Solutions and detailed explanations to GRE numeric questions and problems in sample 1

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?
Solution
Pump A can fill the tank in 4 hours, therefore the quarter of the tank is filled in one hour, hence the rate of pump A in filling the tank is
1 / 4
If T is the number of hours for pump B to fill the tank, then its rate is
1 / T
When working together for 3 hours both pumps are working at their rates to fill 1 tank. Hence
3(1 / 4) + 3(1 / T) = 1
The term 3(1 / 4) in the above equation is due to pump A working at its rate for 3 hours. The term 3(1 / T) is due to pump B and the "1" on the right of the equation corresponds to 1 tank. We now solve the above equation for T
3(1 / T) = 1  3 / 4
3(1 / T) = 1 / 4
1 / T = 1 / 12
T = 12 hours

The values of x and y are related by the equation y = k / x, where k is a constant. If y = 45 when x = 3, what is the value of x when y = 180?
Solution
"y = 45 when x = 3" is used to find the constant k by substituting y and x by their values in the equation y = k / x.
45 = k / 3
Solve for k
k = 3 * 45 = 135
We now use the same equation with known value of k to find x when y = 180 as follows
180 = 135 / x
Solve for x
x = 135 / 180 = 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?
Solution
Let L, W and P be the length, width and perimeter of the rctangle. Hence "a rectangle has a length that is one third of its perimeter" is translated as follows
L = P / 3 = 150 / 3 = 50
The perimeter P is given by the formula
P = 2L + 2W
Substitute P by 150 and L by 50 and solve for W
150 = 2 * 50 + 2 W
W = 25
The area A of the rectangle is given by
A = L W = 50 * 25 = 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?
Solution
Let x and y be the two numbers. "The square of the sum of two numbers is 289" is translated as follows
(x + y)^{2} = 289
Expand the left side of the above equation
x^{2} + y^{2} + 2 x y = 289
x y is the product of the two numbers and is given. Hence
x^{2} + y^{2} + 2 (66) = 289
Which gives
x^{2} + y^{2} = 157
Hence the sum of the square of x and y is 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, in billions, was spent on generating electricity?
Solution
30% of the money spent on energy is given by
30% * 30 = 9

The sum of three consecutive odd integers is 249. Find the largest of these numbers.
Solution
Let x be the smallest of these numbers. x + 2 and x + 4 will the next two odd integers. Hence
x + (x + 2) + (x + 4) = 249
Solve for x the above equation.
3x + 2 + 4 = 249
3x = 243
x = 81
The largest of these numbers is x + 4 and its value is
x + 4 = 81 + 4 = 85

The sum of two numbers is 3.6 and the difference of these numbers is 1.2. Find the largest of these numbers.
Solution
Let x and y be the two numbers. Hence
x + y = 3.6 and x  y = 1.2
Solve the above system of equations by adding the left sides and right sides of the two equations
(x + y) + (x  y) = 3.6 + 1.2
2x = 4.8
x = 2.4
Use equation x + y = 3.6 to find y
y = 3.6  2.4 = 1.2
The largest of these numbers is 2.4

20% of what number is 125?
Solution
Let x the number. Hence
20% x = 125
Solve for x
(20 / 100) x = 125
x = 125 * 100 / 20 = 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 answer to the nearest cent)
Solution
Let x be the original price. The price after the first reduction of 10% is given by
x  10% x = x  (10/100)x = x  0.1x
The price after the second reduction of 15% is given by
(x  0.1x)  15% (x  0.1x) = x  0.1x  (15/100)(x  0.1x)
= x  0.1x  0.15(x  0.1x)
= x  0.1x  0.15x + 0.015x
= 0.765x
The final price is 22 dollars. Hence
0.765x = 22
Solve for x
x = 28.7581
Rounded to the nearest cent, the original price x is equal to
28.76 dollars

The average of 1/2, 1/4, 2/3 and x is equal to 3/4. Find x.
Solution
The average of 1/2, 1/4, 2/3 and x is given by
(1/2 + 1/4 + 2/3 + x) / 4
and is equal to 3/4. Hence
(1/2 + 1/4 + 2/3 + x) / 4 = 3 / 4
Solve for x. First multiply both sides of the equation by 4 ans simplify
(1/2 + 1/4 + 2/3 + x) = 3
x = 3  (1/2 + 1/4 + 2/3)
Set all fractions to common denominator
x = 36/12  (6/12 + 3/12 + 8/12) = 36/12  17/12
x = 19/12
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