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GMAT Test Prep : Quantitative Problem Solving Test VI

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

1. Which of the following cannot be the largest angle of a triangle?
• 57
• 60
• 61
• 64
• 73
Answer: 57
An angle less than 60o cannot be the largest angle of any triangle as proven below.
Let the angles of the triangle measure x, y and z (totalling 180o), with x denoting the largest angle.
Then, x = 180 − yz with y < x and z < x.
x is minimum when the other two angles are maximum (equal to x).
So, xmin = 180 − xminxmin or xmin = 180/3 = 60.
Thus, the minimum value of the largest angle in a triangle is 60o.


2. Three bars of margarine are required to make one batch of chocolate truffles. How many bars of margarine are needed to make enough truffles to occupy 6 shelves in a cake shop, given that each shelf can accommodate 2½ batches of truffles?
• 7½
• 15
• 18
• 45
• 60
Answer: 45
Since 6 shelves of the shop have chocolate truffles and each shelf can accommodate 2½ batches of truffles, the total number of batches to be
made is 6 x 2½ = 15.
Since each batch requires 3 bars of margarine, the total number of bars required is 3 x 15 = 45.


3. If integers x and y are such that 3x + 4y = 25 and x2 + y2 = 25, then the value of y =
• −4
• −3
• 3
• 4
• None of the above
Answer: 4
There are two methods to solve this problem.
Method 1 : Recognize that x and y are 3 and 4 respectively, since the numbers 3, 4 and 5 form the first Pythagorean triplet.
Method 2 : Substitute x = (25 − 4y)/3 from the first equation in the second equation and multiply by 9 throughout to get the following quadratic
equation:
(25 − 4y)2 + 9y2 = 225.
So, (625 − 200y + 16y2) + 9y2 = 225 or 25y2 − 200y + 400 = 0.
Thus, y2 − 8y + 16 = 0 or (y − 4)2 = 0 giving y = 4.



4. If b = 8dc and a = d/3, what is the average (arithmetic mean) of a, b, c, and d ?
• 4a
• 7a
a/7
• 4a + 7
• 7a + 4
Answer: 7a
The average (arithmetic mean) of n numbers is defined as their sum divided by n. In this case,
sum = a + b + c + d = a + 8dc + c + d = a + 9d = a + 9 (3a) = 28a.
average = sum/4 = 7a.


5. In January, 55 percent of a hospital's 300 patients were women. In February, 30 percent of its 440 patients were women. What was the percent change in the number of women patients from January to February ?
• decrease of 20%
• increase of 20%
• decrease of 25%
• increase of 25%
• increase of 46.67%
Answer: decrease of 20%
Women patients in January = 0.55 x 300 = 165.
Women patients in February = 0.3 x 440 = 132.
Change = 132 − 165 = −33.
Percent change = (−33/165) x 100% = (−1/5) x 100% = −20% (decrease of 20%).


6.
trapezium
If AD = 50, BC = 40 and CD = 120 in the figure above, its perimeter is
• 210
• 330
• 360
• 370
• 380
Answer: 360
trapezium
ADE is a right-angled triangle. By the Pythagorean theorem,
AE2 + DE2 = AD2 or AE2 + 402 = 502.
AE = (502 − 402)½ = (2500 − 1600)½ = √900 = 30.
AB = AE + EB = 30 + 120 = 150.
Perimeter = 150 + 40 + 120 + 50 = 360.


7. The price of an ounce of gold was g dollars at the beginning of January. It decreased by 10 percent at the beginning of February and then increased by 20 percent of this new price at the beginning of March. What was the price in dollars per ounce at the beginning of March ?
• 0.88g
• 0.90g
g
• 1.08g
• 1.10g
Answer: 1.08g
Price at beginning of February (after decrease of 10%) = 0.9g.
Price at beginning of March (after increase of 20%) = 1.2 (0.9g) = 1.08g.


8. What is the ratio of (6 + ¾) to (6 x ¾) ?
• 8/243
• 2/3
• 1
• 3/2
• 243/8
Answer: 3/2
6 + ¾ = (24 + 3)/4 = 27/4.
(6 x ¾) = 18/4.
Ratio = (27/4) ÷ (4/18) = (27/4) x (4/18) = 27/18 = 3/2.


