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## GMAT Test Prep : Quantitative Problem Solving Test V

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 Choose the best answer from the choices given. All numbers used are real numbers.

1. What is the value of ^(−7)^, given that ^m^ = m3m2m?
• −399
• −385
• −301
• −287
• 301
(−7)3 = (−7) x (−7) x (−7) = −343
(−7)2 = (−7) x (−7) = 49
∴ Substituting in the given function, ^(−7)^ = −343 − 49 − (−7) = −343 − 49 + 7 = −385

2. How many multiples of both 3 and 6 are there between 27 and 78 exclusive?
• 8
• 9
• 10
• 16
• 18
A multiple of 3 is not necessarily a multiple of 6, e.g., 27.
On the other hand, a multiple of 6 is necessarily a multiple of 3.
So, only the multiples of 6 need to be considered.
The multiples of 6 between 27 and 78 exclusive are
30, 36, 42, 48, 54, 60, 66 and 72.
∴ Number of multiples of both 3 and 6 between 27 and 78 exclusive = 8.

3. How many two-digit numbers can be written using the digits 0 through 5, given that 0 can be the first digit, but no digit can be repeated?
• 11
• 20
• 25
• 30
• 36
There are 6 possibilities (0, 1, 2, 3 and 4, 5) for the first digit of the number, because 0 can be the first digit.
There are 5 possibilities for the second digit, because no digit can be repeated.
∴ Total possibilities (two-digit numbers) = 6 x 5 = 30

4. If ab < 0, which of these must be true?
a > 0
b > 0
a/b < 0
a/b > 0
a + b < 0
Answer: a/b < 0
If ab < 0, either a or b is definitely less than 0, but both cannot be less than 0 (because the product of two negative numbers is always positive).
a/b < 0 must be true (the other choices may or may not be true depending on the actual numbers).

5. What is the value of pq, given that (0.5 x 10p ) / (25 x 10q ) = 0.2 x 1013
• 10
• 12
• 13
• 14
• 15
Evaluating the left-hand side of the equation, (0.5 x 10p ) / (25 x 10q ) = 0.02 x 10pq = 0.2 x 10pq − 1
Thus, 0.2 x 10pq − 1 = 0.2 x 1013 or pq − 1 = 13
pq = 14

6. If c2 − 4c − 45 = 0, which of these could be the value of c?
• −9
• −4
• 4
• 5
• 9
Factoring c2 − 4c − 45 gives(c + 5)(c − 9).
If (c + 5)(c − 9) = 0, then c = −5 or c = 9.
∴ The correct answer is 9, because −5 is not a listed choice.

7. The average (arithmetic mean) of 8 numbers is 7.5. When two more numbers are added to the 8 numbers, the new average of all the 10 numbers is 8.8. What is the average of the two numbers added?
• 1.3
• 2.6
• 3.3
• 14
• 28
The average (arithmetic mean) of nnumbers is their sum divided by n.
Sum of 8 numbers = 8 x 7.5 = 60
Sum of 10 numbers = 10 x 8.8 = 88
Sum of the two numbers added = 88 − 60 = 28
∴ Average of the two numbers added = 28 / 2 = 14

8. Four members of a family want to raise funds for a Hawaiian vacation. They distribute 600 ice-cream packs equally among themselves, and decide that each pack is to be sold for \$4. If two of them manage to sell 75% of their ice-cream packs and the other two manage to sell only 40% of their packs, how many dollars did they collect?
• 225
• 345
• 1020
• 1380
• 2400
Number of packs sold by two family members = 75% x 300 = 225
Number of packs sold by the other two family members = 40% x 300 = 120
Total packs sold by the 4 family members = 345
∴ Funds collected = 345 x \$4 = \$1380

9. If the length of a rectangle is decreased by 25% and its width is increased by 40%, what is the percentage increase or decrease in its area, if any?
• Decrease of 5%
• Increase of 5%
• Decrease of 15%
• Increase of 15%
• There is neither an increase nor a decrease.
Answer: Increase of 5%
Let original length = L and original width = W. Then, original area = LW
Now, new length = 0.75L and new width = 1.4W. So, new area = (0.75 x 1.4) LW = 1.05 LW
∴ Increase = (1.05 LWLW)/(LW) = 1.05 − 1 = 0.05 = 0.05 x 100% = 5%

10. Faucet C fills a tank in 3 hours, whereas faucet D independently fills the same tank in 9 hours. What fraction of the tank will be filled, if both the faucets are kept running for 1½ hours, and the tank is originally one-fourth full?
• 1/6
• 1/3
• 4/9
• 5/9
• 11/12
Faucet C fills 1/3 of the tank in 1 hour.
Faucet D fills 1/9 of the tank in 1 hour.
Fraction of tank filled by both faucets in 1 hour = 1/3 + 1/9 = 3/9 + 1/9 = 4/9
Fraction of tank filled by both faucets in 1½ hours = 4/9 x 3/2 = 2/3
∴ Fraction of tank filled (including original ¼th fraction) = 2/3 + 1/4 = 8/12 + 3/12 = 11/12

11. If the radius of a circle is halved, what is the percentage decrease in its area?
• 10%
• 25%
• 50%
• 75%
• 90%
Let original radius = r. Then, original area = π r2.
Now, new radius = 0.5r. So, new area = π (0.5r)2 = 0.25π r2
∴ Decrease = (π r2 − 0.25π r2 )/(π r2) = 1 − 0.25 = 0.75 = 0.75 x 100% = 75%

12. If x@y = 2x2y3, then (−4)@(−3) =
• −59
• −5
• 5
• 41
• 59
(−4)2 = (−4) x (−4) = 16
(−3)3 = (−3) x (−3) x (−3) = −27
∴ Substituting in the given function, (−4)@(−3) = 2(16) − (−27) = 32 + 27 = 59

13. In how many different ways can the digits 2, 2, 7 and 9 be arranged, so that there is at least one 2 between 7 and 9 always?
• 3
• 4
• 6
• 9
• 12
There are three possibilities (2, 7, 9) for the first digit.
If the first digit is 2, there are two arrangements: 2729 and 2927.
If the first digit is 7, there are two arrangements: 7292 and 7229.
If the first digit is 9, there are two arrangements: 9272 and 9227.
∴ There are 6 arrangements possible.

