GRE
Practice Test 18
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What is the perimeter of a rectangle that is 3 times as long as it is wide and has the same area as a circle of circumference 6?

  • Solution

    Let breadth be x

    ∴ length = 3x

    ∴ Area = 3 × 2

    Circumference = 6

    ∴ 2π r = 6 ∴ r = 3⁄π

    ∴ Area = π r2 = πx = \(\frac{9}{\pi ^{2}}\) = 9⁄π

    ∴ 3x2 = 9⁄π ∴ x2 = 3⁄π

    ∴ x = \(\sqrt{\frac{3}{\pi }}\)

    Perimeter = 2(x + 3x) = 8x = \(\frac{8\sqrt{3}}{\pi }\)

In the figure above, centers of all three circles lie on the same line. Medium sized circle has twice the radius of the smallest circle. Smallest circle has radius 1. What is the length of boundary of the shaded region?

  • Solution

    Smallest circle has radius = 1

    ∴ Medium sized circle has radius = 2

    ∴ Biggest circle i.e. outside circle has radius = 3

    Length of boundary of the shaded region is the sum of the lengths of the three semicircles, i.e. 1π + 2π + 3π = 6π

The negative of sum of 2 consecutive odd numbers is less than -35. Which of the following may be one of the numbers?

  • Solution

    Numbers can be 17 and 19. The sum will be 36 negative of 36 will be -36. Hence number can be 17 or 19 or any odd number after 19.

x and y are positive integers, and x – 2y = 5. Which of the following could be the value of X2– 4y2?

  • Solution

    X2- 4y2 = (x + 2y) (x - 2y) = 5 (x + 2y)

    Clearly X2- 4y2 has to be a multiple of 5.

    ∴ -3, 14,51 are ruled out.

    If result is zero, then x + 2y = 0 => x = -2y.

    In that case x and y both cannot be positive.

    So expression must equal 45.

    i.e.x + 2y = 9 ∴ x = 7,y = 1

How many liters of 50% antifreeze must be mixed with 80 liters of 20% antifreeze to get a mixture that is 40% antifreeze?

  • Solution

    Method 1:

    Let quantity be x liters.

    ∴ 0.5x + 0.2(80) = 004 (x + 80)

    ∴ 0.5x + 16 = 0.4x + 32

    ∴ x = 160

    Method 2:

    You can also solve the problem by allegation method

Bill averaged 70 on his first m exams. After taking n more exams, he had an overall average of 75 for the year. In terms of m and n, his average for last n exams was

  • Solution

    Average for m exams = 70

    ∴ Total for m exams = 70m

    Average for (m + n) exams = 75

    ∴ Total for (m + n) exams = 75m + 75n

    Average for last n exams =

    \(\frac{75m+75n-70m}{m+n-m}=\frac{5m+75n}{n}=\frac{5m}{n}+75\)

If A < 2 - 4B, which of the following is true?

  • Solution

    A < 2 - AB

    ∴ A - 2 < -4B

    ∴ \(\frac{A - 2}{4} < -B\)

    ∴ \(\frac{2 - A}{4} > B\)

Cost of 4 rolls, 6 muffins and 3 loaves of bread is $9.10. Cost of 2 rolls, 3 muffins and a loaf of bread is $3.90. What is the cost of a loaf of bread in dollars?

  • Solution

    4r + 6m + 3b =9:10........(I)

    2r + 3m + b = 3.90........(II) x 2

    4r + 6m + 2b =7.80........(III)

    (I)- (III)gives b = $1.30

At a certain pizzeria, 1⁄8 of pizzas sold in one week were mushrooms and 1⁄3 of remaining pizzas sold were pepperonis. If n number of pizzas sold were pepperoni, how many were mushrooms?

  • Solution

    If the total numbers of pizzas are 24, pizzas with mushrooms are 3.

    ∴ The remaining pizzas are 21. Out of them one third i.e. 7 pizzas are pepperonis, which is equal to n.

    Pizzas with mushroom will be 3⁄7

2,4,6,8,n,3,5,7,9

In the list above, if n is an integer between 1 and 10 inclusive, then the median must be

  • Solution

    Rearranging the eight actual numbers from least to greatest, we get

    2,3,4,5,6,7,8,9 n has to be an integer.

    If n ≤ 5, then 5 is the middle most number among 9 numbers under consideration.

    So median = 5

    If n ≥ 6, then median is 6

    n cannot take values between 5 and 6 as n is an integer.

    ∴ Median is either 5 or 6

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