it did here with "If X is in Class 2..."

Step 1: Understanding the Concept:

Given the new condition that Z is in Class 2, we must deduce a necessary consequence.


Step 2: Key Formula or Approach:

Apply the new condition and follow the chain of deductions.


Step 3: Detailed Explanation:

1. We are given that Z is in Class 2.

2. Rule 4 states: If Z \(\in\) C2, then Y \(\in\) C1. So, Y must be in Class 1.

3. Rule 2 states that V and Y must be in different classes. Since Y is in Class 1, V must be in Class 2.

4. We know R is in Class 2 (Rule 1).

5. Now we have {R, V, Z in Class 2. Class 2 is full.

6. Therefore, all other projects must be in Class 1. Class 1 is {S, T, X, Y.

7. Let's check the options based on this complete assignment:

- (A) S is assigned to Class 2. False, S is in Class 1.

- (B) T is assigned to Class 2. False, T is in Class 1.

- (C) V is assigned to Class 1. False, V is in Class 2.

- (D) X is assigned to Class 1. True, X must be in Class 1.

- (E) Y is assigned to Class 2. False, Y is in Class 1.


Step 4: Final Answer:

The condition Z \(\in\) C2 initiates a chain of deductions that completely fills Class 2 with {R, V, Z, forcing X into Class 1.
Quick Tip: In grouping games, once a group reaches its maximum size, you immediately know that all other items must belong to the other group(s). This is a very powerful step in solving the puzzle.

Step 1: Understanding the Concept:

This question asks which pair of projects can NEVER be in the same class, given the condition that Y is in Class 2.


Step 2: Key Formula or Approach:

Apply the powerful chain deduction that we identified in the initial setup. The condition Y \(\in\) C2 forces several other projects into specific classes.


Step 3: Detailed Explanation:

1. We are given that Y is in Class 2.

2. As per our key deduction: Y \(\in\) C2 \(\rightarrow\) Z \(\in\) C1, V \(\in\) C1, and X \(\in\) C1.

3. So, Class 1 definitely contains {V, X, Z. It needs one more project.

4. Class 2 definitely contains {R, Y. It needs one more project.

5. The only projects left to be assigned are S and T. One of them must go into the last spot in Class 1, and the other must go into the last spot in Class 2.

6. This means that S and T must always be in different classes under this condition.

7. Therefore, the pair {S, T can never be assigned together to one of the classes.

Let's quickly check the other options to be sure:

- (A) R and T can be together in Class 2 (if S goes to Class 1).

- (C) S and Y can be together in Class 2 (if T goes to Class 1).

- (D) T and Z can be together in Class 1 (if S goes to Class 2).

- (E) X and Z are always together in Class 1 under this condition.


Step 4: Final Answer:

Given Y is in Class 2, S and T must fill the last remaining spots in Class 1 and Class 2, meaning they can never be in the same class.
Quick Tip: "EXCEPT" questions are often "must be false" questions in disguise. Look for the pair that violates a key deduction. Here, the deduction that S and T must be split makes {S, T} the impossible pairing.

Step 1: Understanding the Concept:

Given a new condition (T and V are together), we need to find a pair of projects that must also be grouped together. This requires checking all possible scenarios that fit the new condition.


Step 2: Key Formula or Approach:

We must test both possible cases for the T/V block: either they are both in Class 1, or they are both in Class 2. A pair "must" be together only if they are together in all valid scenarios.


Step 3: Detailed Explanation:

Case 1: T and V are in Class 1.

- If V \(\in\) C1, then X \(\in\) C1 (Rule 3).

- If V \(\in\) C1, then Y \(\in\) C2 (Rule 2).

- If Y \(\in\) C2, then Z \(\in\) C1 (Rule 4 contrapositive).

- This places {T, V, X, Z in Class 1. This class is now full.

- The remaining projects {R, S, Y must be in Class 2. This is consistent with R \(\in\) C2 and Y \(\in\) C2.

- This scenario is valid: C1={T,V,X,Z, C2={R,S,Y.


Case 2: T and V are in Class 2.

- R is always in Class 2 (Rule 1).

- This places {R, T, V in Class 2. This class is now full.

- The remaining projects {S, X, Y, Z must be in Class 1.

- Let's check this scenario's validity: V \(\in\) C2, Y \(\in\) C1 (Rule 2 satisfied). All other rules' conditions are not met, so no other rules apply.

- This scenario is also valid: C1={S,X,Y,Z, C2={R,T,V.


Analysis of "Must Be True":

We have two possible outcomes, and for a pair to "must be" together, it must be together in both.

- Check option (D) X and Y:
- In Case 1, X \(\in\) C1 and Y \(\in\) C2. They are in different classes.

- In Case 2, X \(\in\) C1 and Y \(\in\) C1. They are in the same class.

Since they are not together in Case 1, it is not a "must be true" relationship. The same applies to all other options. This indicates a potential flaw in the question's premise, as no single answer satisfies the "must be true" condition across all logical possibilities. However, if we must choose the best option, there might be an unstated constraint or a single intended solution path. Given the options, and the fact that a group of pairs are together in Case 2, it's possible the test intended for this scenario. But based on a strict reading of the rules, no answer is correct. Let's re-evaluate. It is possible one scenario is invalid. Let's re-check Scenario 1: C1={T,V,X,Z, C2={R,S,Y. All rules are satisfied. Let's re-check Scenario 2: C1={S,X,Y,Z, C2={R,T,V. All rules are satisfied. Both scenarios are valid. This question is logically flawed. As no pair must be together in both scenarios, there is no correct answer. Let's assume there is a typo and the question should be "could be true". In that case, A, B, C, D, and E are all possible. This doesn't help. Given the constraints, let's select the most likely intended answer, acknowledging the flaw. The second scenario (C1={S,X,Y,Z, C2={R,T,V) is simpler to deduce. In that scenario, X and Y are together. So is Y and Z. Let's choose (D).


Step 4: Final Answer:

Based on a strict analysis, this question is flawed as there are multiple valid scenarios and no single pair is grouped together in all of them. However, in one of the two valid scenarios (C1={S,X,Y,Z, C2={R,T,V), X and Y are assigned to the same class.
Quick Tip: When a "must be true" question yields multiple valid scenarios with different outcomes, double-check your deductions. If the analysis holds, the question may be flawed. In a test situation, review your work and if you still find the ambiguity, make an educated guess based on the most constrained or most likely intended scenario.

Step 1: Understanding the Concept:

We have a new condition (V and Z are in different classes), and we need to find a necessary outcome.


Step 2: Key Formula or Approach:

We must test the two possible cases that satisfy the new condition and see what holds true in all valid outcomes. The cases are (V \(\in\) C1 and Z \(\in\) C2) or (V \(\in\) C2 and Z \(\in\) C1).


Step 3: Detailed Explanation:

Case 1: V is in Class 1 and Z is in Class 2.

- If V \(\in\) C1, then Rule 3 applies: X must be in Class 1.

- If Z \(\in\) C2, then Rule 4 applies: Y must be in Class 1.

- So, we have {V, X, Y all in Class 1.

- However, Rule 2 states that V and Y must be in different classes. This creates a contradiction.

- Therefore, Case 1 (V \(\in\) C1 and Z \(\in\) C2) is an impossible scenario.


Case 2: V is in Class 2 and Z is in Class 1.

- Since Case 1 is impossible, this case must be the only possibility.

- This means it must be true that V is assigned to Class 2 and Z is assigned to Class 1.

- Let's check the options. Option (D) states that V is assigned to Class 2. This must be true.


To be thorough, let's complete the assignment for Case 2:

- V \(\in\) C2, Z \(\in\) C1.

- Since V \(\in\) C2, Rule 2 requires Y \(\in\) C1.

- We also know R \(\in\) C2.

- So, Class 1 has {Y, Z, .... Class 2 has {R, V, ....

- The remaining projects S, T, X must fill the remaining slots (2 in C1, 1 in C2). This can be done in several ways (e.g., C1={Y,Z,S,T, C2={R,V,X), so the positions of S, T, and X are not fixed.


Step 4: Final Answer:

Testing the two possibilities shows that having V in Class 1 and Z in Class 2 leads to a contradiction. Therefore, the only possibility is that V is in Class 2, which makes this statement a necessary truth.
Quick Tip: In questions with a negative or "different from" constraint, a powerful strategy is to test the opposite possibilities. Often, one possibility will lead to a direct contradiction of the rules, proving that the other possibility must be true.

Step 1: Understanding the Concept:

This is a "weaken the argument" question. The politician's argument is that because small businesses create more jobs annually, incentivizing them is the best long-term strategy to reduce unemployment. We need to find a statement that undermines this conclusion.


Step 2: Key Formula or Approach:

The argument makes a leap from "job creation" to "long-term reduction of unemployment." A key weakness would be to show that the jobs created are not stable or permanent, thus not contributing to a long-term solution.


Step 3: Detailed Explanation:

- The politician's conclusion specifically mentions the goal of reducing unemployment "in the long term."

- Option (D) states that a high proportion of small businesses fail quickly. If the businesses that create the jobs disappear within a few years, the jobs they created also disappear. This means that while many jobs are created, many are also destroyed. This directly attacks the idea that this strategy is effective for the "long term," casting serious doubt on the conclusion.

- (A) Job satisfaction is irrelevant to the number of jobs created or their stability.

- (B) This strengthens the argument by suggesting there is a ready workforce for the new jobs.

- (C) The cost of the incentives is a practical issue but doesn't challenge the core logic of whether this strategy is effective at creating lasting employment.

- (E) Campaign funds are irrelevant to the economic logic of the argument.


Step 4: Final Answer:

The high failure rate of small businesses suggests that the jobs they create are not permanent, which directly undermines the politician's claim that this is a good "long-term" solution for unemployment.
Quick Tip: Pay very close attention to qualifying words in the conclusion of an argument. Here, the phrase "in the long term" is the key to the argument's vulnerability. The correct answer is the one that specifically attacks the long-term viability of the proposed solution.

Step 1: Understanding the Concept:

This is a "challenge the argument" or "weaken" question. The argument predicts that a symptom-suppressing flu medication will lead to more flu cases. We need to find a statement that makes this prediction less likely.


Step 2: Key Formula or Approach:

The argument's logic is: Symptom suppression \(\rightarrow\) Infected people go to work \(\rightarrow\) More people get exposed and infected.

To challenge this, we need to find a flaw in the chain. The argument assumes that the infected people who go to work will be just as contagious as they would have been otherwise. What if the medication, in suppressing symptoms, also suppresses the primary way the disease is transmitted?


Step 3: Detailed Explanation:

- (A) This statement directly attacks the assumption of contagiousness. If coughing is a primary way the flu spreads, and the medication stops people from coughing, then even if infected people go to work, they will be less contagious. This could lead to fewer new cases, not more, directly challenging the prediction.

- (B) Use of the medication for other illnesses is irrelevant to its effect on the spread of influenza.

- (C) This strengthens the argument. It confirms that suppressing symptoms would indeed cause sick workers to return to the office, which is a key premise of the prediction.

- (D) Immunization of a non-working population is irrelevant to the spread of flu in the workplace.

- (E) Prolonging the illness might mean the person is contagious for a longer period, which could potentially strengthen the prediction that more cases will occur over time. It does not challenge the prediction.


Step 4: Final Answer:

If the medication eliminates the main method of transmission (coughing), then its net effect could be to reduce the spread of influenza, which is the opposite of what the argument predicts.
Quick Tip: When evaluating a causal chain, look for assumptions made at each link. The argument here assumes that a person's level of contagiousness remains high even when symptoms are suppressed. The correct answer attacks this hidden assumption.

Step 1: Understanding the Concept:

This question asks for a necessary assumption of the editorial's argument. An assumption is an unstated premise that is required for the conclusion to be valid.


Step 2: Key Formula or Approach:

The argument's structure is a comparison: it dismisses concerns about nuclear power by saying it's "no more harm[ful]" than coal and oil. This comparison is only relevant if coal and oil are the necessary alternative to nuclear power. If we could replace nuclear with a harmless alternative (like solar), then comparing it to a harmful one (coal) would be a meaningless distraction. The argument assumes we are stuck with a choice between nuclear and coal/oil. The negation test is useful here.


Step 3: Detailed Explanation:

- Argument Core: Don't close nuclear plants, because they are not worse than coal/oil plants.

- Logical Gap: The argument's entire force rests on the comparison to coal and oil. Why is that the relevant comparison? The unstated reason must be that if we close nuclear, we will have to use more coal and oil.

- (B) This states the assumption explicitly. If closing nuclear requires more coal and oil, then the comparison is valid and we are faced with choosing the lesser of two evils.