14. What is the minimum value of b for which [b3 + 8][(64/b) + 16] = 0?
• − 4
• − 2
• − ¼
• ¼
• 2
Taking each factor of the left-hand side of the equation as 0 and evaluating :
b3 + 8 = 0 or b3 = −8 giving b = −2
(64/b) + 16 = 0 or 64/b = −16 giving b = 64/(−16) = −4
∴ The minimum value of b is −4.

15. If 22 x 33 = 32 x n, then n =
• 4
• 8
• 12
• 21
• 36
Evaluating the given equation, we get:
4 x 27 = 9 x n
So, n = (4 x 27) / 9 = 4 x 3 = 12

16. Corporation A hired only chemical engineers and mechanical engineers in the year 2003. If 55% of the employees hired were chemical engineers, and the remaining 450 employees hired were mechanical engineers, how many employees were hired totally?
• 100
• 450
• 505
• 550
• 1000
Percentage of mechanical engineers hired = 100% − 55% = 45%
Taking the employees hired totally as x, we get(45/100)x = 450
So, x = (450 x 100) / 45 = 1000

17. If a tractor covers 4 miles in 20 minutes, what is its speed in miles per hour?
• 1/5
• 1/4
• 5
• 12
• 60
Speed = Distance / Time
So, Speed of tractor (in miles per minute) = 4/20 = 1/5
To convert to miles per hour, multiply by 60, and get:
(1/5) x 60 = 12 miles per hour

18. If m is an even integer, which of these must be an odd integer?
m + 2
m + 4
• 2m − 1
• 2m − 2
• 3m
Answer: 2m − 1
2m is an even integer and (2m − 1) is an odd integer.
Alternatively, taking an even integer (say, 2), and evaluating all the choices given, we get:
m + 2 = 2 + 2 = 4 (even)
m + 4 = 2 + 4 = 6 (even)
2m − 1 = 4 − 1 = 3 (odd)
2m − 2 = 4 − 2 = 2 (even)
3m = 3 x 2 = 6 (even)

19. 'The Golf Company' purchased clubs and balls in the ratio 2 : 9. If it purchased a total of 34 clubs, how many balls did it purchase?
• 68
• 102
• 136
• 153
• 187
Each part of the ratio = 34 / 2 = 17
∴ Number of balls purchased = 17 x 9 = 153

20. Brian took a loan of \$4000 from his friend at a simple interest of 6% per year. If he has to pay off the entire loan amount and the interest in equal installments within 8 months, how much would he have to pay every month?
• \$480
• \$500
• \$520
• \$660
• \$740
Loan amount to be paid every month = \$4000 / 8 = \$500
Interest payable per month = (6/100) x \$4000 / 12 = \$20
∴ Payment every month = \$500 + \$20 = \$520.

21. A certain integer n when divided by 5 yields a remainder of 4. Which of these cannot be an integer?
n / 9
n / 10
n / 14
n / 19
n / 24
Answer: n / 10
Since n when divided by 5 yields a remainder of 4, n must be the sum of 4 and a multiple of 5, i.e.,
n could be 9 (4 + 5 x 1), 14 (4 + 5 x 2), 19 (4 + 5 x 3), 24 (4 + 5 x 4), etc.
Thus, n has either a 4 or 9 in the units place.
Integers (except multiples of 10 with a 0 in the units place) when divided by 10 give rise to decimal numbers and not integers.
∴ (n / 10) cannot be an integer (since no n is a multiple of 10).

Alternatively, substitute the following values for n in the choices given, to form integers:
If n = 9 (4 + 5 x 1), then n / 9 = 1 (an integer)
If n = 14 (4 + 5 x 2), then n / 14 = 1 (an integer)
If n = 19 (4 + 5 x 3), then n / 19 = 1 (an integer)
If n = 24 (4 + 5 x 4), then n / 24 = 1 (an integer)

22. 'The Dry-Fruit Company' sells a dry-fruit mixture containing 12 pounds of raisins, 4 pounds of almonds, and 4 pounds of walnuts. If the price of raisins, almonds and walnuts per pound is \$a, \$b, and \$c respectively, what is the price (in dollars) of 1 pound of the dry-fruit mixture?
• (a + 3b + 3c) / 5
• (3a + b + c) / 5
• (a + 4b + 4c) / 20
• (12a + b + 4c) / 20
• (12a + 4b + c) / 20
Answer: (3a + b + c) / 5
Weight of mixture produced = 20 pounds
Price of raisins = \$(12a); Price of almonds = \$(4b); Price of walnuts = \$(4c)
Total price of mixture produced = \$(12a + 4b + 4c)
Price (in dollars) of 1 pound of mixture = (12a + 4b + 4c) / 20
∴ On simplifying (dividing the numerator and denominator by 4), we get: (3a + b + c) / 5

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