Solutions and detailed explanations to questions and problems similar to the questions in the GRE test. sample 1

A car covered 130 miles using 4 gallons of diesel. What distance would the same car cover, under similar conditions, on 6.7 gallons?
Solution
The number of miles covered using 1 gallon is given by
(130 / 4) = 32.5 miles/gallon
Using 6.5 gallons, the car will cover
6.7 * 32.5 = 217.75 miles

If plotted in the same rectangular system of axis, the graphs of f(x) =  x  4  and g(x) = 2 will have
A) no points of intersection
B) 1 point of intersection
C) 2 points of intersection
D) 3 points of intersection
E) 4 points of intersection
Solution
Shown below is the graph of x4 by steps. First top left is the Vshaped graph of x, then below is the graph of x4 which is the graph of x shifted 4 units down. Top right is the graph of x  4 which is the graph of x4 where the part of the graph below the x axis is reflected on the x axis to make it positive or zero. Finally the graph of x  4 and that of y = 2 shows that there are 4 points of intersection.
.

Which of the following is the largest?
A) 125%
B) 1.25
C) 1 + 1/3
D) 4/3
E) 0.0015/0.001
Solution
Convert all given quantities into decimal numbers
A) 125% = 125/100 = 1.25
B) 1.25 = 1.25
C) 1 + 1/3 = 1.333...
D) 4/3 = 1.333...
E) 0.0015/0.001 = 1.5/1 = 1.5
The largest of all given quantities is 0.0015/0.001

  10  19  20  = ?
Solution
First calculate  10  19
 10  19 = 29 = 29
Substitute  10  19 by 29 in the given expression and calculate
  10  19  20  = 29  20 = 9 = 9

The algebraic expression x / (x + 2) is undefined if x =
Solution
Since division by 0 is not allowed in mathematics, the given expression is undefined for any value that makes its denominator equal to 0. The denominator of the given expression is equal to 0 if
x + 2 =0 or x =  2
The given expression is undefined if x = 2

(√5  √7)(√5 + √7) = ?
Solution
We first expand the given expression using the identity (x  y)(x + y) = x^{2}  x^{2} and then simplify
(√5  √7)(√5 + √7) = (√5)^{2}  (√7)^{2} = 5  7 =  2

In the figure below the sides AB and AC of triangle ABC have equal lengths. Find the size of angle ABC.
.
Solution
The size of angle BAC is equal to 46° since it is a vertical angle to the given angle of size 46°. We now use the fact that the sum of all angles in a triangle is equal to 180°.
size of angle ABC + size of angle ACB + 46° = 180°
Since AB and AC have equal lengths, triangle ABC is isosceles and therefore angles ABC and ACB are equal in size. Hence
2 (size of angle ABC) + 46° = 180°
size of angle ABC = (180  46) / 2 = 67°

If y = 10° in the figure below, what is the value of x?
.
Solution
The sum of all three angles in a triangle is equal to 180°. Hence
(4x + 4y) + (x + 3y) + (x + 2y) = 180
Group like terms
6x + 9y = 180
Substitute y by 10 in the equation and solve for x.
6x + 9(10) = 180
6x = 180  90 = 90
x = 15

By what percent will the volume of a rectangular solid increase if its length, width and height increase by 25% each?
Solution
Let L, W and H be the length, width and height of the rectangular solid. The volume V is given by
V = L W H
When L, W and H are increased by 25%, they become
L = L + 25%L = L(1 + 25/100) = L(1 + 0.25) = 1.25 L
W = 1.25 W
H = 1.25 H
The volume V2 after the increase is given by
V2 = (1.25 L) (1.25 W) (1.25 H) = 1.25^{3} L W H
Find the percentage P of increase of the volume as follows
P = (V2  V) / V = (1.25^{3} L W H  L W H) / L W H =
L W H (1.25^{3}  1) / L W H
= 1.25^{3}  1 = 0.95 (rounded to the nearest hundredth)
= 95%

What is the average of all prime numbers between 20 and 40?
Solution
The prime numbers between 20 and 40 are
23, 29, 31 and 37
Their average is equal to
(23 + 29 + 31 + 37) / 4 = 30
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