GRE 2024 Quantitative Reasoning Practice Test Set 1 Question Paper with Solutions PDF

Step 1: Analyzing the equation.

We are given that \( 10^x = 10,000,000,000 \). First, express \( 10,000,000,000 \) as a power of 10: \[ 10^x = 10^{10}. \]

Step 2: Compare Quantity A and Quantity B.

This implies \( x = 10 \), so \[ Quantity A = 10, \quad Quantity B = 12. \]

Step 3: Conclusion.

Clearly, \( 10 < 12 \), hence Quantity B is greater. Quick Tip: When comparing quantities involving powers of 10, express the numbers in terms of the same base to simplify comparisons.

Step 1: Analyze the equation.

We are given the equation \( 10y + 20x = 50 \). This is the equation of a line in the standard form. To find the y-intercept and slope, we will rewrite it in slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
\[ 10y = -20x + 50 \quad \Rightarrow \quad y = -2x + 5. \]

Step 2: Identify the quantities.

From the equation \( y = -2x + 5 \), we see that: \[ Quantity A (y-intercept) = 5, \quad Quantity B (slope) = -2. \]

Step 3: Conclusion.

The relationship between the y-intercept and the slope cannot be determined just from this equation without additional context. Thus, we cannot conclude a definitive comparison between the two quantities. Quick Tip: When comparing quantities involving equations of lines, identify the key components (slope and y-intercept) and remember that without further information, their relationship may not be clear.

Step 1: Analyzing the figure.

We are given that the figure is not drawn to scale. This means we cannot use visual cues to compare the quantities.

Step 2: Identify the quantities.

- Quantity A is \( \frac{y}{x} \),
- Quantity B is 3.

Step 3: Conclusion.

Since the figure is not drawn to scale and we do not have further details about the relationship between \( x \) and \( y \), the relationship between the two quantities cannot be determined. Quick Tip: When comparing quantities based on figures, always ensure that the figure is drawn to scale or that you have enough numerical data to make a comparison.

Step 1: Calculate the total tax paid by Jane.

To calculate the tax paid by Jane, we need to compute the tax paid in each month and sum them up. The tax paid in each month is:
- January: \( 10,000 \times 10% = 1,000 \)
- February: \( 50,000 \times 30% = 15,000 \)
- March: \( 20,000 \times 20% = 4,000 \)
- April: \( 10,000 \times 10% = 1,000 \)
- May: \( 30,000 \times 20% = 6,000 \)
- June: \( 90,000 \times 40% = 36,000 \)

The total tax paid is: \[ 1,000 + 15,000 + 4,000 + 1,000 + 6,000 + 36,000 = 63,000. \]

Step 2: Find the average tax paid.

The total income earned is: \[ 10,000 + 50,000 + 20,000 + 10,000 + 30,000 + 90,000 = 210,000. \]
The average tax paid is: \[ \frac{63,000}{6} = 10,500. \]

Step 3: Compare with Quantity B.

Quantity B is 22% of the average income: \[ 22% of 35,000 = 22% \times 35,000 = 7,700. \]

Step 4: Conclusion.

The average tax paid by Jane is 10,500, which is greater than 7,700. Hence, Quantity B is greater. Quick Tip: To calculate averages and percentages accurately, remember to first calculate individual values before summing them up for the average.

Step 1: Identify the smallest composite number greater than 2.

The smallest composite number greater than 2 is 4.

Step 2: Identify the smallest prime number less than 10.

The smallest prime number less than 10 is 2.

Step 3: Calculate \( p \times q \).

We have \( p = 2 \) and \( q = 4 \), so: \[ p \times q = 2 \times 4 = 8. \]

Step 4: Conclusion.

The correct answer is \( 8 \), so there might be an issue with the options or extraction. The correct solution is as described, so please check the options. Quick Tip: Remember to check the definitions of prime and composite numbers to avoid confusion when identifying them.

Step 1: Analyze the inequality.

We are given the inequality \( \frac{1}{2^n} > 1 \), which implies that \( 2^n < 1 \).

Step 2: Solve for \( n \).

Since \( 2^n < 1 \), it follows that \( n < 0 \).

Step 3: Conclusion.

The value \( n = -1/2 \) satisfies the inequality, so the correct answer is \( (B) \). Quick Tip: To solve inequalities involving exponents, consider the behavior of the base when the exponent is negative or fractional.

Step 1: Analyze the figure.

The lines \( l \) and \( m \) are parallel, and the angle \( d \) is given as 45°. The angle formed by line \( RS \) is a right angle with line \( m \).

Step 2: Conclusion.

Since the two lines are parallel, angle \( a \) is supplementary to angle \( d \), and hence, angle \( a \) must be 90°. Quick Tip: When dealing with parallel lines and angles, remember that alternate interior angles and supplementary angles play an important role.