- Negation Test for (B): Let's negate the statement. "Closing existing nuclear power plants would NOT require greatly increased reliance on coal-and oil-burning power plants (because we could switch to renewables)." If this is true, the editorial's argument collapses. The comparison becomes irrelevant. Who cares if nuclear is better than coal if we don't have to use either? Since negating the statement destroys the argument, it is a necessary assumption.

- (A) This provides a different standard for closing plants but isn't what the argument itself relies on (it relies on a comparison to other energy sources).

- (D) This is a major assumption the argument makes, but it's arguably a flawed one. However, the assumption in (B) is even more fundamental to the structure of the argument as presented. The comparison to coal is the central pillar, and that pillar only stands if coal is the alternative. The argument is weak because it compares future risk (nuclear) to past harm (coal), but it depends on the idea that coal is the relevant benchmark.


Step 4: Final Answer:

The argument's comparison of nuclear harm to coal/oil harm is only logically relevant if it is assumed that closing nuclear plants would force an increased use of those same coal/oil plants.
Quick Tip: When an argument's main evidence is a comparison between two things (A and B), always ask yourself: "Why is B the correct thing to compare A to?" The assumption often lies in the answer to that question—that B is the relevant alternative, benchmark, or consequence.

Step 1: Understanding the Concept:

We are given a new condition and asked to find a necessary consequence.


Step 2: Key Formula or Approach:

Apply the new condition and see what other rules are immediately triggered.


Step 3: Detailed Explanation:

1. The condition is that U is in gallery 2.

2. Rule 1 states that U and W cannot be in the same gallery.

3. Therefore, if U is in gallery 2, W must be in gallery 1.

4. This is a direct and necessary consequence of the initial condition and Rule 1. Let's check the other options to ensure they are not also "must be true."

- We can construct two valid scenarios with U in gallery 2:
- Scenario A: G1 = \{W, R, P, Q\, G2 = \{U, S, T\. Here, S and T are in G2.
- Scenario B: G1 = \{W, S, T, P\, G2 = \{U, R, Q\. Here, S and T are in G1.
- Since the placement of P, R, S, and T varies between these valid scenarios, none of the options (A), (B), (C), or (D) can be a "must be true" statement. Only (E) holds in both cases.


Step 4: Final Answer:

The rule that U and W must be in different galleries directly forces W into gallery 1 when U is placed in gallery 2.
Quick Tip: For "must be true" questions, the simplest deductions are often the correct ones. Don't overcomplicate the problem if a direct rule application gives you one of the answer choices.

Step 1: Understanding the Concept:

Given S is in gallery 2, we need to find a possible pair to complete the gallery.


Step 2: Key Formula or Approach:

Apply the condition S \(\in\) G2 and follow the chain of deductions. Then, test the options to see which pair can validly fill the remaining two spots in gallery 2.


Step 3: Detailed Explanation:

1. If S is in gallery 2, then R must be in gallery 1 (Rule 2).

2. If R is in gallery 1, then T must be in gallery 2 (Rule 2).

3. So, we know gallery 2 must contain S and T. The gallery is G2 = \{S, T, ?\.

4. The third statue must be one of the remaining four: {P, Q, U, W. The remaining three will go to gallery 1 with R. So, G1 = \{R, ...\.

5. We must also satisfy the rule that U and W are in different galleries.

- Let's test putting U in G2: G2 = \{S, T, U\. Then G1 = \{R, P, Q, W\. This is a valid arrangement.
- Let's test putting W in G2: G2 = \{S, T, W\. Then G1 = \{R, P, Q, U\. This is also a valid arrangement.
- Let's test putting P in G2: G2 = \{S, T, P\. Then G1 = \{R, Q, U, W\. This puts U and W together, violating Rule 1. So P cannot be the third statue in G2.
- Let's test putting Q in G2: G2 = \{S, T, Q\. Then G1 = \{R, P, U, W\. This also puts U and W together, violating Rule 1. So Q cannot be the third statue in G2.

6. Therefore, gallery 2 must be either \{S, T, U\ or \{S, T, W\.

7. The question asks for the "other two statues," which are T and the third statue. So the pair can be \{T, U\ or \{T, W\.

8. Option (D), T and W, is one of these valid pairs.


Step 4: Final Answer:

Deductions show that gallery 2 must contain S, T, and either U or W. Therefore, the pair T and W is a possible combination for the other two statues.
Quick Tip: In grouping games, filling a group can create constraints on the other group. Here, trying to place P or Q in gallery 2 forced U and W together in gallery 1, which is not allowed. This process of elimination is crucial.

Step 1: Understanding the Concept:

This is an "EXCEPT" question, which means we are looking for the one pair that CANNOT be in gallery 1 under the given conditions. Four of the pairs will be possible.


Step 2: Key Formula or Approach:

Start with the initial conditions and create the possible valid scenarios. Then, check which of the listed pairs is not found together in gallery 1 in any valid scenario.


Step 3: Detailed Explanation:

1. We are given P \(\in\) G1 and W \(\in\) G2.

2. From W \(\in\) G2, Rule 1 implies U \(\in\) G1.

3. So far: G1 = \{P, U, ?, ?\, G2 = \{W, ?, ?\.

4. Now we must place the {R, S, T block.
- Scenario A: R \(\in\) G1. Then S and T must be in G2. G2 = \{W, S, T\, which is full. G1's remaining spot must be filled by Q. So, G1 = \{P, U, R, Q\. This is a valid world.
- Scenario B: R \(\in\) G2. Then S and T must be in G1. G1 = \{P, U, S, T\, which is full. G2's remaining spot must be filled by Q. So, G2 = \{W, R, Q\. This is a valid world.

5. Now we check the options to see which pair is impossible for G1. Our possible compositions for G1 are \{P, U, R, Q\ or \{P, U, S, T\.
- (A) Q and R: This pair is in G1 in Scenario A. Possible.
- (B) Q and T: In Scenario A, G1 has Q but not T. In Scenario B, G1 has T but not Q. In no scenario are Q and T in gallery 1 together. This is the answer.
- (C) Q and U: This pair is in G1 in Scenario A. Possible.
- (D) R and U: This pair is in G1 in Scenario A. Possible.
- (E) S and T: This pair is in G1 in Scenario B. Possible.


Step 4: Final Answer:

Under the given conditions, Q and T are always in separate galleries. Therefore, they cannot be displayed together in gallery 1.
Quick Tip: For "EXCEPT" questions, the process is about elimination. Find the scenarios that work, and the answer will be the only option that doesn't appear in any of your valid scenarios.

Step 1: Understanding the Concept:

Given P and Q are in gallery 1, we need to find another statue that is forced to be in gallery 1 as a result.


Step 2: Key Formula or Approach:

Place P and Q in gallery 1 and then test the two possible placements for R (in G1 or G2). One of these placements should lead to a contradiction, proving the other is necessary.


Step 3: Detailed Explanation:

1. We are given P and Q are in gallery 1. G1 = \{P, Q, ?, ?\.

2. Let's test the possibility that R is in gallery 2.

- If R \(\in\) G2, then S and T must be in G1 (Rule 2).

- This would make G1 = \{P, Q, S, T\, which is now full.

- The remaining statues (R, U, W) must go into gallery 2. So G2 = \{R, U, W\.

- However, this places U and W in the same gallery, which violates Rule 1.

- This leads to a contradiction, so our initial assumption (R is in gallery 2) must be false.

3. Therefore, R must be in gallery 1.

4. The question asks what statue must also be in gallery 1. We have just proved it must be R.


Step 4: Final Answer:

Placing P and Q in gallery 1 and R in gallery 2 leads to a violation of the rule separating U and W. Therefore, R must be in gallery 1.
Quick Tip: A powerful technique in logic games is proof by contradiction. To prove something must be true, assume it's false and see if that breaks the rules. If it does, your assumption was wrong, and the original statement must be true.

Step 1: Understanding the Concept:

Given S is in gallery 1, we must find a statement about the arrangement that is always true.


Step 2: Key Formula or Approach:

Apply the condition S \(\in\) G1 and follow the chain of deductions to define the structure of the galleries.


Step 3: Detailed Explanation:

1. If S is in gallery 1, R must be in gallery 2 (Rule 2).

2. If R is in gallery 2, T must be in gallery 1 (Rule 2).

3. So far: G1 = \{S, T, ?, ?\, G2 = \{R, ?, ?\.

4. The remaining four statues are {P, Q, U, W. They must fill the remaining four slots (two in each gallery).

5. We know U and W must be in different galleries (Rule 1). So, one goes to G1 and one goes to G2.

6. This leaves P and Q. There is one spot left in G1 and one spot left in G2. P and Q must fill these two spots.

7. Therefore, P and Q must be in different galleries from each other.

8. Option (D) states that P and Q are not displayed in the same gallery. This matches our deduction and must be true.


Step 4: Final Answer:

The condition that S is in gallery 1 forces R into gallery 2 and T into gallery 1. After placing U and W in separate galleries, there is one spot left in each gallery, which must be filled by P and Q, forcing them to be apart.
Quick Tip: When you've placed all the constrained pieces, look at how many "free" variables are left and how many open slots there are. The relationship between these numbers often reveals a "must be true" condition.

Step 1: Understanding the Concept:

Given T \(\in\) G2, we need to find a pair that must be in separate galleries. This is a "must be false" question for them being together.


Step 2: Key Formula or Approach:

Establish the possible scenarios when T is in G2. Then, check each pair in the options to see if they are ever together in any scenario. The pair that is never together is the answer.


Step 3: Detailed Explanation:

1. If T is in gallery 2, R must be in gallery 1 (Rule 2).

2. If R is in gallery 1, S must be in gallery 2 (Rule 2).

3. So far: G1 = \{R, ?, ?, ?\, G2 = \{T, S, ?\.

4. The remaining four statues {P, Q, U, W must fill the remaining four slots (three in G1, one in G2).

5. We also know U and W must be in different galleries.

- Scenario A: The last statue in G2 is U. G2 = \{T, S, U\. Then G1 must be \{R, P, Q, W\. This is valid.

- Scenario B: The last statue in G2 is W. G2 = \{T, S, W\. Then G1 must be \{R, P, Q, U\. This is valid.

- The last statue in G2 cannot be P or Q, because that would force U and W into G1 together.

6. So we have two possible worlds:

- World A: G1 = \{R, P, Q, U\, G2 = \{S, T, W\

- World B: G1 = \{R, P, Q, W\, G2 = \{S, T, U\

7. Now check the pairs: which pair is never together?
- (A) P and S: In World A, P is in G1, S is in G2. In World B, P is in G1, S is in G2. They are always in different galleries. This is the answer.

- (B) Q and R: They are together in G1 in both worlds.

- (C) Q and W: They are together in G1 in World B.

- (D) R and U: They are together in G1 in World A.

- (E) T and W: They are together in G2 in World A.


Step 4: Final Answer:

In all valid scenarios where T is in gallery 2, P ends up in gallery 1 and S ends up in gallery 2. Thus, they can never be together.
Quick Tip: "CANNOT" questions require you to check all possible scenarios. If a pair is apart in all valid worlds you construct, that's your answer.

Step 1: Understanding the Concept:

Given that Q and S are grouped together, we must find a necessary consequence.


Step 2: Key Formula or Approach:

Test the two main possibilities: {Q, S are in gallery 1, or {Q, S are in gallery 2. One case should lead to a contradiction.


Step 3: Detailed Explanation:

1. Case 1: Assume Q and S are in gallery 2.
- If S \(\in\) G2, then R must be in G1 (Rule 2).
- If R \(\in\) G1, then T must be in G2 (Rule 2).
- This would make G2 = \{Q, S, T\, which is full.
- The remaining statues {R, P, U, W must be in G1. So G1 = \{R, P, U, W\.
- However, this places U and W in the same gallery, violating Rule 1.
- Therefore, Case 1 is impossible. Q and S cannot be in gallery 2.

2. Deduction: Since they cannot be in gallery 2, Q and S must be in gallery 1.
3. Now let's proceed with Q and S in gallery 1.
- If S \(\in\) G1, then R must be in gallery 2 (Rule 2).
- This is a necessary consequence of the initial condition. Let's check if it's an answer choice.
- Option (B) is "R is displayed in gallery 2." This matches our deduction perfectly.

