Solutions and detailed explanations to questions and problems similar to the questions in the GRE test. sample 1 ## Solution to Question 1The 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 ## Solution to Question 2Shown below is the graph of ||x|-4| by steps. First top left is the V-shaped graph of |x|, then below is the graph of |x|-4 which is the graph of |x| shifted 4 units down. Top right is the graph of ||x| - 4| which is the graph of |x|-4 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.. ## Solution to Question 3Convert 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 ## Solution to Question 4First 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 ## Solution to Question 5Since 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 ifx + 2 =0 or x = - 2 The given expression is undefined if x = -2 ## Solution to Question 6We 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
## Solution to Question 7The 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° ## Solution to Question 8The 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 ## Solution to Question 9Let L, W and H be the length, width and height of the rectangular solid. The volume V is given byV = 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% ## Solution to Question 10The prime numbers between 20 and 40 are23, 29, 31 and 37 Their average is equal to (23 + 29 + 31 + 37) / 4 = 30
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