Step 1: Analyze the given information.

We are given that lines \( l \) and \( m \) are parallel, \( O \) is the center of the circle, and the measure of angle \( d \) is 45°. The length of line \( RS \) is \( \frac{\sqrt{2}}{2} \) and line \( RS \) forms a right angle with line \( m \).

Step 2: Conclusion.

Using basic trigonometric relationships in the right triangle, we can deduce that the length of line \( PR \) is \( \sqrt{2} \). Quick Tip: In right triangles, the length of the hypotenuse is often related to the legs by the Pythagorean theorem or trigonometric ratios like sine, cosine, and tangent.

Step 1: Analyze the figure.

We know that line \( RS \) is a radius of the circle, and it forms a right angle with line \( m \). Using the properties of a right triangle and the given length of line \( RS \), we can deduce that the diameter is \( \sqrt{2} \).

Step 2: Conclusion.

The diameter of the circle is \( \sqrt{2} \). Quick Tip: When dealing with right triangles and circles, always remember the relationship between the radius and diameter, and use trigonometry when applicable.

Step 1: Understanding \( A \cup B \).

The notation \( A \cup B \) refers to the union of the two sets \( A \) and \( B \), meaning all students who are either in liberal arts or life science majors.

Step 2: Conclusion.

The correct interpretation of \( A \cup B \) is all students who are either liberal arts or life science majors. Quick Tip: In set theory, the union \( A \cup B \) represents all elements that belong to either set \( A \) or set \( B \), or both.

Step 1: Understanding \( A \cap B \).

The notation \( A \cap B \) refers to the intersection of the two sets \( A \) and \( B \), meaning all students who are both liberal arts and life science majors.

Step 2: Conclusion.

The correct interpretation of \( A \cap B \) is the set of students who are double-majors in both liberal arts and life sciences. Quick Tip: In set theory, the intersection \( A \cap B \) represents all elements that belong to both sets \( A \) and \( B \).

Step 1: Analyze the given expression.

We are given the expression \( A \times B = (AB)^2 + (A + B)^2 \), but without specific values for \( A \) and \( B \), it is not possible to determine which quantity is greater.

Step 2: Conclusion.

Since no specific values are provided for \( A \) and \( B \), the relationship between the two quantities cannot be determined. Quick Tip: When comparing quantities with variables, always check if you have enough information to solve for specific values before drawing conclusions.

Step 1: Assign variables.

Let the number of female students be \( f \). Then male students = \( 2f \).
Let the number of US citizen students be \( x \), then international students = \( 3x \).

Step 2: Total students equation.
\[ f + 2f = 3f = total students = 36 \Rightarrow f = 12, \quad 2f = 24. \]

So, female students = 12.

Step 3: Now apply the international/US citizen relationship.
\[ x + 3x = 4x = 36 \Rightarrow x = 9. \]

So, US citizen students = 9.

Step 4: Comparison.

Quantity A = 9, Quantity B = 12 \[ \Rightarrow Quantity B is greater. \]

Wait! There seems to be a contradiction. Since both variables depend on the same total of 36, and give different values (female = 12, US = 9), we must **reconcile the constraint**.

Let's try solving the equation combining both relationships:
- Let \( f \) be number of female students,
- Then male = \( 2f \)
- Total students = \( f + 2f = 3f \)
- Also, let US students = \( x \), international = \( 3x \)
- Total students = \( x + 3x = 4x \)

So, both \( 3f = 36 \) and \( 4x = 36 \) hold. \[ f = 12, \quad x = 9 \Rightarrow Quantity A (US) = 9, Quantity B (female) = 12. \]

Step 5: Final conclusion.

So, \[ \boxed{Quantity B is greater.} \] Quick Tip: When working with multiple relationships in a word problem, assign variables, translate each constraint into an equation, and solve systematically.

Step 1: Analyze the given.

We are not given the radius of the circle nor any dimensions or angles related to triangle QRS.

Step 2: Formula review.

- Half the circumference = \( \frac{1}{2} \cdot 2\pi r = \pi r \)
- Area of triangle = \( \frac{1}{2}ab\sin C \), or other forms—but no sides or angles are given

Step 3: Conclusion.

Since we lack numerical or geometric data, we cannot compare the two quantities. Quick Tip: If quantities depend on unknown values with no further constraints or relationships, the correct choice is usually: cannot be determined.

Step 1: Analyze the information.

We are told a graph shows GDP data, but the actual graph is missing or not provided.

Step 2: Attempting estimation.

We have no numerical values or bar heights. Thus, it’s impossible to determine the actual GDP values or compare them.

Step 3: Conclusion.

Since no data is visible or provided, we cannot determine the relationship. Quick Tip: For questions referencing graphs or tables, always ensure you have access to the actual data before making a comparison.