Below is a set of 10 GRE-style numeric entry questions designed to help you practice and prepare for the GRE Quantitative Reasoning section, complete with step-by-step solutions and explanations.
Working together, two water pumps \(A\) and \(B\) can fill a water tank in \(3\) hours. Working alone, pump \(A\) can fill the tank in \(4\) hours. How long does it take pump \(B\), working alone, to fill the same tank?
Let the total volume of the tank be 1 unit.
Solving for \(\dfrac{1}{b}\):
\[ \dfrac{1}{b} = \dfrac{1}{3} - \dfrac{1}{4} = \dfrac{4 - 3}{12} = \dfrac{1}{12} \]Therefore, \(b = 12\) hours.
Answer: 12 hours
The values of \(x\) and \(y\) are related by the equation
\[ y = \dfrac{k}{x}, \]where \(k\) is a constant. If \(y = 45\) when \(x = 3\), what is the value of \(x\) when \(y = 180\)?
First, find the constant \(k\) using the given values \(y = 45\) and \(x = 3\):
\[ 45 = \dfrac{k}{3} \implies k = 45 \times 3 = 135 \]Next, find \(x\) when \(y = 180\):
\[ 180 = \dfrac{135}{x} \implies x = \dfrac{135}{180} = \dfrac{3}{4} = 0.75 \]Answer: 0.75 (or \( \dfrac{3}{4}\))
A rectangle has a length that is one third of its perimeter. The perimeter is \(150\) meters. What is the area of the rectangle?
Substitute \(P = 150\) and \(L = 50\):
\[ 150 = 2(50 + W) \implies 75 = 50 + W \implies W = 25 \text{ meters} \]Area = \(L \times W = 50 \times 25 = 1250\) square meters.
Answer: 1250
The square of the sum of two numbers is \(289\). The product of the two numbers is \(66\). What is the sum of the squares of the two numbers?
Let the two numbers be \(a\) and \(b\).
We use the algebraic identity:
\[ (a + b)^2 = a^2 + b^2 + 2ab \]Substitute the known values into the identity:
\[ 289 = a^2 + b^2 + 2(66) \] \[ 289 = a^2 + b^2 + 132 \implies a^2 + b^2 = 289 - 132 = 157 \]Answer: 157
In a certain country, in one year, \(30\%\) of the total money spent on energy was spent on generating electricity. If \(30\) billion dollars were spent on energy, how much money was spent on generating electricity?
Calculate \(30\%\) of 30 billion dollars:
\[ 30\% \text{ of } 30 = \dfrac{30}{100} \times 30 = 9 \text{ billion dollars} \]Answer: 9
The sum of three consecutive odd integers is \(249\). Find the largest of these integers.
Let the three consecutive odd integers be represented as \(x - 2\), \(x\), and \(x + 2\).
Their sum is:
\[ (x - 2) + x + (x + 2) = 249 \implies 3x = 249 \implies x = 83 \]The three integers are \(81\), \(83\), and \(85\). The largest of these integers is \(85\).
Answer: 85
The sum of two numbers is \(3.6\) and the difference of these numbers is \(1.2\). Find the larger of the two numbers.
Let the two numbers be \(x\) and \(y\) with \(x > y\):
\[ x + y = 3.6 \] \[ x - y = 1.2 \]Add both equations together to eliminate \(y\):
\[ (x + y) + (x - y) = 3.6 + 1.2 \implies 2x = 4.8 \implies x = 2.4 \]Answer: 2.4
\(20\%\) of what number is \(125\)?
Let the unknown number be \(N\):
\[ 0.20 \times N = 125 \implies N = \dfrac{125}{0.20} = 125 \times 5 = 625 \]Answer: 625
The price of a shirt was first decreased by \(10\%\) and then decreased a second time by \(15\%\). What was the original price of the shirt if the final price is \(22\) dollars?
(Round your answer to the nearest cent.)
Let \(P\) be the original price.
Set this equal to the final price of \(22\) dollars:
\[ 0.85 \times 0.90P = 22 \implies 0.765P = 22 \] \[ P = \dfrac{22}{0.765} \approx 28.758 \dots \]Rounding to the nearest cent gives \(28.76\).
Answer: 28.76
The average of
\[ \dfrac{1}{2}, \quad \dfrac{1}{4}, \quad \dfrac{2}{3}, \quad \text{and } x \]is equal to
\[ \dfrac{3}{4}. \]Find the value of \(x\).
Since there are 4 numbers, their sum divided by 4 equals \(\dfrac{3}{4}\). Thus, their total sum is:
\[ \text{Sum} = 4 \times \dfrac{3}{4} = 3 \]Now, add the given fractions and \(x\):
\[ \dfrac{1}{2} + \dfrac{1}{4} + \dfrac{2}{3} + x = 3 \]Find a common denominator (which is 12) for the fractions:
\[ \dfrac{6}{12} + \dfrac{3}{12} + \dfrac{8}{12} + x = 3 \implies \dfrac{17}{12} + x = 3 \]Solve for \(x\):
\[ x = 3 - \dfrac{17}{12} = \dfrac{36}{12} - \dfrac{17}{12} = \dfrac{19}{12} \]Answer: \( \dfrac{19}{12}\) (or approximately 1.583)