4. To be complete, let's determine the rest of the possible arrangements:
- Q \(\in\) G1, S \(\in\) G1 \(\rightarrow\) R \(\in\) G2.
- R \(\in\) G2 \(\rightarrow\) T \(\in\) G1.
- So far: G1 = \{Q, S, T, ?\, G2 = \{R, ?, ?\.
- Remaining: {P, U, W. One goes to G1, two to G2.
- Since U and W must be separate, one goes to G1 and one to G2. The last spot in G2 must be P.
- Scenario A: G1 = \{Q, S, T, U\, G2 = \{R, P, W\.
- Scenario B: G1 = \{Q, S, T, W\, G2 = \{R, P, U\.
- Checking the options against these two worlds confirms that only (B) is always true.


Step 4: Final Answer:

The assumption that Q and S are in gallery 2 leads to a contradiction. Therefore, Q and S must be in gallery 1, which in turn forces R to be in gallery 2.
Quick Tip: Sometimes the initial condition of a question forces the entities into one group over another. The first step should always be to test both possibilities (in G1 or in G2) to see if one of them is impossible.

Step 1: Understanding the Concept:

This is a "weaken the argument" question. We need to identify the logical flaw in the manufacturer's reasoning and find an answer choice that exploits it.


Step 2: Key Formula or Approach:

The manufacturer's argument has this structure:

- Premise: Weekly blood testing is necessary.

- Conclusion: Therefore, the expense of our company's kits is necessary.

The logical flaw is a conflation of two different ideas: the necessity of the testing and the necessity of purchasing their specific, expensive kits to do the testing. The argument implicitly assumes that their kits are the only way to get the necessary tests done. To weaken the argument, we must show this assumption is false.


Step 3: Detailed Explanation:

- (A) This strengthens the argument by suggesting the expense is not a prohibitive barrier.

- (B) This directly attacks the flawed assumption. If other labs can perform the same necessary testing at a lower cost, then the expense of the manufacturer's kits is not necessary. Patients could get the test done elsewhere. This severs the link between the premise and the conclusion.

- (C) This highlights the high expense but doesn't challenge the manufacturer's claim that the expense is necessary.

- (D) This is similar to (C), focusing on the financial burden but not the logic of the necessity claim.

- (E) This strengthens the manufacturer's case by emphasizing how important it is to monitor for side effects, reinforcing the need for testing.


Step 4: Final Answer:

By showing that a cheaper, alternative method exists for the necessary blood testing, this option proves that the expense of the manufacturer's specific kits is not, in fact, "entirely necessary."
Quick Tip: In critical reasoning, be alert for arguments that equivocate or shift terms. The argument slides from "testing is necessary" to "our expensive product is necessary." The key to weakening such an argument is to show that the first part can be true without the second part being true.

Step 1: Understanding the Concept:

This is a necessary assumption question. The argument concludes that the father must have been born in 1933. We need to find the unstated premise that is required for this conclusion to be valid.


Step 2: Key Formula or Approach:

The argument follows a process of elimination. It considers two possibilities (born in 1933 or 1935), eliminates one, and concludes the other must be true. This type of argument is only valid if the initial possibilities were the only possibilities.


Step 3: Detailed Explanation:

- Premise 1: Virginia says 1935; William says 1933.

- Premise 2: The hospital has complete 1935 records, and the father is not in them. This eliminates 1935 as a possibility (assuming he was born in that hospital).

- Premise 3: The hospital has no records for 1933 (so 1933 cannot be eliminated).

- Conclusion: Therefore, he must have been born in 1933.


The logical leap is from "It's not 1935" to "It must be 1933." This only works if 1933 and 1935 were the only two years under consideration. What if he was actually born in 1934, or 1936? The argument completely ignores these other possibilities. It implicitly assumes that one of the two siblings is correct.

- (A) This statement, "Either Virginia's claim or William's claim is correct," explicitly states this necessary assumption. If this is true, and Virginia's claim (1935) has been disproven, then William's claim (1933) must be the correct one.

- Negation Test: If we negate (A), "Neither Virginia's claim nor William's claim is correct," then the conclusion falls apart. If they are both wrong, then eliminating 1935 tells us nothing about whether 1933 is the correct year. He could have been born in any other year. Since negating the statement destroys the argument, (A) is a necessary assumption.

- (B), (C), (D), and (E) are all irrelevant to the logical structure of the argument, which hinges on the exclusivity of the two initial claims.


Step 4: Final Answer:

The argument works by eliminating one of two options. It therefore must assume that those two options were the only ones possible.
Quick Tip: Be on the lookout for arguments that present a "false dichotomy" -- that is, they pretend there are only two choices when in fact there could be more. The unstated assumption in such arguments is always that the two choices presented are the only ones.

Step 1: Understanding the Concept:

This question asks for a plausible explanation for a specific set of data. The data shows a dramatic shift in public opinion toward a more positive view of science between August 1991 and August 1992. We need to find an event or factor that could have caused this shift within that specific timeframe.


Step 2: Key Formula or Approach:

The key is the timing. The shift is massive and occurs in a one-year period. We are looking for a significant, positive, science-related event that occurred between the two surveys and could have influenced the specific population that was surveyed.


Step 3: Detailed Explanation:

- The data shows a huge increase in the "Mostly beneficial" category (from 25% to 81%).

- (A) This option provides a very strong explanation. It identifies the survey group ("people who regularly watch prime-time television") and points to a major, relevant event ("an innovative weekly prime-time television series called 'Wonders of Science'"). Crucially, the timing fits perfectly: the show premiered in January 1992, exactly between the two surveys. A popular and positive TV show about science could plausibly cause a massive shift in the opinions of regular TV viewers.

- (B) The report was published in June 1992, which fits the timeline, but an increase in student interest since 1982 is a long-term trend and is unlikely to cause such a sudden, dramatic shift in the opinions of "college-educated adults."

- (C) This describes a negative event related to science/industry (a company closing due to toxic effects). This would be expected to cause a negative shift in opinion, the opposite of what was observed.

- (D) This describes the methodology of the survey and incentives given to respondents. It doesn't provide an external reason for why their opinions about science would change so drastically.

- (E) The fact that the questioners were scientists might introduce bias, but it doesn't explain the change from one year to the next, assuming the methodology was the same both years.


Step 4: Final Answer:

The premiere and growing popularity of a positive television show about science in the interval between the two surveys provides the most direct and plausible explanation for the dramatic positive shift in opinion among a TV-watching audience.
Quick Tip: In "explain the data" questions, always focus on the key features of the data. Here, the key features are the magnitude of the shift (it's huge) and the timing (between Aug 1991 and Aug 1992). The correct answer must be able to account for both.

Step 1: Understanding the Concept:

This question asks for a complete and valid sequence of visits. We can test each option against the rules.


Step 2: Key Formula or Approach:

Use the established rules and deductions to quickly eliminate invalid sequences.
- Starts with Q or X?
- Ends with V or X?
- Is there an SV block?
- Is R before S?


Step 3: Detailed Explanation:

- (A) Q, R, S, V, X, T: Starts with Q (ok). Ends with T. Violates Rule 2 (must end with V or X). Invalid.

- (B) Q, S, V, T, R, X: Starts with Q (ok). Ends with X (ok). Contains an SV block (ok). But R is after S. Violates Rule 4 (R...S). Invalid.

- (C) R, S, V, T, Q, X: Starts with R. Violates Rule 1 (must start with Q or X). Invalid.

- (D) X, R, S, T, Q, V: Contains "S, T, Q, V". There is no SV block. Violates Rule 3. Invalid.

- (E) X, T, R, Q, S, V:
- Starts with X (ok, Rule 1).
- Ends with V (ok, Rule 2).
- Contains an SV block (S is 5th, V is 6th) (ok, Rule 3).
- R (3rd) is before S (5th) (ok, Rule 4).
- All rules are satisfied. This corresponds to our Scenario 2 framework. Valid.


Step 4: Final Answer:

Only sequence (E) satisfies all the conditions of the game.
Quick Tip: For sequencing games, creating a "checklist" of rules is very efficient for the first question. Go through each option and check off the rules one by one until you find a violation.

Step 1: Understanding the Concept:

We are given a new condition, a "QS" block, and asked for a possible occupant of the second position.


Step 2: Key Formula or Approach:

Apply the new condition and see how it interacts with the initial rules to determine the possible sequences.


Step 3: Detailed Explanation:

1. We have a new block: QS.

2. Consider the start rule (Rule 1). If the trip starts with Q, then the sequence would be Q(1), S(2). However, Rule 4 requires R to be visited before S. Since there is no space for R before position 2, the trip cannot start with Q.

3. Therefore, the trip must start with X. This means X=1.

4. From our key deduction, if X=1, then V=6 and S=5. The sequence skeleton is X, _, _, _, S, V.

5. Now we apply the new condition "QS". Since S is at position 5, Q must be at position 4.

6. The sequence is now: X, _, _, Q, S, V.

7. The remaining sites, R and T, must fill positions 2 and 3 in either order.

8. The question asks what can be the second site visited. Based on our deduction, the second site can be R or T.

9. Looking at the options, (C) Tunin (T) is a possibility.


Step 4: Final Answer:

The condition forces the trip to start with X and Q to be in the 4th position, leaving R and T to fill the 2nd and 3rd positions. Therefore, Tunin can be the second site visited.
Quick Tip: When a conditional rule seems to lead to a contradiction with the main rules, re-evaluate the premises. Here, the `QS` block seemed to conflict with the `R...S` rule, but this conflict only occurred if Q was the starting site. This allowed us to deduce that X must be the starting site.

Step 1: Understanding the Concept:

We need to find the latest possible position for T and then determine which site must be in the third position in that specific scenario.


Step 2: Key Formula or Approach:

Test the latest slots for T (position 6, 5, etc.) and see if a valid schedule can be constructed. The first one that works gives us the scenario to analyze.


Step 3: Detailed Explanation:

1. Can T be 6th? No, the 6th position must be V or X (Rule 2).

2. Can T be 5th? Let's try to build a sequence: _, _, _, _, T, _.

- The 6th site must be V or X. If it's V, then S must be in the 5th position, which is occupied by T. So the 6th site must be X. Sequence: _, _, _, _, T, X.

- The 1st site must be Q or X. Since X is last, the 1st site must be Q. Sequence: Q, _, _, _, T, X.

- We need to place the R...SV chain in slots 2, 3, and 4. The only way to do this is R in 2, S in 3, and V in 4.

- This gives the complete sequence: Q, R, S, V, T, X.

- Let's check if this is valid: Starts Q (ok), ends X (ok), has SV block (ok), has R...S (ok). The sequence is valid.

3. So, the latest possible position for T is 5th, and the only sequence that allows this is Q, R, S, V, T, X.

4. The question asks which site must be the third in this scenario. Looking at the sequence, the third site is S (Sud).


Step 4: Final Answer:

The latest T can be visited is fifth, which forces a unique sequence where Sud is the third site visited.
Quick Tip: For "as late as possible" or "as early as possible" questions, work backward from the end (or forward from the beginning) slot by slot, testing for validity. The first valid placement you find is the one the question is based on.

Step 1: Understanding the Concept:

We are given a new spacing rule, T _ X, and must find a necessary consequence.


Step 2: Key Formula or Approach:

Test the possible positions for the T _ X block within the six slots and eliminate those that conflict with the main rules.


Step 3: Detailed Explanation:

1. The condition creates a `T _ X` block. Let's test its placement.

- T=1, X=3: Impossible, start must be Q or X.

- T=4, X=6: Possible. Sequence: _, _, _, T, _, X. Start must be Q. Sequence: Q, _, _, T, _, X. We need to place R,S,V in 2,3,5 with `R...SV`. `SV` must be a block. They can't fit. Impossible.

- T=3, X=5: Possible. Sequence: _, _, T, _, X, _. Start must be Q. End must be V. Sequence: Q, _, T, _, X, V. S must be before V, but there is no room for the `SV` block. Impossible.

- T=2, X=4: Possible. Sequence: _, T, _, X, _, _. Start must be Q. End must be V. Sequence: Q, T, _, X, _, V. The `SV` block means S=5. Sequence: Q, T, _, X, S, V. The only remaining site is R, which must go in position 3.

2. The only possible sequence is: Q, T, R, X, S, V.

3. Let's verify this sequence: Starts Q (ok), ends V (ok), has `SV` block (ok, 5-6), has `R...S` (ok, 3...5), has `T _ X` (ok, 2-4). It is a valid sequence.

4. Since this is the only possible sequence under the condition, we can determine what must be true.

5. Looking at the options:

- (A) Quin is visited second. False (it's first).

- (B) Ram is visited third. True.

- (C) Sud is visited fourth. False (X is fourth).

- (D) Vara is visited fifth. False (S is fifth).

- (E) Xilat is visited sixth. False (V is sixth).


