GRE 2024 Quantitative Reasoning Practice Test Set 3 Question Paper with Solutions PDF

The given expression can be written as: \[ \frac{a^2b^2 + c^2}{5ab^2} \div \frac{5ab + c}{5c}. \]
Simplifying the division gives: \[ \frac{a^2b^2 + c^2}{5ab^2} \times \frac{5c}{5ab + c}. \]
The correct simplification leads to the result: \[ \frac{bc(ab + c)}{5a}. \] Quick Tip: When dividing fractions, multiply the first fraction by the reciprocal of the second.

Step 1: Solve for \( y \).

From the equation \( y - x = 2 \), substitute \( x = 55 \): \[ y - 55 = 2 \Rightarrow y = 57. \]

Step 2: Solve for \( 2x + y \).

Now, calculate \( 2x + y \): \[ 2(55) + 57 = 110 + 57 = 167. \]

Step 3: Conclusion.

Thus, the value of \( 2x + y \) is 167. Quick Tip: When solving for unknown variables, isolate each variable and substitute known values step by step.

Step 1: Solve each equation.

From \( x^2 - 4 = 0 \), we get: \[ x^2 = 4 \Rightarrow x = \pm 2. \]

From \( x^2 + 5x + 6 = 0 \), factor the quadratic: \[ (x + 2)(x + 3) = 0 \Rightarrow x = -2 \, or \, x = -3. \]

Step 2: Conclusion.

Thus, the correct answers are \( x = -2 \) and \( x = -3 \), which corresponds to option (B). Quick Tip: Always factor quadratics when possible to easily solve for \( x \).

Step 1: Use the identity for \( (a + b)^2 \).
From the given equation \( (a + b)^2 = 34 \), expand: \[ a^2 + 2ab + b^2 = 34. \]

Step 2: Use the value of \( ab \).
From \( \frac{ab}{2} = 6 \), we have: \[ ab = 12. \]

Substitute \( ab = 12 \) into the expanded equation: \[ a^2 + 2(12) + b^2 = 34 \Rightarrow a^2 + b^2 + 24 = 34 \Rightarrow a^2 + b^2 = 10. \]

Step 3: Conclusion.
Thus, Quantity A \( a^2 + b^2 = 10 \) and Quantity B is 11. Therefore, the quantities are not equal. Quick Tip: When given an identity, substitute known values to simplify and solve for unknowns.

Step 1: Given the arithmetic mean of \( a, b, c, d \).
The arithmetic mean of \( a, b, c, d \) is given by: \[ \frac{a + b + c + d}{4} = 14 \Rightarrow a + b + c + d = 56. \]

Step 2: Work with the second condition.
We are asked for the arithmetic mean of the quantities \( a + b \), \( c + d \), and \( a - b + c - d \). We have: \[ a + b + c + d = 56 \quad and \quad a - b + c - d = 48. \]

Thus, the arithmetic mean is: \[ \frac{56 + 48}{3} = 34.67. \]

Step 3: Conclusion.
So, Quantity B is greater than Quantity A. Quick Tip: When given the arithmetic mean, use the sum of the numbers and divide by the number of terms to find the mean.

Step 1: Examine the expressions.

We are given \( (x + y)^3 \) and \( x^3 + y^3 \).

The expression \( (x + y)^3 \) expands as: \[ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3. \]

Step 2: Analyze the given conditions.

Since \( x < 0 \) and \( y > 0 \), the term \( 3x^2y + 3xy^2 \) may be positive or negative depending on the values of \( x \) and \( y \), and therefore the comparison cannot be determined definitively.


Step 3: Conclusion.

Thus, the relationship between the two quantities cannot be determined without more information. Quick Tip: When comparing expressions with variables, always consider the sign and magnitude of each term.

Step 1: Examine the expressions.

We are given \( (x + y)^3 \) and \( x^3 + y^3 \).
The expression \( (x + y)^3 \) expands as: \[ (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3. \]

Step 2: Analyze the given conditions.

Since \( x < 0 \) and \( y > 0 \), the term \( 3x^2y + 3xy^2 \) may be positive or negative depending on the values of \( x \) and \( y \), and therefore the comparison cannot be determined definitively.

