# Free GRE Solutions and Explanations to Sample 1

 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 ||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. . 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) = x2 - x2 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) = 1.253 L W H Find the percentage P of increase of the volume as follows P = (V2 - V) / V = (1.253 L W H - L W H) / L W H = L W H (1.253 - 1) / L W H = 1.253 - 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 More Math Practice Tests Free GRE Quantitative for Practice Free Practice for GAMT Math tests Free Compass Math tests Practice Free Practice for SAT, ACT Math tests Free AP Calculus Questions (AB and BC) with Answers