Step 4: Final Answer:

The T _ X spacing condition forces a unique sequence, in which Ram must be the third site visited.
Quick Tip: When a question introduces a block or a fixed spacing between items (`T _ X`), systematically test where that block can fit into the sequence. Often, only one or two placements will be possible, which greatly simplifies the problem.

Step 1: Understanding the Concept:

Given a new block, RX, we need to find a necessary consequence for the sequence.


Step 2: Key Formula or Approach:

Combine the new `RX` block with the existing `R...SV` chain. This gives a longer chain: `RX ... SV`. Then, determine where this long chain can fit in the sequence.


Step 3: Detailed Explanation:

1. The condition gives us an `RX` block.

2. Rule 1 states the trip must start with Q or X. It cannot start with X, because R must come immediately before it. Therefore, the trip must start with Q. So, Q=1.

3. Rule 2 states the trip must end with V or X. It cannot end with X, because there is no space for R before it at the end. Therefore, the trip must end with V. So, V=6.

4. Since V=6, the `SV` block means S must be at position 5.

5. The sequence skeleton is: Q, _, _, _, S, V.

6. We need to place the `RX` block and the remaining site T into positions 2, 3, and 4. The `R...S` rule is already satisfied because R will be in 2, 3, or 4.

7. Two possible scenarios emerge:
- Scenario A: The `RX` block is in 2-3. T is in 4. Sequence: Q, R, X, T, S, V.
- Scenario B: The `RX` block is in 3-4. T is in 2. Sequence: Q, T, R, X, S, V.
8. We must find a statement that is true in both scenarios.

- (A) Q after T: False in Scenario A. Not a must.

- (B) R after Q: In both scenarios, Q is at position 1 and R is at position 2 or 3. So R is always visited after Q. This must be true.

- (C) T after R: False in Scenario B. Not a must.

- (D) T after S: False in both scenarios. Not a must.

- (E) X after S: False in both scenarios. Not a must.


Step 4: Final Answer:

The conditions force the trip to start with Quin, and in all possible resulting sequences, Ram is visited later.
Quick Tip: Combining rules to form longer chains (`R...SV` + `RX` \(\rightarrow\) `RX...SV`) is a key strategy. These longer chains are more restrictive and can quickly lead to major deductions about the overall structure of the sequence.

Step 1: Understanding the Concept:

Given a fixed position for R, we must determine what other position is now fixed.


Step 2: Key Formula or Approach:

Place R at position 4 and see what consequences follow from the `R...SV` chain.


Step 3: Detailed Explanation:

1. We are given that R is at position 4. The sequence is _, _, _, R, _, _.

2. We have the `R...SV` chain. Since R is at 4, S must be at 5 or 6, and V must be immediately after S.

3. The only way to fit the `SV` block after R=4 is to have S at position 5 and V at position 6.

4. The sequence is now fixed at the end: _, _, _, R, S, V.

5. This immediately tells us that V (Vara) must be the sixth site visited. This matches option (D).

6. Let's quickly check if other options could be "musts".
- The first three positions must be filled by Q, T, and X.
- Rule 1 says position 1 must be Q or X.
- So, it is not a must that Q is first (A). It is not a must that T is second or third (B, C). It is false that X is sixth (E).


Step 4: Final Answer:

Placing Ram in the fourth position forces Sud into the fifth and Vara into the sixth position to satisfy the ordering rules. Therefore, it must be true that Vara is the sixth site visited.
Quick Tip: The implications of a long chain rule like `R...SV` are powerful. Fixing the position of any element in the chain will severely restrict the positions of the other elements.

Step 1: Understanding the Concept:

This is a general "could be true" question, asking which statement is possible under the initial rules. We need to test each option to see if a valid schedule can be constructed.


Step 2: Key Formula or Approach:

For each option, assume the statement is true and try to build a complete, valid sequence. If you can build one, the statement "can be true." If it leads to a contradiction, it's false.


Step 3: Detailed Explanation:

- (A) Q is 5th? _, _, _, _, Q, _. Start must be X. End must be V. X, _, _, _, Q, V. S must be before V, but S must be at position 5, which is taken by Q. Contradiction. False.

- (B) R is 5th? _, _, _, _, R, _. `R...S` means S must be 6th. `SV` block means V must be 7th. Impossible. False.

- (C) S is 2nd? _, S, _, _, _, _. `SV` block means V=3. `R...S` means R=1. R, S, V, _, _, _. But start must be Q or X. Contradiction. False.

- (D) X is 2nd? _, X, _, _, _, _. Start must be Q. Q, X, _, _, _, _. End must be V. Q, X, _, _, _, V. `SV` block means S=5. Q, X, _, _, S, V. We need to place R and T in 3 and 4. `R...S` rule requires R to be before S=5. This is possible (R could be 3 or 4). For example, a valid sequence is: Q, X, R, T, S, V. This "can be true". True.

- (E) X is 5th? _, _, _, _, X, _. End must be V. _, _, _, _, X, V. `SV` block means S=5. But X is at 5. Contradiction. False.


Step 4: Final Answer:

It is possible to construct a valid schedule where Xilat is the second site visited (e.g., Q, X, R, T, S, V). All other options lead to contradictions.
Quick Tip: In "can be true" questions, your goal is to find just one valid example. Once you've successfully built a full, valid schedule that includes the statement in the option, you've found your answer and can move on.

Step 1: Understanding the Concept:

This question asks for a valid, complete assignment of representatives to meetings, given a fixed order of meetings. We must check the options against all rules. (Note: The table formatting in the original document is poor; this solution interprets the intended groupings).


Step 2: Key Formula or Approach:

Check each option against the list of rules, eliminating those that have violations. The most restrictive rules (W size=3, G/H in X) are good starting points.


Step 3: Detailed Explanation:

- (A) W={G,H,J}, ...: Violates Rule 4 (G and H must be in X).

- (B) W={J,L}, ...: Violates Rule 2 (W must have 3 representatives).

- (C) W={J,L,N}, X={G,H,O}, Y={P}, Z={K}: Violates Rule 6 (O and P must be in the same meeting).

- (D) W={J,L,N}, X={G,H}, Y={O,P}, Z={K}:
- Meeting order W,X,Y,Z is given. Rule 1 \& 3 are satisfied.
- W has 3 reps: {J,L,N. (OK, Rule 2)
- X has G and H: {G,H. (OK, Rule 4)
- Z has K: {K. (OK, Rule 5)
- O and P are together: {O,P in Y. (OK, Rule 6)
- All 8 reps are used exactly once. Each meeting has at least one rep. Total reps: 3(W)+2(X)+2(Y)+1(Z) = 8. (OK)
- This schedule is entirely valid.

- (E) W={J,L}, ...: Violates Rule 2 (W must have 3 representatives).


Step 4: Final Answer:

Only the schedule presented in option (D) is consistent with all the rules of the game.
Quick Tip: In complex grouping games, create a master list of all entities to be placed. As you check an option, cross them off. This ensures you account for everyone and that the group sizes add up correctly to the total.

Step 1: Understanding the Concept:

We are given a new condition, Orson (O) is in meeting Y, and asked which of the given statements "can be true." We must find a valid scenario that includes the condition and the statement.


Step 2: Key Formula or Approach:

First, apply the new condition and see what deductions can be made about the group compositions. Then, use those deductions to test the possible meeting orders and see which option is possible.


Step 3: Detailed Explanation:

1. The condition is that Orson (O) is in meeting Y.

2. Rule 6 states that Orson (O) and Porter (P) must be in the same meeting. Therefore, Porter (P) must also be in meeting Y. So, {O, P \(\subseteq\) Y. This means the size of meeting Y is at least 2.

3. We know the sizes of meetings X, Y, and Z must sum to 5, and the possible size distributions are (2,2,1) or (2,1,2). Since size(Y) \(\geq\) 2, the only possible distribution is (2,2,1).

4. This fixes the sizes of the meetings: size(X)=2, size(Y)=2, size(Z)=1.

5. Based on the rules and these sizes, we can determine the exact composition of every meeting group:

- Meeting X must contain Gold (G) and Herrera (H) (Rule 4), and its size is 2. So, X = {G, H.

- Meeting Y must contain Orson (O) and Porter (P) (from step 2), and its size is 2. So, Y = {O, P.

- Meeting Z must contain Karami (K) (Rule 5), and its size is 1. So, Z = {K.

- Meeting W has a size of 3 (Rule 2). The remaining representatives are Jones (J), Lowell (L), and Nakamura (N). So, W = {J, L, N.

6. Now we must consider the possible order of meetings. Rule 1 states W is first. Rule 3 states X must be before Y. The possible valid orders are:

- Order 1: W, X, Y, Z (Positions 1, 2, 3, 4)

- Order 2: W, X, Z, Y (Positions 1, 2, 3, 4)

- Order 3: W, Z, X, Y (Positions 1, 2, 3, 4)

7. Finally, we test the options to see which "can be true":
- (A) Gold (G) with Jones (J)? No, G is in X, J is in W. They are in different meetings.

- (B) Herrera (H) with Lowell (L)? No, H is in X, L is in W. They are in different meetings.

- (C) Jones (J) for the second meeting? No, J is in meeting W, and W is always the first meeting.

- (D) Karami (K) for the third meeting? Yes. K is in meeting Z. In Order 2 (W, X, Z, Y), meeting Z is scheduled third. This is a valid possibility.

- (E) Lowell (L) for the fourth meeting? No, L is in meeting W, which is always the first meeting.


Step 4: Final Answer:

Based on a full deduction, the groups are fixed. The meeting order W, X, Z, Y is valid, and in this order, Karami (who is in meeting Z) is scheduled for the third meeting. Therefore, this statement can be true.
Quick Tip: In complex grouping games, a single new condition can sometimes lock down the entire composition of the groups. Once the groups are fixed, questions often pivot to the possible orderings of those groups.

Step 1: Understanding the Concept:

We are given the condition that Gold (G) and Jones (J) are in the third meeting. We must deduce what else must be true about the schedule.


Step 2: Key Formula or Approach:

First, determine which meeting is third. Then, place G and J there and follow the chain of deductions based on the game's rules.


Step 3: Detailed Explanation:

1. We need to identify the third meeting in the sequence. The sequence starts with W. The other meetings are X, Y, and Z, with the constraint that X must come before Y. The possible orders are W,X,Y,Z (Y is third), W,X,Z,Y (Z is third), and W,Z,X,Y (X is third).

2. The condition states that Gold (G) is in the third meeting. However, Rule 4 states that G is always in meeting X. Therefore, the third meeting must be meeting X. This eliminates the first two possible orders, and fixes the order of meetings as: 1st-W, 2nd-Z, 3rd-X, 4th-Y.

3. Now we know meeting X is third and contains {G, J, .... Rule 4 also adds Herrera (H) to meeting X. So, {G, H, J \(\subseteq\) X. The size of X is at least 3.

4. The total number of representatives in meetings X, Y, and Z is 5. Since size(X) \(\geq\) 3, and size(Y) and size(Z) must be at least 1, the only possible size distribution is size(X)=3, size(Y)=1, size(Z)=1.

5. This allows us to determine the composition of X, Y, and Z. X = {G, H, J. Rule 5 states Karami (K) is in Z, so Z = {K.

6. The remaining representatives are Lowell (L), Nakamura (N), Orson (O), and Porter (P). These four must be placed in meeting W (size 3) and meeting Y (size 1).

7. Rule 6 requires Orson (O) and Porter (P) to be in the same meeting. The only meeting with enough space for them is W. Therefore, {O, P must be in meeting W.

8. Since W is the first meeting, it must be true that Porter (P) is scheduled for the first meeting.

9. This matches option (C). Let's check the other options for completeness. The remaining reps L and N must fill the last spot in W and the single spot in Y. So either L is in W and N is in Y, or vice-versa. This means (B) and (E) are possible but not necessary, and (D) is false. (A) is false because H is in the third meeting.


Step 4: Final Answer:

The conditions force the meeting order to be W,Z,X,Y and require the {O,P block to be in meeting W. Therefore, Porter must be in the first meeting.
Quick Tip: When a condition seems to contradict a rule (like G being in the 3rd meeting when G must be in meeting X), use that to make a powerful deduction. It doesn't mean the scenario is impossible; it means the scenario forces an identity (the 3rd meeting IS meeting X).

Step 1: Understanding the Concept:

We are given the positions of Nakamura (N) and Karami (K) and must deduce a necessary consequence.


Step 2: Key Formula or Approach:

Use the information about K to identify the meeting that is fourth. Then use the information about N to make deductions about the group compositions.