9. A merchant sold two gold coins for $450 each. He sold the first coin for 25 percent less than its cost and the second coin for 50 percent more than its cost. What was his total net gain or loss, if any?
• Loss of $112.50
• Gain of $112.50
• Loss of $337.50
• Gain of $337.50
• There was NO total net gain or loss.
Answer: There was NO total net gain or loss.
Let the cost of the first coin be f and the cost of the second coin be s.
0.75f = 450 or f = 450/0.75 = (4/3) x 450 = 600
1.5s = 450 or s = 450/1.5 = (2/3) x 450 = 300
Difference on first coin = $450 − $600 = −$150
Difference on second coin = $450 − $300 = $150
No total net gain or loss!


10. If the positive integers m and n have the same two digits, but in reverse order, then the difference between m and n cannot be
• 24
• 36
• 54
• 63
• 72
Answer: 24
Note that 51 − 15 = 36; 71 − 17 = 54; 92 − 29 = 63; 91 − 19 = 72; etc.
If x and y are the two digits, then m = 10x + y.
With the digits in reverse order, n = 10y + x.
Thus, mn = 9 (xy).
Since (mn) must be divisible by 9 (specifically, 9, 18,27, 36, 45, 54, 63 and 72),
the difference cannot be 24.


11. A snail crawls at the rate of y inches per hour. How many minutes will it take the snail to crawl a distance of x inches ?
x / y
y / x
• 60x / y
• 60y / x
y / (60x)
Answer: 60x / y
Speed = Distance / Time
Time = Distance / Speed = x / y hr = 60x / y minutes.


12. How many integers m are there that satisfy −13 < 3m + 4 < 13 ?
• Five
• Six
• Seven
• Eight
• Nine
Answer: Eight
Given: −13 < 3m + 4 < 13
Subtracting 4 throughout: −17 < 3m < 9
Dividing by 3 throughout: −5 s2/3 < m < 3
The integers are −5, −4, −3, −2, −1, 0, 1, 2.
Thus, there are eight integers that satisfy the given inequality.
Alternatively, by trial and error, one may show that −5 and 2 satisfy the given inequality, but −6 and 3 do not.


13. A pizzeria sells small pizzas for $7.99 and large pizzas for $12.99. If its total sales last Saturday amounted to $949.00 and it sold 30 large pizzas, how many small pizzas did it sell ?
• 65
• 70
• 75
• 80
• 85
Answer: 70
Sales of large pizzas = $12.99 x 30 = $389.70
Sales of small pizzas = $949 − $389.70 = $559.30
Number of small pizzas sold = $559.30/$7.99 = 70.
Note that it is convenient to approximate $559.30/$7.99 by $560/$8.


14. Jill cycled a total of 3 hours and covered a total distance of 33 miles over two days. Jill's average speed on the first day was 4 mph (miles per hour) slower than that on the second day; however, she cycled 1½ hours longer on the first day than she cycled on the second day. What was her average speed on the second day ?
• 10 mph
• 14 mph
• 18 mph
• 22 mph
• 26 mph
Answer: 14 mph
If Jill cycled for t hours on the second day, then she cycled for (t + 1½) hours on the first day.
So, t + t + 1½ = 3 or 2t = 1½ giving t = ¾ h.
Now Speed = Distance / Time or Distance = Time x Speed.
If Jill's average speed for the ¾ hour on the second day is u mph, then her average speed for 2¼ hours on the first day is (u − 4) mph.
So, (3/4) u + (9/4) (u − 4) = 33 or 3u + 9u − 36 = 132.
Thus, 12u = 168 giving u = 168/12 = 14 mph.


15. What fraction is equivalent to the decimal 0.3125 ?
• 1/6
• 3/8
• 3/16
• 5/16
• 21/80
Answer: 5/16
0.3125 = 3125/10000 = 125/400 = 5/16


16. Colin has paid $455 toward the purchase of his computer. If 65 percent of the price remains to be paid, how many dollars does Colin still have to pay ?
• 445
• 455
• 845
• 855
• 1300
Answer: 845
Let x be the price of the computer.
Since $455 is 35% of the price, 0.35x = 455
So, x = 455/0.35 = 45500/35 = 1300.
Dollars still to be paid = 1300 − 455 = 845.