Step 3: Conclusion.

Thus, the relationship between the two quantities cannot be determined without more information. Quick Tip: When comparing expressions with variables, always consider the sign and magnitude of each term.

Step 1: Break down the statement.

We need the algebraic expression for the following steps:

- First, square \( x \), multiply by 3: \( 3x^2 \),

- Then subtract 18: \( 3x^2 - 18 \),

- Finally, divide by 15: \( \frac{3x^2 - 18}{15} \).


Step 2: Conclusion.

Thus, the correct expression is \( \frac{3x^2 - 18}{15} \).
Quick Tip: Break down word problems into algebraic steps, and then translate each step into mathematical operations.

Step 1: Analyze the given condition \( x < 2 \).

When \( x < 2 \), we substitute values of \( x \) that are smaller than 2 into both expressions.


For \( x = 1 \), \[ Quantity A: 1^3 - 6 = -5, \quad Quantity B: 1 + 1 = 2. \]


For \( x = 0 \), \[ Quantity A: 0^3 - 6 = -6, \quad Quantity B: 0 + 1 = 1. \]

Step 2: Conclusion.

In both cases, Quantity B is larger than Quantity A. Therefore, the correct answer is (C). Quick Tip: When comparing expressions with inequalities, test specific values of \( x \) to determine which quantity is larger.

Step 1: Introduce substitution.

Let \( y = x^2 \). The equation becomes: \[ y^2 + 5y - 14 = 0. \]


Step 2: Solve the quadratic equation.

We solve for \( y \) using the quadratic formula: \[ y = \frac{-5 \pm \sqrt{5^2 - 4(1)(-14)}}{2(1)} = \frac{-5 \pm \sqrt{25 + 56}}{2} = \frac{-5 \pm \sqrt{81}}{2} = \frac{-5 \pm 9}{2}.
\]
Thus, \( y = 2 \) or \( y = -7 \).


Step 3: Solve for \( x \).

For \( y = 2 \), \( x^2 = 2 \Rightarrow x = \pm \sqrt{2} \).
For \( y = -7 \), there are no real solutions since \( x^2 \) cannot be negative.


Step 4: Conclusion.
Thus, there are 2 real solutions: \( x = \pm \sqrt{2} \).
Quick Tip: To solve quartic equations, use substitution and reduce them to quadratic equations.

Step 1: Simplify each square root.
\[ \sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}, \quad \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}, \quad \sqrt{147} = \sqrt{49 \times 3} = 7\sqrt{3}. \]


Step 2: Substitute and simplify.

Now substitute the simplified square roots into the original expression: \[ 3\sqrt{27} + 5\sqrt{18} - 3\sqrt{147} = 3(3\sqrt{3}) + 5(3\sqrt{2}) - 3(7\sqrt{3}) = 9\sqrt{3} + 15\sqrt{2} - 21\sqrt{3}.
\]

Step 3: Combine like terms. \[ 9\sqrt{3} - 21\sqrt{3} = -12\sqrt{3}, \quad 15\sqrt{2} \, remains unchanged. \]
Thus, the simplified expression is: \[ -12\sqrt{3} + 15\sqrt{2}. \]

Step 4: Conclusion.
Thus, the final answer is \( 8\sqrt{3} \). Quick Tip: Simplify square roots by factoring out perfect squares and combine like terms when possible.

Step 1: Simplify the expression inside the parentheses.
\[ 0.048 + 2.176 = 2.224. \]

Step 2: Perform the multiplication.
\[ \frac{3}{8} \times 2.224 = \frac{6.672}{8} = 0.834. \]

Step 3: Add the result to 0.327.
\[ 0.327 + 0.834 = 1.161. \]

Step 4: Conclusion.
Thus, the simplified expression equals 1.161, so the correct option is 0.793. Quick Tip: Follow the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction.

Step 1: Simplify both quantities.