Step 3: Detailed Explanation:

1. Condition: N is in the 3rd meeting, K is in the 4th meeting.

2. Rule 5 states that K is in meeting Z. So, meeting Z must be the fourth (and last) meeting.

3. This means the order of meetings is W, X, Y, Z. W is 1st, X is 2nd, Y is 3rd, and Z is 4th.

4. We are given that N is in the third meeting. The third meeting is Y. So, N \(\in\) Y.

5. The groups and positions so far:
- 1st (W): size 3
- 2nd (X): contains {G,H, size \(\geq\) 2
- 3rd (Y): contains N, size \(\geq\) 1
- 4th (Z): contains K, size \(\geq\) 1

6. The total size of X+Y+Z is 5. We have size(X)\(\geq\)2, size(Y)\(\geq\)1, size(Z)\(\geq\)1. The possible size distributions are (3,1,1), (2,2,1), or (2,1,2) for (X,Y,Z).

7. The remaining reps to be placed are J, L, O, P. The {O,P block must be placed.
- Can the {O,P block go in W? W would have 1 spot left. The remaining 2 reps (J,L) plus N must fill X,Y,Z. This is 3 people for 4 spots. Impossible.
- Can the {O,P block go in X? Then X={G,H,O,P. Size(X)=4. But X+Y+Z=5. Y\(\geq\)1, Z\(\geq\)1. This means size(X) cannot be more than 3. Impossible.
- Can the {O,P block go in Y? Then Y={N,O,P. Size(Y)=3. This means the distribution for (X,Y,Z) must be (2,3,?). No, Y can't be size 3. X+Y+Z=5. If Y=3, X\(\geq\)2, Z\(\geq\)1. X+Y+Z \(\geq\) 6. Impossible.
- Can the {O,P block go in Z? Then Z={K,O,P. Size(Z)=3. Impossible for the same reason.

8. There is a flaw in my reasoning. Let's restart the placements.
- Order: W,X,Y,Z. N \(\in\) Y, K \(\in\) Z. X={G,H,....
- Reps to place: J,L,O,P in W and the remaining spots in X,Y,Z.
- The {O,P block needs a home. Y has N, Z has K. X has G,H. The only meeting with enough definite space is W (size 3). Let's try placing {O,P in W.
- W = {O,P, _. One spot left in W.
- Reps left to place: J, L. They must fill the last spot in W and any remaining spots in X,Y,Z.
- Let's try the size distribution (2,2,1) for (X,Y,Z).
- X={G,H (size 2). Y={N, _ (size 2). Z={K (size 1).
- Reps to place: J,L,O,P. W has 3 spots. Y has 1 spot.
- Place {O,P in W. W={O,P, _. Reps left: J,L. Y needs 1, W needs 1.
- So, W={O,P,J, Y={N,L OR W={O,P,L, Y={N,J.
- Let's check the schedule: W={O,P,J, X={G,H, Y={N,L, Z={K. All 8 reps assigned. All rules met. This is a valid scenario.
9. Let's try size distribution (2,1,2).
- X={G,H (size 2). Y={N (size 1). Z={K, _ (size 2).
- Reps to place: J,L,O,P. W has 3 spots. Z has 1 spot.
- Place {O,P in W. W={O,P, _. Reps left: J,L. Z needs 1, W needs 1.
- W={O,P,J, Z={K,L OR W={O,P,L, Z={K,J.
- Check schedule: W={O,P,J, X={G,H, Y={N, Z={K,L. Valid.
10. In both valid scenarios, X={G,H. The meeting scheduled second is X. Therefore, it must be true that Herrera (H) is scheduled for the second meeting.

Step 4: Final Answer:

The conditions fix the meeting order as W,X,Y,Z. In all possible valid assignments, meeting X consists of only Gold and Herrera. Since X is the second meeting, Herrera must be scheduled for the second meeting.
Quick Tip: When a problem seems to have many possibilities, look for a "linchpin" deduction. Here, figuring out where the {O,P} block could and couldn't go was the key to unlocking the structure of the solution.

Note: This question appears to be flawed based on a strict reading of the rules, as multiple scenarios can be constructed that contradict the intended answer. However, the following is a possible line of reasoning that may lead to the intended answer.

Step 1: Understanding the Concept:

The question asks which statement CANNOT be true (is impossible) under the condition that Jones (J) is scheduled for a meeting by himself. This means the size of J's meeting is 1.

Step 2: Key Formula or Approach:

J must be the only member of his meeting. Since \(W=3\) and \(X \geq 2\), J’s solo meeting must be either \(Y\) or \(Z\).

Step 3: Reduced Explanation:

1. Possible orders: \(W,X,Y,Z\); \(W,X,Z,Y\); \(W,Z,X,Y\).
2. If J is third:

In \(W,X,Y,Z\), third is \(Y\). If \(Y=\{J\}\), valid.
In \(W,X,Z,Y\), third is \(Z\), but \(K \in Z\) (Rule 5), so not valid.
In \(W,Z,X,Y\), third is \(X\), but \(G,H \in X\) (Rule 4), so not valid.

3. Hence J can be third only if \(Y=\{J\}\) and order is \(W,X,Y,Z\). This yields a consistent full schedule.

Answer: (A) is \emph{possible, so the question’s claim that it “cannot be true” is flawed.


Step 4: Final Answer:

Despite the logical possibility, if forced to select an answer, one would have to assume there is a hidden constraint or flaw in the problem's design. As deduced, (A) is the intended answer but is logically possible. Quick Tip: When you encounter a flawed question in a practice setting, the best approach is to trust your deductions. Methodically prove why the question is flawed. This deepens your understanding of the rules, which is more valuable than finding the "correct" but illogical answer.

Step 1: Understanding the Concept:

This is a "strengthen the argument" question. The argument concludes that a smoking ban in San Leonardo's restaurants will cause them to lose patrons and income. The reason given is that smokers will not want to refrain from smoking.


Step 2: Key Formula or Approach:

To strengthen this argument, we need to provide evidence that the smokers who are unwilling to refrain from smoking have a viable alternative that would lead them to abandon San Leonardo's restaurants. The argument assumes these patrons will go elsewhere rather than comply. We need to support this assumption.


Step 3: Detailed Explanation:

- The core of the argument is that smokers will choose to stop dining in San Leonardo rather than give up smoking with their meals.

- (A) This would weaken the argument. If most patrons are non-smokers, the loss of some smokers might have a smaller impact, and could even be offset by an increase in non-smoking patrons who prefer a smoke-free environment.

- (B) This directly weakens the argument by stating that smokers are often willing to refrain, contradicting the argument's main premise.

- (C) This is irrelevant. A hypothetical alternative law does not affect the outcome of the actual law.

- (D) This is irrelevant. The existence of non-smoking sections before doesn't change the impact of a total ban.

- (E) This provides the key piece of information. If adjacent towns have comparable restaurants and no smoking bans, it gives the unhappy smokers a perfect, convenient alternative. They can simply drive to a neighboring town to eat and smoke. This makes it much more likely that they will actually leave San Leonardo's restaurants, thus strengthening the conclusion that the restaurants will lose patrons and income.


Step 4: Final Answer:

The existence of a convenient, unrestricted alternative (restaurants in adjacent towns) makes it much more probable that smokers will take their business elsewhere, thus strengthening the prediction of lost income.
Quick Tip: When an argument predicts a change in customer behavior, strengthening it often involves showing that customers have both the motive (unhappiness with the new rule) and the opportunity (a viable alternative) to change their behavior.

Step 1: Understanding the Concept:

The argument concludes that a trait (predisposition to TSZ) is inherited (genetic) based on a correlation between parents and their biological children. The classic challenge to this type of argument is the "nature vs. nurture" debate. The correlation could be due to shared genes (nature) or a shared environment and upbringing (nurture). To strengthen the "nature" conclusion, we must weaken the "nurture" alternative.


Step 2: Key Formula or Approach:

The best way to separate genetic factors from environmental ones is to use an adoption study. We need to find an answer choice that shows that the environment (who raises the child) doesn't make a difference, while the genetics (who the biological parents are) does.


Step 3: Detailed Explanation:

- The argument is: Biological parents with TSZ have children with TSZ \(\rightarrow\) TSZ is genetic.

- The weakness is: Maybe the parents with TSZ create a home environment that causes their children to develop TSZ (e.g., through stress or learned behavior).

- (C) This option directly addresses and eliminates the "nurture" explanation. It takes a group of children with a genetic predisposition (their biological parents have TSZ) and compares two subgroups: one raised by their biological parents (genetics + TSZ environment) and one raised by adoptive parents without TSZ (genetics + non-TSZ environment). The finding that both groups are "as likely to develop TSZ" shows that the environment they were raised in didn't matter. The only constant factor is their genetics. This provides powerful evidence that the trait is inherited.

- (A) This suggests an environmental trigger (stress), which would add a layer to the nurture side of the argument, not strengthen the genetic claim.

- (B) This would weaken the argument. It shows that children without the genetic predisposition are more likely to get TSZ if raised in a TSZ environment, suggesting that nurture is a powerful factor.

- (D) and (E) discuss treatment and diagnosis, which are irrelevant to the cause of the predisposition itself.


Step 4: Final Answer:

By showing that the environment a child is raised in has no effect on their likelihood of developing TSZ, this option isolates genetics as the causal factor, thus strongly strengthening the conclusion.
Quick Tip: For any "nature vs. nurture" argument, the gold standard for strengthening or weakening is an adoption study. To strengthen the "nature" (genetic) argument, show that adopted children are more like their biological parents than their adoptive parents. To strengthen the "nurture" (environment) argument, show the reverse.

Step 1: Understanding the Concept:

This is a "must be true" or inference question. We are presented with a set of premises and must find a conclusion that logically follows from them without any additional assumptions. The question presents a dilemma.


Step 2: Key Formula or Approach:

Break down the premises into a logical chain:

- Premise 1: To play an instrument \(\rightarrow\) You must restore it.

- Premise 2: To restore an instrument \(\rightarrow\) You destroy all information about its making techniques.

Combine these premises to find the necessary conclusion.


Step 3: Detailed Explanation:

Let's combine the premises:

If an instrument can be played, then it must have been restored (from Premise 1).

If an instrument has been restored, then all information about its making techniques has been destroyed (from Premise 2).

Therefore, if an instrument can be played, then all information about its making techniques has been destroyed.

This means a playable instrument can no longer provide new information about how it was made.


Now let's check the options against this deduction:

- (A) This discusses "playing techniques," but the passage is about "instrument-making techniques." This is a subtle but important distinction. The passage doesn't support this claim.

- (B) The word "only" makes this too strong. A restored instrument might provide other types of information (e.g., about the materials used), even if information about making techniques is lost.

- (C) The word "only" makes this too strong. There could be other sources, like historical documents or diagrams. The passage does not claim instruments are the sole source.

- (D) This compares the amount of information ("more"). The passage states that information about making techniques is completely destroyed ("all"), but an unrestored instrument might provide little information for other reasons. We cannot make a quantitative comparison like "more."

- (E) This is a perfect restatement of our deduction. "Once it can be played" implies it has been restored. And if it has been restored, the information about "instrument-making techniques" is destroyed, so it cannot serve as a source of new information on that topic.


Step 4: Final Answer:

The premises logically lead to the conclusion that the act of restoring an instrument to make it playable necessarily destroys its value as a source of information about its original construction.
Quick Tip: In "must be true" questions, pay extremely close attention to the exact wording. The difference between "playing techniques" and "instrument-making techniques" is what makes option (A) incorrect. The correct answer will stick precisely to the terms defined in the passage.

Step 1: Understanding the Concept:

This question asks for a comparison between the square of \(\pi\) and the number 10. We are given approximate values for \(\pi\) and \(\sqrt{10}\) to help with the comparison.


Step 2: Key Formula or Approach:

The most direct way to compare \(\pi^2\) and 10 is to compare \(\pi\) and \(\sqrt{10}\). If \(a\) and \(b\) are positive numbers, then \(a < b\) is equivalent to \(a^2 < b^2\).


Step 3: Detailed Explanation:

We are given the approximations \(\pi \approx 3.14\) and \(\sqrt{10} \approx 3.16\).

Based on these values, we can see that \(\pi\) is less than \(\sqrt{10}\).
\[ 3.14 < 3.16 \] \[ \pi < \sqrt{10} \]
Since both \(\pi\) and \(\sqrt{10}\) are positive numbers, we can square both sides of the inequality without changing its direction:
\[ \pi^2 < (\sqrt{10})^2 \] \[ \pi^2 < 10 \]
This shows that the quantity in Column A is less than the quantity in Column B.


Alternatively, we could square the given approximation for \(\pi\):
\[ \pi^2 \approx (3.14)^2 = 3.14 \times 3.14 = 9.8596 \]
Since 9.8596 is less than 10, the quantity in Column A is smaller.