17. A plane flies at an average speed of 400 miles per hour from London to Toronto. It departs London at 11:45 in the morning, London time. The distance between London and Toronto is about 3600 miles, and Toronto time is 5 hours earlier than London time. At what hour in the evening, Toronto time, will the plane reach Toronto?
• 3:45
• 4:45
• 6:45
• 7:45
• 8:45
Answer: 3:45
Speed = Distance / Time or Time = Distance / Speed.
So, Flight time = 3600 / 400 = 9 hours.
Since it departs London at 11:45 in the morning, London time, it will reach Toronto after a 9-hour flight at 8:45 in the evening, London time.
Since Toronto time is 5 hours earlier than London time, it reaches at 3:45 in the evening, Toronto time.


18. For any integer n greater than 0, n! denotes the product of all the integers from 1 to n, inclusive. How many multiples of 3 are there between 6! − 6 and 6! + 6, inclusive ?
• One
• Two
• Three
• Four
• Five
Answer: Five
6! = 1 x 2 x 3 x 4 x 5 x 6 = 720.
Since 714, 717, 720, 723 and 726 are divisible by 3, there are 5 multiples of 3 between 714 and 726.
Note a number is divisible by 3 if the sum of its digits is divisible by 3.


19. How many prime numbers are there between 781 and 785, inclusive ?
• None
• One
• Two
• Three
• Four
Answer: None
782 and 784 are divisible by 2.
783 is divisible by 3.
785 is divisible by 5.
781 is divisible by 11.
So, there are no prime numbers between 781 and 785.


20. Working independently, machines A, B, C and D can do a certain assignment, in 4, 6, 8 and10 hours, respectively. If two machines work together at their individual rates, what is the ratio of the time it takes machines A and B to do the assignment to the time it takes machines C and D to do the assignment ?
• 5/9
• 27/50
• 50/27
• 3/32
• 32/3
Answer: 27/50
In one hour, the four machines can independently do one-fourth, one-sixth, one-eighth and one-tenth of the assignment. If the machines work together to do the assignment, their individual rates may be added. Thus,
(1/4) + (1/6) = (3 + 2)/12 = 5/12 (for machines A and B together)
(1/8) + (1/10) = (5 + 4)/40 = 9/40 (for machines C and D together)
Machines A and B together do (5/12)th of the assignment in 1 hour.
So, Machines A and B together do the entire assignment in (12/5) hours.
Similarly, Machines C and D together do the entire assignment in (40/9) hours.
Required ratio = (12/5) ÷ (40/9) = (12/5) x (9/40) = 27/50.


21. A room is 16 feet in length, 15 feet in width, and 12 feet in height. What is the maximum distance possible in feet along a straight line between any two points in the room ?
• 15
• 20
• 25
• 10√2
• 10√3
Answer: 25
The room is in the shape of a cuboid with rectangular faces (walls).The length of the diagonal inside the cuboid (not along the faces) specifies the maximum distance along a straight line between any two points.
Thus,Diagonal = (162 + 152 + 122)½ = (256 + 225 + 144)½ = √625 = 25.
For a cuboid, a2 + b2 + c2= d2, where a, b and c are the lengths of its sides and d is the length of its diagonal. The formula may be derived by applying the Pythagorean theorem twice (first for the diagonal along the face and then for the diagonal inside the cuboid).


22. Which fraction has the smallest value ?
• 8/(34 x 73)
• 27/(35 x 73)
• 12/(33 x 73)
• 2/(33 x 72)
• 98/(32 x 74)
Answer: 8/(34 x 73)
The first step is to cancel the common factors by multiplying throughout by (32 x 72). The second step is to express the fractions with the same denominator for purposes of comparison. Thus,
8/(34 x 73) x (32 x 72) = 8/(32 x 7) = 8/63
27/(35 x 73) x (32 x 72) = 27/(33 x 7) = 1/7 = 9/63
12/(33 x 73) x (32 x 72) = 12/(3 x 7) = 4/7 = 36/63
2/(33 x 72) x (32 x 72) = 2/3 = 42/63
98/(32 x 74) x (32 x 72) = 98/72 = 2 = 126/63



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