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

  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

  2. 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.

    gre solution problem 2 .


  3. 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


  4. | |- 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

  5. 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

  6. (√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

  7. In the figure below the sides AB and AC of triangle ABC have equal lengths. Find the size of angle ABC.

    gre problem 7.



    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

  8. If y = 10 in the figure below, what is the value of x?

    gre problem 8.

    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

  9. 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%

  10. 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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