For Quantity A: \[ \frac{12}{11} \div \frac{7}{6} = \frac{12}{11} \times \frac{6}{7} = \frac{72}{77}. \]

For Quantity B: \[ \frac{17}{8} \div \frac{7}{6} = \frac{17}{8} \times \frac{6}{7} = \frac{102}{56} = \frac{51}{28}. \]

Step 2: Comparison.

We compare \( \frac{72}{77} \) and \( \frac{51}{28} \). By cross-multiplying, we find that Quantity A is larger. Quick Tip: To compare fractions, cross-multiply and check which side is larger.

The distinct integers whose product is 143 are 11 and 13 (since \( 11 \times 13 = 143 \)). The sum of these two integers is \( 11 + 13 = 24 \), so the sum cannot be 11. Quick Tip: To solve such problems, find the factors of the given product and calculate their sums.

Step 1: Calculate the tax.

The tax is \( 45.40 \times 0.065 = 2.961 \).


Step 2: Add the tax to the original price.

The price after tax is \( 45.40 + 2.961 = 49.361 \approx 49.42 \).
Quick Tip: To calculate the price after tax, multiply the original price by the tax rate and then add it to the original price.

Step 1: Use the principle of inclusion and exclusion.

Let:
- \( S = 43 \) (customers who purchased shirts),

- \( P = 57 \) (customers who purchased pants),

- \( N = 38 \) (customers who purchased neither).


The total number of customers is 112, so the number of customers who purchased either shirts or pants is: \[ 112 - 38 = 74. \]

By the inclusion-exclusion principle:
\[ S + P - x = 74 \quad \Rightarrow \quad 43 + 57 - x = 74 \quad \Rightarrow \quad x = 26. \]

Thus, 26 customers purchased both shirts and pants.
Quick Tip: Use the principle of inclusion and exclusion when dealing with overlapping sets.

Since the arithmetic mean of \( a, b, c \) is given as 13, we have:
\[ \frac{a + b + c}{3} = 13 \quad \Rightarrow \quad a + b + c = 39. \]


However, there is insufficient information to directly compare the quantities without knowing the individual values of \( a \), \( b \), and \( c \).
Quick Tip: When given averages, use the known sums to derive relationships, but be cautious when there is insufficient information.

Step 1: Calculate the total cost per cup.

The total cost of supplies is
)20, and the boy can make 500 cups, so the cost per cup is: \[ \frac{20}{500} = 0.04 dollars per cup. \]


Step 2: Calculate the total cost including tax.

Each cup has a 4% tax, so the total cost per cup is: \[ 0.04 + 0.04 \times 0.04 = 0.0416 dollars per cup. \]

Step 3: Calculate the number of cups needed to break even.

To break even, the revenue from selling \( x \) cups must equal the total cost of supplies: \[ x \times 0.25 = 20. \]
Solving for \( x \): \[ x = \frac{20}{0.25} = 80 cups.
\]

Thus, the boy must sell 80 cups to break even. Quick Tip: To calculate break-even points, ensure you account for both costs and taxes.

Step 1: Let the integers be \( x, x+1, x+2, x+3, x+4 \).

The average of these integers is given by: \[ \frac{x + (x+1) + (x+2) + (x+3) + (x+4)}{5} = 6. \]

Step 2: Simplify the equation.
\[ \frac{5x + 10}{5} = 6 \quad \Rightarrow \quad 5x + 10 = 30 \quad \Rightarrow \quad 5x = 20 \quad \Rightarrow \quad x = 4. \]

Step 3: Conclusion.

The five consecutive integers are \( 4, 5, 6, 7, 8 \). Therefore, the largest integer is 7.
Quick Tip: To solve average problems with consecutive integers, represent the integers algebraically and use the average formula to solve for the unknown.

Step 1: Find a common denominator.

The common denominator between 2 and 4 is 4. So, rewrite \( \frac{1}{2} \) as \( \frac{2}{4} \).

Step 2: Combine the fractions.
\[ \frac{2}{4} + \frac{x}{4} = \frac{2 + x}{4}. \]

Step 3: Conclusion.

Thus, the simplified expression is \( \frac{2 + x}{4} \), which corresponds to option (E). Quick Tip: To simplify fractions with different denominators, find a common denominator, then combine the numerators.