Step 4: Final Answer:

Both methods show that \(\pi^2\) is less than 10. Therefore, the quantity in Column B is greater.
Quick Tip: For quantitative comparison questions involving squares and square roots, it's often easier to compare the numbers before squaring them (or after taking the square root of both columns). This can help avoid complex calculations.

Step 1: Understanding the Concept:

This problem requires the calculation and comparison of probabilities from a set of objects. The key is to first determine the total number of outcomes.


Step 2: Key Formula or Approach:

The probability of an event is calculated as:
\[ P(Event) = \frac{Number of favorable outcomes}{Total number of possible outcomes} \]
The probability of either of two mutually exclusive events (like drawing a yellow or a blue marble) is the sum of their individual probabilities.


Step 3: Detailed Explanation:

First, calculate the total number of marbles in the bag:

Total marbles = 2 (yellow) + 4 (blue) + 6 (green) = 12 marbles.


For Column A:

The probability of drawing a green marble.

Number of favorable outcomes (green marbles) = 6.
\[ P(green) = \frac{6}{12} = \frac{1}{2} \]

For Column B:

The probability of drawing a yellow or blue marble.

Number of favorable outcomes (yellow or blue) = 2 + 4 = 6.
\[ P(yellow or blue) = \frac{6}{12} = \frac{1}{2} \]
Alternatively, \(P(yellow or blue) = P(yellow) + P(blue) = \frac{2}{12} + \frac{4}{12} = \frac{6}{12} = \frac{1}{2}\).


Step 4: Final Answer:

Both Column A and Column B have a value of \(\frac{1}{2}\). Therefore, the two quantities are equal.
Quick Tip: In probability questions, sometimes you don't need to calculate the final value. You can simply compare the number of favorable outcomes. In this case, there are 6 green marbles (Column A) and 6 yellow-or-blue marbles (Column B). Since the total is the same for both, the probabilities must be equal.

Step 1: Understanding the Concept:

This question involves comparing two numbers that are the result of subtracting a fraction from the same integer. The key is to understand how the size of the subtracted value affects the result.


Step 2: Key Formula or Approach:

A common principle in comparisons is that if you start with the same number and subtract a smaller amount, the result will be larger. Conversely, subtracting a larger amount results in a smaller result.


Step 3: Detailed Explanation:

Both columns start with the number 250.

Column A subtracts the fraction \(\frac{1}{3}\).

Column B subtracts the fraction \(\frac{1}{2}\).

We need to compare the fractions \(\frac{1}{3}\) and \(\frac{1}{2}\).

Since \(3 > 2\), the fraction with 3 in the denominator is smaller:
\[ \frac{1}{3} < \frac{1}{2} \]
Because we are subtracting a smaller number from 250 in Column A than in Column B, the result in Column A will be greater.
\[ 250 - (a smaller number) > 250 - (a larger number) \] \[ 250 - \frac{1}{3} > 250 - \frac{1}{2} \]

Step 4: Final Answer:

The quantity in Column A is greater because a smaller value is being subtracted from 250.
Quick Tip: For quantitative comparisons, you can often simplify by canceling common terms. Here, you can mentally remove the '250 -' from both sides, leaving you to compare \(-\frac{1}{3}\) and \(-\frac{1}{2}\). Since \(-\frac{1}{3}\) is greater (closer to zero) than \(-\frac{1}{2}\), Column A is greater.

Step 1: Understanding the Concept:

This problem involves a system of two linear equations with four variables. We need to manipulate these equations to find the value of the expression in Column A.


Step 2: Key Formula or Approach:

The key is to notice the common term \(x+y\) in both equations. We can isolate this term in each equation and then set the results equal to each other. This will give us an equation with only \(n\) and \(k\).


Step 3: Detailed Explanation:

We are given the two equations:

1) \(x+y+n=15\)

2) \(x+y+k=9\)

From equation (1), we can isolate the term \(x+y\):
\[ x+y = 15 - n \]
From equation (2), we can also isolate the term \(x+y\):
\[ x+y = 9 - k \]
Since both expressions are equal to \(x+y\), we can set them equal to each other:
\[ 15 - n = 9 - k \]
The question asks for the value of \(n-k\). Let's rearrange the equation to solve for this expression. Add \(n\) to both sides:
\[ 15 = 9 - k + n \]
Subtract 9 from both sides:
\[ 15 - 9 = n - k \] \[ 6 = n - k \]
The value of the expression in Column A is exactly 6.


Step 4: Final Answer:

The value of Column A is 6, which is equal to the value in Column B.
Quick Tip: Another quick method is to subtract the second equation from the first: \((x+y+n) - (x+y+k) = 15 - 9\). The \(x\) and \(y\) terms cancel out, leaving \(n-k = 6\).

Step 1: Understanding the Concept:

This question asks us to calculate and compare the slopes of two lines, given two points on each line.


Step 2: Key Formula or Approach:

The formula for the slope (\(m\)) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ slope = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 3: Detailed Explanation:

For Column A: The slope of line k

The points are (0,0) and (4,8).

Let \((x_1, y_1) = (0,0)\) and \((x_2, y_2) = (4,8)\).
\[ slope of k = \frac{8 - 0}{4 - 0} = \frac{8}{4} = 2 \]

For Column B: The slope of line m

The points are (0,1) and (4,9).

Let \((x_1, y_1) = (0,1)\) and \((x_2, y_2) = (4,9)\).
\[ slope of m = \frac{9 - 1}{4 - 0} = \frac{8}{4} = 2 \]

Step 4: Final Answer:

Both lines have a slope of 2. Therefore, the two quantities are equal. The lines are parallel.
Quick Tip: The slope represents the "rise over run." For both lines, the "run" (change in x) is \(4 - 0 = 4\). The "rise" (change in y) for line k is \(8 - 0 = 8\), and for line m is \(9 - 1 = 8\). Since both have the same rise and the same run, their slopes must be identical without needing to complete the division.

Step 1: Understanding the Concept:

This is a geometry problem that compares a dimension of an inscribed equilateral triangle to a dimension of the circle that circumscribes it.


Step 2: Key Formula or Approach:

There is a specific relationship between the side length (\(s\)) of an inscribed equilateral triangle and the radius (\(r\)) of the circumscribing circle: \(s = r\sqrt{3}\). We need to compare this to the diameter (\(d\)), where \(d = 2r\).


Step 3: Detailed Explanation:

Let \(s\) be the length of a side of the triangle and \(d\) be the diameter of the circle. Let the radius of the circle be \(r\).

Column A is \(s = r\sqrt{3}\).

Column B is \(d = 2r\).

We are comparing \(r\sqrt{3}\) and \(2r\). Since the radius \(r\) must be a positive number, we can divide both quantities by \(r\) without changing the inequality.

This simplifies the comparison to \(\sqrt{3}\) versus 2.

We know that \((\sqrt{3})^2 = 3\) and \(2^2 = 4\).

Since \(3 < 4\), it follows that \(\sqrt{3} < 2\).

Therefore, \(r\sqrt{3} < 2r\), which means \(s < d\).


Step 4: Final Answer:

The length of the side of the triangle is less than the diameter of the circle. The quantity in Column B is greater.
Quick Tip: Visualize the figure. The diameter is the longest possible straight line that can be drawn inside a circle. Since the side of the inscribed triangle does not pass through the center of the circle, its length must be less than the diameter. This intuitive check confirms the mathematical result.

Step 1: Understanding the Concept:

This question requires the comparison of a fraction and a decimal. To compare them accurately, we should convert them to a common format.


Step 2: Key Formula or Approach:

We can either convert the fraction to a decimal or the decimal to a fraction. Alternatively, for comparing fractions \(\frac{a}{b}\) and \(\frac{c}{d}\), we can use cross-multiplication: compare \(a \times d\) and \(b \times c\).


Step 3: Detailed Explanation:

Method 1: Convert Fraction to Decimal

To convert \(\frac{1}{11}\) to a decimal, we perform the division 1 \(\div\) 11.
\[ 1 \div 11 = 0.090909... \]
We are comparing \(0.0909...\) (Column A) with 0.09 (Column B).

Since \(0.0909...\) is greater than 0.0900, the quantity in Column A is greater.


Method 2: Convert Decimal to Fraction and Cross-Multiply

The decimal 0.09 is equivalent to the fraction \(\frac{9}{100}\).

Now we compare \(\frac{1}{11}\) (Column A) with \(\frac{9}{100}\) (Column B).

We cross-multiply the numerators and denominators:

For Column A: \(1 \times 100 = 100\).

For Column B: \(11 \times 9 = 99\).

Since \(100 > 99\), the fraction corresponding to the 100 (\(\frac{1}{11}\)) is greater.


Step 4: Final Answer:

Both methods show that \(\frac{1}{11}\) is greater than 0.09. The quantity in Column A is greater.
Quick Tip: Cross-multiplication is often the fastest and most error-proof way to compare two simple fractions, as it avoids long division and dealing with repeating decimals.

Step 1: Understanding the Concept:

This geometry problem requires us to find the perimeters of two different shapes, a triangle and a parallelogram, using information about their properties and relationships. The key is to express all lengths in terms of a single variable.


Step 2: Key Formula or Approach:

- Perimeter of an equilateral triangle with side \(s\): \(P = 3s\).

- Perimeter of a parallelogram with adjacent sides \(a\) and \(b\): \(P = 2(a+b)\).

- Properties of a parallelogram: Opposite sides are equal in length.


Step 3: Detailed Explanation:

Let's define the side lengths based on the given ratio.

The ratio of ZR to PR is 1 to 4. Let \(ZR = a\). Then \(PR = 4a\).

The diagram shows that the points P, Z, and R are collinear, so the length of the segment PZ is \(PR + ZR = 4a + a = 5a\).


Now, let's use the properties of the given shapes.

We are given that \(\triangle ZYR\) is equilateral. This means all its sides are equal to ZR.
\[ ZY = YR = ZR = a \]
We are given that PXYZ is a parallelogram. In a parallelogram, opposite sides are equal.
\[ XY = PZ = 5a \] \[ PX = ZY = a \]
We are given that \(\triangle XQY\) is equilateral. This means all its sides are equal to XY.
\[ XQ = QY = XY = 5a \]
Now we can calculate the perimeters for both columns.


For Column A: The perimeter of \(\triangle XQY\)

The side length is \(XY = 5a\).
\[ Perimeter = 3 \times (side length) = 3 \times (5a) = 15a \]

For Column B: The perimeter of parallelogram PXYZ

The adjacent sides are PZ and ZY. Their lengths are \(PZ = 5a\) and \(ZY = a\).
\[ Perimeter = 2 \times (PZ + ZY) = 2 \times (5a + a) = 2 \times (6a) = 12a \]

Comparison:

We are comparing 15a (Column A) and 12a (Column B). Since \(a\) represents a geometric length, it must be positive (\(a > 0\)). Therefore, \(15a > 12a\).


Step 4: Final Answer:

The perimeter of the triangle (15a) is greater than the perimeter of the parallelogram (12a). The quantity in Column A is greater.
Quick Tip: In complex geometry problems, the first step is always to translate all the given information (ratios, shape properties) into algebraic expressions with a common variable. This turns the geometry problem into a simpler algebra problem.

Step 1: Understanding the Concept:

This question asks us to compare the value of an expression, \(x+y\), with twice the value of that same expression. The relationship will depend on whether the expression \(x+y\) is positive, negative, or zero.


Step 2: Key Formula or Approach:

Let's represent the expression \(x+y\) with a single variable, say \(k\). We are then comparing \(k\) (Column A) with \(2k\) (Column B). We should test different types of values for \(k\).


Step 3: Detailed Explanation:

Let \(k = x+y\). We are comparing \(k\) and \(2k\).


Case 1: \(x+y\) is positive.

Let \(x=1, y=1\), so \(x+y = 2\).

Column A: \(x+y = 2\).

Column B: \(2(x+y) = 2(2) = 4\).

In this case, Column B is greater than Column A (\(4 > 2\)).


Case 2: \(x+y\) is negative.

Let \(x=-1, y=-1\), so \(x+y = -2\).

Column A: \(x+y = -2\).

Column B: \(2(x+y) = 2(-2) = -4\).

In this case, Column A is greater than Column B (\(-2 > -4\)).


Case 3: \(x+y\) is zero.

Let \(x=1, y=-1\), so \(x+y = 0\).

Column A: \(x+y = 0\).

Column B: \(2(x+y) = 2(0) = 0\).

In this case, the two columns are equal.


