A set of 10 questions similar to GRE quantitative comparison reasoning questions, complete with detailed solutions and explanations, is presented below.
When solving quantitative comparison questions, you are asked to compare two quantities – Quantity A and Quantity B – and then determine which of the following statements describes the relationship:
Evaluate expressions in A and B:
A: \[ 2(3 + 4)^2 + 10 = 2(7)^2 + 10 = 2(49) + 10 = 98 + 10 = 108 \]
B: \[ 2 + (3 + 4)^2 + 10 = 2 + 49 + 10 = 61 \]
Quantity A is greater than Quantity B.
Answer (A)
Simplify expressions in A and B:
A: \[ 2 \times 3^{10} + 2 \times 3^{10} + 2 \times 3^{10} = 2 \times \left( 3^{10} + 3^{10} + 3^{10} \right) = 2 \times \left( 3 \times 3^{10} \right) = 2 \times 3^{11} \]
B: \[ 3 \times 3^{10} = 3^{11} \]
Quantity A is greater than Quantity B.
Answer (A)
Evaluate expressions in A and B:
A: \[ 25\% \text{ of } 300 = \dfrac{25}{100} \times 300 = \dfrac{25 \times 300}{100} = 75 \]
B: \[ \text{three quarters of } 200 = \dfrac{3}{4} \times 200 = \dfrac{3 \times 200}{4} = 150 \]
Quantity B is greater than Quantity A.
Answer (B)
ABC is a triangle such that the measure of angle A is \(45^\circ\). The measure of angle C is twice the measure of angle B.
\[ \begin{array}{|l|l|} \hline \textbf{Quantity A} & \textbf{Quantity B} \\ \hline \text{The measure of angle } A & \text{The measure of angle } B \\ \hline \end{array} \]Quantity A is known (\(45^\circ\)). Let us calculate Quantity B. The measure of angle C is twice the measure of angle B:
\[ C = 2B \]The sum of the angles in triangle ABC is \(180^\circ\):
\[ A + B + C = 180 \]Substitute \(A = 45\) and \(C = 2B\):
\[ 45 + B + 2B = 180 \implies 3B = 135 \implies B = \dfrac{135}{3} = 45^\circ \]The measures of angles A and B are equal.
Answer (C)
\(x\) is a variable that may take any real value.
\[ \begin{array}{|l|l|} \hline \textbf{Quantity A} & \textbf{Quantity B} \\ \hline x^2 + 1 & 100x + 1 \\ \hline \end{array} \]Evaluate expressions in A and B for different values of \(x\):
For \(x = 0\):
For \(x = 1\):
Different values of \(x\) yield different comparisons. Therefore, the relationship cannot be determined.
Answer (D)
Given only the perimeter of a rectangle, its dimensions (length and width) can vary arbitrarily, meaning its area cannot be uniquely determined. Thus, the relationship cannot be determined from the information given.
Answer (D)
\(x = -10^{-2}\)
\[ \begin{array}{|c|c|} \hline \textbf{Quantity A} & \textbf{Quantity B} \\ \hline x^3 & x^2 \\ \hline \end{array} \]Evaluate expressions in A and B for the given value of \(x\):
A: \[ x^3 = (-10^{-2})^3 = (-1)^3(10^{-2})^3 = -10^{-6} \]
B: \[ x^2 = (-10^{-2})^2 = (-1)^2(10^{-2})^2 = 10^{-4} \]
Since negative numbers are always less than positive numbers, Quantity B is greater than Quantity A.
Answer (B)
Convert units to centimeters (cm) first:
Calculate volume using \(V = \pi \times \text{radius}^2 \times \text{length}\):
A: \[ \pi \times 1^2 \times 10^4 = 10^4\pi \]
B: \[ \pi \times 10^2 \times 10^3 = 10^2 \times 10^3\pi = 10^5\pi \]
The volume in B is greater than the volume in A.
Answer (B)
Since \(x\) is positive, multiplying all terms of \(0 < x < 1\) by \(x\) gives:
\[ 0 < x^2 < x \]Taking the reciprocals of positive inequality terms reverses the inequality sign:
\[ \dfrac{1}{x^2} > \dfrac{1}{x} \]Quantity A is greater than Quantity B.
Answer (A)
A bag contains green, blue, and yellow balls. The ratio of green to blue balls is \(2:7\). The ratio of green to yellow balls is \(3:5\).
\[ \begin{array}{|c|c|} \hline \textbf{Quantity A} & \textbf{Quantity B} \\ \hline \text{Number of blue balls} & \text{Number of yellow balls} \\ \hline \end{array} \]Let \(g\), \(b\), and \(y\) be the numbers of green, blue, and yellow balls. From the given ratios:
\[ \dfrac{g}{b} = \dfrac{2}{7} \implies \dfrac{b}{g} = \dfrac{7}{2} \] \[ \dfrac{g}{y} = \dfrac{3}{5} \]Multiply the fractions \(\dfrac{g}{y}\) and \(\dfrac{b}{g}\):
\[ \dfrac{g}{y} \cdot \dfrac{b}{g} = \dfrac{3}{5} \cdot \dfrac{7}{2} \implies \dfrac{b}{y} = \dfrac{21}{10} \]Since \(\dfrac{b}{y} = \frac{21}{10} > 1\), the number of blue balls (A) is greater than the number of yellow balls (B).
Answer (A)