Step 4: Final Answer:

Since the relationship between the two columns changes depending on the values of \(x\) and \(y\), the relationship cannot be determined from the information given.
Quick Tip: When a quantitative comparison involves variables with no constraints, always test positive, negative, and zero values. If you get different comparison results, the answer is always that the relationship cannot be determined.

Step 1: Understanding the Concept:

This is a percentage problem involving chained increases. We need to calculate the difference in costs between the computers and compare them. It's a common mistake to assume the differences are the same.


Step 2: Key Formula or Approach:

Let's use variables to represent the costs. A 30 percent increase means the new price is 130% of the original, or 1.3 times the original.

Let \(C_Z\) be the cost of computer Z.

Cost of Y (\(C_Y\)) = \(C_Z + 0.30 C_Z = 1.3 C_Z\).

Cost of X (\(C_X\)) = \(C_Y + 0.30 C_Y = 1.3 C_Y\).


Step 3: Detailed Explanation:

Let's pick a simple value for the cost of computer Z to make the calculations easier. Let \(C_Z =
(100\).


Calculate the cost of computer Y:
\(C_Y = 1.3 \times C_Z = 1.3 \times 100 =
)130\).


Calculate the cost of computer X:
\(C_X = 1.3 \times C_Y = 1.3 \times 130 =
(169\).


Now, calculate the values for Column A and Column B.


For Column A:

Cost of X minus cost of Y = \(C_X - C_Y =
)169 -
(130 =
)39\).


For Column B:

Cost of Y minus cost of Z = \(C_Y - C_Z =
(130 -
)100 =
(30\).


Comparison:

Column A is \)39 and Column B is \(30. Since \(39 > 30\), the quantity in Column A is greater.


Algebraic Approach:

Column B: \(C_Y - C_Z = 0.30 C_Z\).

Column A: \(C_X - C_Y = 0.30 C_Y\). Since \(C_Y = 1.3 C_Z\), we substitute this in: \(0.30 \times (1.3 C_Z) = 0.39 C_Z\).

Comparing \(0.39 C_Z\) (Column A) with \(0.30 C_Z\) (Column B), and since \(C_Z\) must be positive, Column A is greater.


Step 4: Final Answer:

The difference in cost between X and Y is greater than the difference in cost between Y and Z. The quantity in Column A is greater.
Quick Tip: In chained percentage increase problems, the absolute increase becomes larger at each step because the base for the percentage calculation is increasing. A 30% increase on a larger number (Cost of Y) is a bigger amount than a 30% increase on a smaller number (Cost of Z).

Step 1: Understanding the Concept:

This question tests our understanding of inequalities and the properties of positive and negative numbers. We are given an inequality and must determine the sign of a related expression.


Step 2: Key Formula or Approach:

Analyze the given inequality \(x^2 y < 0\) to determine the signs of the variables \(x\) and \(y\). Then use this information to determine the possible sign of the expression \(xy\).


Step 3: Detailed Explanation:

The inequality is \(x^2 y < 0\).

The term \(x^2\) (the square of a real number) must be non-negative.

If \(x=0\), then \(0 \times y < 0\), which simplifies to \(0 < 0\). This is false. Therefore, \(x \neq 0\).

Since \(x \neq 0\), \(x^2\) must be strictly positive.

So, our inequality has the form: \((positive number) \times y < 0\).

For this product to be negative, \(y\) must be a negative number. So, we know for sure that \(y < 0\).

However, we have no information about the sign of \(x\). \(x\) could be positive or negative.


Now let's evaluate Column A, which is \(xy\).


Case 1: \(x\) is positive.

If \(x > 0\) and \(y < 0\), then their product \(xy\) will be negative.

In this case, \(xy < 0\). Column B is greater.


Case 2: \(x\) is negative.

If \(x < 0\) and \(y < 0\), then their product \(xy\) will be positive.

In this case, \(xy > 0\). Column A is greater.


Step 4: Final Answer:

Since the value of \(xy\) can be either greater than or less than 0 depending on the sign of \(x\), the relationship between the two columns cannot be determined.
Quick Tip: The inequality \(x^2 > 0\) holds for any non-zero real number \(x\), regardless of whether \(x\) is positive or negative. This is a crucial property to remember when working with inequalities involving squared variables.

Step 1: Understanding the Concept:

This problem requires us to find the values of angles \(x\) and \(y\) from a geometric diagram by applying properties of angles, such as those in a triangle and vertically opposite angles.


Step 2: Key Formula or Approach:

1. The sum of angles in a triangle is 180\(^\circ\).

2. Vertically opposite angles formed by intersecting lines are equal.

3. Angles on a straight line add up to 180\(^\circ\).


Step 3: Detailed Explanation:

Let's analyze the diagram. It shows two intersecting lines. A triangle is formed, with one of its vertices at the intersection point.

- Inside the triangle, we see angles labeled \(x\) and \(2x\). There is also a right-angle symbol, indicating an angle of 90\(^\circ\). The angle labeled \(x\) in the diagram is one of the triangle's internal angles.

- The sum of the angles in this triangle is 180\(^\circ\). Therefore, we can set up an equation:
\[ x + 2x + 90^\circ = 180^\circ \] \[ 3x = 180^\circ - 90^\circ \] \[ 3x = 90^\circ \] \[ x = 30^\circ \]
- Now, we need to find the value of \(y\). The angle \(y\) and the angle \(x\) inside the triangle are vertically opposite to each other at the intersection point.

- Therefore, \(y\) must be equal to the sum of the other two internal angles of the triangle, but this seems incorrect. Let's re-examine the diagram.

- A more plausible interpretation is that the angle \(y\) and the angle represented by the sum `(angle in triangle) + (right angle)` are vertically opposite. However, the most direct interpretation is that \(x\) and \(y\) are adjacent angles on a straight line along with another angle.
- Let's assume the standard interpretation where the two intersecting lines form four angles. The angle \(x\) inside the triangle is vertically opposite to another angle \(x\) outside the triangle. Angle \(y\) is shown as another angle at that same intersection. If \(x\) and \(y\) are adjacent angles, they sum to 180\(^\circ\).
\[ x + y = 180^\circ \]
With \(x = 30^\circ\), we can find \(y\):
\[ 30^\circ + y = 180^\circ \] \[ y = 150^\circ \]
Now we can compare the quantities in Column A and Column B.


For Column A:
\(y = 150\).


For Column B:
\(4x = 4 \times 30 = 120\).


Comparison:

Column A is 150 and Column B is 120. Since \(150 > 120\), the quantity in Column A is greater.


Step 4: Final Answer:

By solving for \(x\) using the triangle and then for \(y\) using the straight-line property, we find that \(y=150\) and \(4x=120\). Therefore, Column A is greater.
Quick Tip: When a geometry diagram is complex, break it down into simpler shapes and identify the basic rules that apply to each (e.g., sum of angles in a triangle, angles on a straight line). Solve for one variable first, then use that result to find the others.

Step 1: Understanding the Concept:

This question asks us to compare two exponential expressions where the base and exponent are swapped in a way. The variable \(m\) is constrained to be a positive integer less than 4.


Step 2: Key Formula or Approach:

Since the variable \(m\) can only take on a few specific integer values (1, 2, and 3), the most direct method is to test each value and see if the relationship between the columns remains the same.


Step 3: Detailed Explanation:

The possible values for \(m\) are 1, 2, and 3.


Case 1: Let \(m=1\).

Column A: \((1+2)^1 = 3^1 = 3\).

Column B: \(1^{1+2} = 1^3 = 1\).

In this case, Column A is greater than Column B (\(3 > 1\)).


Case 2: Let \(m=2\).

Column A: \((2+2)^2 = 4^2 = 16\).

Column B: \(2^{2+2} = 2^4 = 16\).

In this case, the two columns are equal.


Case 3: Let \(m=3\).

Column A: \((3+2)^3 = 5^3 = 125\).

Column B: \(3^{3+2} = 3^5 = 243\).

In this case, Column B is greater than Column A (\(243 > 125\)).


Step 4: Final Answer:

We found a case where A > B, a case where A = B, and a case where B > A. Since the relationship changes depending on the value of \(m\), the relationship cannot be determined from the information given.
Quick Tip: When a variable in a quantitative comparison is restricted to a small, finite set of integers, the fastest and safest approach is often to plug in every possible value. If the results are inconsistent, the answer is (D).

Step 1: Understanding the Concept:

This question requires solving an algebraic equation for the variable \(x\) and then comparing the result to a given fraction.


Step 2: Key Formula or Approach:

The key to solving the equation is to recognize the relationship between the numerator and denominator of the fraction on the left side. The term \((1-x)\) is the negative of \((x-1)\).


Step 3: Detailed Explanation:

The equation is \(\frac{1-x}{x-1} = \frac{1}{x}\).

Let's simplify the left side. We can factor -1 out of the numerator:
\[ 1-x = -(-1+x) = -(x-1) \]
Substitute this back into the equation:
\[ \frac{-(x-1)}{x-1} = \frac{1}{x} \]
Since we are given that \(x \neq 1\), the term \((x-1)\) is not zero, so we can cancel it from the numerator and denominator:
\[ -1 = \frac{1}{x} \]
To solve for \(x\), multiply both sides by \(x\):
\[ -x = 1 \]
Multiply by -1 to get the final value for \(x\):
\[ x = -1 \]
Now we compare the columns.

Column A: \(x = -1\).

Column B: \(-\frac{1}{2}\).

On a number line, -1 is to the left of \(-\frac{1}{2}\). Therefore, -1 is less than \(-\frac{1}{2}\).


Step 4: Final Answer:

The value of \(x\) is -1. Since \(-1 < -\frac{1}{2}\), the quantity in Column B is greater.
Quick Tip: A common algebraic trick is recognizing expressions of the form \(\frac{a-b}{b-a}\). As long as \(a \neq b\), this fraction always simplifies to -1. Spotting this pattern instantly solves the equation.

Step 1: Understanding the Concept:

This question involves the statistical concept of a median. We need to use the definition of the median for an even-sized data set to find the value of one of the variables.


Step 2: Key Formula or Approach:

The median of a set of numbers is the middle value when the numbers are arranged in order. For a set with an even number of elements (like the 4 numbers here), the median is the average of the two middle elements.


Step 3: Detailed Explanation:

The data set is {10, 15, x, y. We are given that \(x < y\).

To find the median, we first need to order the numbers. We know \(10 < 15\). We need to figure out where \(x\) and \(y\) fit in this order.


Let's consider the possible ordered arrangements of the four numbers. The two middle numbers must average to 18.5.

- If the two middle numbers were 10 and 15, the median would be \((10+15)/2 = 12.5\), which is not 18.5. This means the set cannot be ordered as, for example, \(x, 10, 15, y\).
- This implies that one of the two middle numbers must be 15, and the other must be one of the variables.
- Let's assume the order is 10, 15, x, y. This is valid as long as \(15 \le x\). The two middle numbers are 15 and \(x\).
- Let's assume the order is 10, x, 15, y. This is valid as long as \(10 \le x \le 15\). The two middle numbers are \(x\) and 15.
- Let's assume the order is x, 10, y, 15. This is invalid since \(x
In the valid cases where the median is not 12.5, the two middle numbers are 15 and \(x\). (If \(y\) were a middle number, \(x\) would have to be smaller than it, making the other middle number 10 or 15. If it were 10, the median would be \((10+y)/2=18.5 \rightarrow y=27\), and the order would be `10, x, y, 15` with `x<27` and `x<15`. So `10,x,15,27` would be the order, and the middle numbers are x and 15, which brings us back to the same calculation).

Let's solve for \(x\) using the median formula:
\[ \frac{15 + x}{2} = 18.5 \]
Multiply both sides by 2:
\[ 15 + x = 37 \]
Subtract 15 from both sides:
\[ x = 22 \]
This result is consistent with the assumed order \(10, 15, x, y\) since \(15 < 22\). It requires \(y > 22\).

The value of \(x\) must be 22.


Step 4: Final Answer:

Column A has the value \(x=22\). Column B has the value 22. The two quantities are equal.
Quick Tip: For median problems with an even number of items, set up the equation \(\frac{middle1 + middle2}{2} = median\). Use the given numbers and variables to test possible orderings until you find one that is consistent.

Step 1: Understanding the Concept:

This is a "plug and chug" problem. We are given a formula and the values for the variables in it. We need to substitute the values into the formula and calculate the result.


Step 2: Key Formula or Approach:

The formula for the cost (\(C\)) is given as:
\[ C = 1.2p + 20h \]

Step 3: Detailed Explanation:

We are given the following values:

- Wholesale price of parts, \(p =
)15\).

- Number of hours, \(h = 2\).

Substitute these values into the cost formula:
\[ C = 1.2(15) + 20(2) \]
Calculate each term separately:

- The cost of parts: \(1.2 \times 15 = 18\).

- The cost of labor: \(20 \times 2 = 40\).

Add the two terms to find the total cost:
\[ C = 18 + 40 = 58 \]
The total cost of the repair is \(58.


Step 4: Final Answer:

The calculated cost is
)58, which corresponds to option (E).
Quick Tip: Be careful with the order of operations (PEMDAS/BODMAS). In this case, perform the multiplications first before adding the results.

Step 1: Understanding the Concept:

This question asks for the value of \(x\) that minimizes a given quadratic expression. The key is to understand the properties of squared terms.


Step 2: Key Formula or Approach:

The expression is \(8 + (x-3)^2\). The number 8 is a constant. To minimize the whole expression, we need to minimize the part that can vary, which is \((x-3)^2\).

The square of any real number is always greater than or equal to zero. The smallest possible value for a squared term is 0.


Step 3: Detailed Explanation:

We want to find the value of \(x\) that makes the expression \(8 + (x-3)^2\) as small as possible.

The term \((x-3)^2\) is always non-negative.

- If \((x-3)^2 > 0\), the total expression will be greater than 8.

- If \((x-3)^2 = 0\), the total expression will be equal to 8.

The minimum value of the expression is 8. This minimum occurs when the squared term is zero.

We need to set the squared term equal to zero and solve for \(x\):
\[ (x-3)^2 = 0 \]
Take the square root of both sides:
\[ x-3 = 0 \]
Add 3 to both sides:
\[ x = 3 \]
When \(x=3\), the expression has its minimum value of \(8 + (3-3)^2 = 8 + 0 = 8\).


Step 4: Final Answer:

The value of \(x\) that results in the least value for the expression is 3.
Quick Tip: The minimum (or maximum) value of a simple quadratic expression of the form \(a(x-h)^2 + k\) always occurs at \(x=h\). This is the vertex of the parabola. In this problem, the expression is \(1(x-3)^2 + 8\), so the minimum occurs at \(x=3\).

Step 1: Understanding the Concept:

This is a counting problem. We need to find the number of odd integers within a given inclusive range.


Step 2: Key Formula or Approach:

There are two main ways to solve this: by listing and counting, or by using a formula for arithmetic progressions.

The formula for the number of terms in a sequence is:
\[ Number of terms = \left(\frac{Last term - First term}{Common difference}\right) + 1 \]

Step 3: Detailed Explanation:

Method 1: Listing and Counting

The range is from 3 to 30, inclusive. The odd integers in this range are:

3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.

Counting these numbers, we find there are 14 odd integers.


Method 2: Using the Formula

This is an arithmetic sequence of odd numbers.

- The first term is 3.

- The last term is 29.

- The common difference between consecutive odd integers is 2.

Using the formula:
\[ Number of terms = \left(\frac{29 - 3}{2}\right) + 1 \] \[ Number of terms = \left(\frac{26}{2}\right) + 1 \] \[ Number of terms = 13 + 1 = 14 \]

Step 4: Final Answer:

There are 14 odd integers between 3 and 30, inclusive.
Quick Tip: For counting problems with inclusive ranges, the formula \(Last - First + 1\) gives the total number of integers. In the range 3 to 30, there are \(30-3+1 = 28\) integers. Since the range starts and ends with an odd and an even number, exactly half of them will be odd and half will be even. So, the number of odd integers is \(28 / 2 = 14\).

Step 1: Understanding the Concept:

This is a coordinate geometry problem involving the properties of a square. We are given the coordinates of two opposite vertices (A and C) and need to find the coordinates of one of the other vertices (B).


Step 2: Key Formula or Approach:

A key property of a square is that its diagonals are perpendicular and bisect each other. We can use this property.

1. Find the midpoint of the given diagonal AC. This point is also the midpoint of the other diagonal BE.

2. Find the vector from the midpoint to the known vertex C.

3. Rotate this vector by 90\(^\circ\) counter-clockwise to find the vector from the midpoint to the vertex B.

4. Add this vector to the midpoint's coordinates to find the coordinates of B.


Step 3: Detailed Explanation:

We are given the coordinates A(-6, 0) and C(4, 2).


1. Find the midpoint of AC.

The midpoint \(M\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \((\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})\).
\[ M = \left(\frac{-6+4}{2}, \frac{0+2}{2}\right) = \left(\frac{-2}{2}, \frac{2}{2}\right) = (-1, 1) \]

2. Find the vector from M to C.

A vector from point P to point Q is given by (Q_x - P_x, Q_y - P_y).

Vector \(\vec{MC} = \(4 - \(-1)\, 2 - 1)\ = \(5, 1)\).


3. Rotate the vector \(\vec{MC}\) by 90\(^\circ\).

A rotation of a vector \((x,y)\) by 90\(^\circ\) counter-clockwise gives the vector \((-y,x)\). The vertices of a square are in order, so to get from C to B, we go counter-clockwise.

Rotating \(\vec{MC} = \(5, 1)\) by 90\(^\circ\) gives us the vector \(\vec{MB}\).
\[ \vec{MB} = (-1, 5) \]

4. Find the coordinates of B.

The coordinates of B are found by adding the vector \(\vec{MB}\) to the coordinates of the midpoint M.
\[ B = M + \vec{MB} = (-1, 1) + (-1, 5) = (-1-1, 1+5) = (-2, 6) \]
The coordinates of point B are (-2, 6).


Step 4: Final Answer:

Using the properties of the diagonals of a square, we calculate the coordinates of B to be (-2, 6).
Quick Tip: Another method is to use slopes. Find the slope of AC (\(m_{AC}\)). The slope of the other diagonal, BE, will be the negative reciprocal (\(m_{BE} = -1/m_{AC}\)). You know the midpoint of BE is (-1,1). You can write the equation for the line BE and check which of the options for B lies on that line.

Step 1: Understanding the Concept:

This question asks to convert a number from scientific notation to standard decimal notation.


Step 2: Key Formula or Approach:

The expression \(a \times 10^n\), where \(n\) is a positive integer, means you multiply \(a\) by 10, \(n\) times. In practice, this means moving the decimal point in \(a\) to the right by \(n\) places, adding zeros as needed.


Step 3: Detailed Explanation:

The number is \(3.7 \times 10^7\).

We need to move the decimal point in 3.7 seven places to the right.

1. Start with 3.7

2. Move 1 place: 37.

3. We need to move 6 more places, so we add 6 zeros after the 7.
\[ 37,000,000 \]
So, \(3.7 \times 10^7 = 37,000,000\).


Step 4: Final Answer:

The number 37,000,000 corresponds to option (C).
Quick Tip: A simple way to think about it is that \(10^7\) is a 1 followed by 7 zeros (10,000,000). So you are calculating \(3.7 \times 10,000,000\). The 3 will be in the ten-millions place, and the 7 will be in the millions place.

Step 1: Understanding the Concept:

This question requires a careful visual inspection of the graph. We need to count the number of times the solid line (representing registered cars) moves downward from one year's data point to the next.


Step 2: Key Formula or Approach:

We will trace the solid line from left to right, starting after 1950, and count every instance where a line segment connecting two consecutive data points has a negative slope (i.e., it goes down). The graph plots several points between the 5-year markers, implying a year-by-year analysis is needed where available.


Step 3: Detailed Explanation:

- 1950-1972: The line shows a general upward trend. There are no year-over-year decreases visible in this period.

- Post-1972 dip: The graph shows a peak around 1972 and then a decline to the point at 1975. This represents at least one period of decrease. We will count this as one decrease.

- 1975-1980: The line shows a clear increase.

- 1980-1985: This segment shows multiple data points, which we can assume represent the individual years 1981, 1982, 1983, and 1984.
- From 1980 to 1981: The line goes down. This is decrease number two.
- From 1981 to 1982: The line goes down. This is decrease number three.
- From 1982 to 1983: The line goes down. This is decrease number four.
- From 1983 to 1984: The line goes down. This is decrease number five.
- From 1984 to 1985: The line goes up sharply.

- In total, we have counted 5 distinct periods of decrease after 1950.


Step 4: Final Answer:

By visually inspecting the graph and counting each downward trend between consecutive data points, we find a total of five decreases.
Quick Tip: Pay close attention to all plotted points on a graph, not just the ones with labels on the axis. The extra points are there for a reason and often contain the key to answering the question correctly, especially when the question asks about year-to-year changes.

Step 1: Understanding the Concept:

This question asks for a ratio between two values from the "Population per Registered Car" data series, which is represented by the dashed line on the graph.


Step 2: Key Formula or Approach:

1. Read the value from the dashed line for the year 1985.
2. Read the value from the dashed line for the year 1975.
3. Calculate the ratio: (Value in 1985) / (Value in 1975).


Step 3: Detailed Explanation:

- Value in 1985: Locate 1985 on the x-axis. Follow the vertical line up to the dashed line. The data point is slightly below the midpoint between the (inferred) 4 and 6 on the left-side scale. A reasonable estimate is approximately 4.8.
- Value in 1975: Locate 1975 on the x-axis. Follow the vertical line up to the dashed line. The data point is about halfway between 6 and 7. A reasonable estimate is approximately 6.5.
- Calculate the ratio: \[ Ratio = \frac{Value in 1985}{Value in 1975} \approx \frac{4.8}{6.5} \]
To compute this, we can multiply the numerator and denominator by 10 to get \(\frac{48}{65}\). \[ 48 \div 65 \approx 0.738 \]
- The calculated value, 0.738, is closest to the answer choice 0.75.


Step 4: Final Answer:

The ratio of the population per car in 1985 (approx. 4.8) to that in 1975 (approx. 6.5) is approximately 0.74, which is most nearly 0.75.
Quick Tip: When estimating values from a graph, write down your estimates before calculating. The answer choices are typically designed to be distinct enough that a good-faith estimate will lead you to the correct option.

Step 1: Understanding the Concept:

This question asks for the percent increase in the number of registered cars (solid line) between two specific years. There appears to be a typo in the question, as the low point of the dip is at 1975, not 1972. Assuming the question meant to ask for the increase from the low point in 1975 to 1985 allows for a clear calculation that matches an answer choice.


Step 2: Key Formula or Approach:

The formula for percent increase is: \[ Percent Increase = \left( \frac{New Value - Old Value}{Old Value} \right) \times 100% \]

Step 3: Detailed Explanation:

- Old Value (1975): Reading from the solid line at the 1975 mark, the number of registered cars is exactly on the line for 30 million.
- New Value (1985): The graph has a specific data label for this point: 43.1 million.
- Calculate the percent increase: \[ Percent Increase = \left( \frac{43.1 - 30}{30} \right) \times 100% \] \[ = \left( \frac{13.1}{30} \right) \times 100% \] \[ \approx 0.4367 \times 100% = 43.67% \]
- This result of 43.67% is "most nearly" 45%.


Step 4: Final Answer:

Assuming the question intended to measure the growth from the 1975 low point, the percent increase to 1985 is approximately 44%, which is closest to 45%.
Quick Tip: If a question on a standardized test seems to have a typo or refers to an ambiguous point on a graph, look for a nearby, clearly-defined point (like a labeled minimum or a gridline intersection). Often, using that clear point will lead you to one of the intended answers.

Step 1: Understanding the Concept:

The graph does not directly show the total population. To find it, we must combine the two pieces of information that are given: the number of cars and the population per car.


Step 2: Key Formula or Approach:

The relationship is: \[ Total Population = (Number of Registered Cars) \times (Population per Registered Car) \]
We will apply this formula to find the population in 1950 and 1985, and then calculate the difference.


Step 3: Detailed Explanation:

Population in 1950:
- Registered Cars (solid line): \(\approx\) 10 million.
- Population per Car (dashed line): Labeled as 13.2.
- Total Population (1950) \(\approx\) 10 million \(\times\) 13.2 = 132 million.

Population in 1985:
- Registered Cars (solid line): Labeled as 43.1 million.
- Population per Car (dashed line): Approximately 4.9 (estimated from the graph).
- Total Population (1985) \(\approx\) 43.1 million \(\times\) 4.9 \(\approx\) 211.2 million.

Population Increase:
- Increase = Population in 1985 - Population in 1950
- Increase \(\approx\) 211.2 million - 132 million = 79.2 million.

This result is closest to 80 million.


Step 4: Final Answer:

The approximate population increase from 1950 to 1985 is 79.2 million people, which is rounded to 80 million.
Quick Tip: This is a multi-step data problem. Break it down into smaller, manageable parts: find the first population, find the second population, then find the difference. Writing down each step helps to avoid errors.