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GMAT Test Prep : Quantitative Problem Solving Test IV

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

1. If m and n are negative integers, which of the following must be true?
I. m n < 0
II. m + n < 0
III. mn < 0
• I only
• II only
• III only
• I and III
• II and III
Answer: II only
I is not true because the product of two negative integers is a positive integer.
II is true because the sum of two negative integers is a negative integer.
III may be true (if m < n, e.g., m = −5 and n = −3 gives mn = −2) or
III may not be true (if m > n, e.g., m = −3 and n = −5 gives mn = +2).


2. If each fraction below is expressed as a decimal, which would have the least number of nonzero digits to the right of the decimal point?
• 3/8
• 3/5
• 3/4
• 4/3
• 5/3
Answer: 3/5
3/8 = 0.375 (3 decimal places)
3/5 = 0.6 (1 decimal place)
3/4 = 0.75 (2 decimal places)
4/3 = 1.333... (recurring or non-terminating decimal)
5/3 = 1.666... (recurring or non-terminating decimal)
So, the decimal with fewest number of decimals is 3/5 = 0.6 (1 decimal place).


3. If the price of a car is reduced by $2000 to a new price of $8000, then the percentage change in the car price is
• 20%
• 25%
• 33.33%
• 66.67%
• 75%
Answer: 20%
Original price = $8000 + $2000 = $10000
Percentage change in price = ($2000/$10000) x 100 % = 20%



4. What is the value of n if 642n + 5 = 165n + 2?
• −1
• 1
• 2¼
• 2¾
• 3¼
Answer:
Converting to same base: (43)2n + 5 = (42)5n + 2
∴ 46n + 15 = 410n + 4
Equating the exponents: 6n + 15 = 10n + 4
Solving the equation: 4n = 11 or n = 11/4 = 2¾.


5. Which expression below cannot equal zero when a2 + a = 6?
a2 − 4
a2 − 9
a2 − 4a + 4
a2 − 5a + 6
a2a − 6
Answer: a2a − 6
Solving given expression: a2 + a − 6 = 0
Factoring: (a + 3) (a − 2) = 0
Thus, a = −3 or a = 2.
Substituting above values of a in each choice shows thata2a − 6 = 6 or −4 (not zero).
Alternatively, factor each choice as follows.
a2 − 4 = (a + 2) (a − 2)
a2 − 9 = (a + 3) (a − 3)
a2 − 4a + 4 = (a − 2) (a − 2)
a2 − 5a + 6 = (a − 2) (a − 3)
a2a − 6 = (a + 2) (a − 3)
(a + 3) or (a − 2) is a factor in all but the last expression.


6. Susan paid 30% of her salary for school fees, 20% for apartment rent, and 25% for other expenses. If the remaining $350 were her monthly savings, how much was her apartment rent?
• $280
• $350
• $420
• $1050
• $1400
Answer: $280
Now, 30% + 20% + 25% = 75%. So, monthly savings = 100% − 75% = 25%.
Since 25% (i.e., ¼) of salary is $350, Salary = $350 / ¼ = $1400.
Apartment rent = 20% of salary = 20/100 x $1400 = $1400/5 = $280.


7. The arithmetic mean of 3x − 8 and another number is 5x. The arithmetic mean of the other number and 3x is
• 4x + 4
• 5x + 4
• 2x + 8
• 5x + 8
• 10x + 8
Answer: 5x + 4
Let the other number be y.
Given: Arithmetic Mean (Average) = (3x − 8 + y)/2 = 5x.
So, y = 10x − 3x + 8 = 7x + 8.
Required Arithmetic Mean = (7x + 8 + 3x)/2 = 5x + 4.


8. Janet gets a commission of 5% on the computers she sells. How many computers at $1150 each does Janet need to minimally sell to make a total commission of more than $600?
• 10
• 11
• 12
• 110
• 120
Answer: 11
10% of $1150 = $115. So, 5% of $1150 = $115/2 = $57.50
Commission is $57.50 on 1 computer, $575 on 10 computers, and $632.50 on 11 computers.


9. The radius of a circle is decreased by 10%. By what percent will its area decrease?
• 10%
• 11%
• 19%
• 20%
• 21%
Answer: 19%
If the radius of the circle is r, its area is πr2.
If the new radius is 0.9r, the new area will be π(0.9r)2 = 0.81 πr2.
Decrease (in area) = 1 − 0.81 = 0.19 = 19%


10. Alan can inspect four cars in 2 hours, while Bill can inspect four cars in 3 hours. How long will it take them to inspect twelve cars if they both work together?
• 2 hours 12 minutes
• 2 hours 30 minutes
• 3 hours 36 minutes
• 7 hours 12 minutes
• 7 hours 30 minutes
Answer: 3 hours 36 minutes
In 2 hours, Alan can inspect 4 cars. So, in 1 hour, he can inspect 2 cars.
In 3 hours, Bill can inspect 4 cars. So, in 1 hour, he can inspect 4/3 cars.
Thus, they can together inspect (2 + 4/3) = 10/3 cars in 1 hour.
To inspect 12 cars, it will take them 12 ÷ 10/3 = 12 x 3/10 = 18/5 hours.
18/5 hours = (3 + 3/5) hours = 3 hours 36 minutes.


11. In a right-angled triangle, the two unknown angles are in the ratio of 1:4. The smaller angle is
• 18o
• 20o
• 22o
• 22.5o
• 30o
Answer: 18o
The angles of a triangle add up to 180o.
So the two unknown angles of the right-angled triangle add up to 90o.
If the smaller angle is xo, then the other unknown angle is 4xo.
So, x + 4x = 90 or x = 90/5 = 18.


12. If x * y = 2x3 + 5y2 − 3, then 3 * 2 =
• 35
• 68
• 70
• 71
• 74
Answer: 71
Taking x as 3 and y as 2 in the given function,
3 * 2 = (2 x 33) + (5 x 22) − 3
∴ 3 * 2 = (2 x 27) + (5 x 4) − 3 = 54 + 20 − 3 = 71


13. Bottles of juice were bought for an engineering conference at a rate of $3.80 per bottle. Only ¼ of the bottles were consumed. If the cost of the remaining 150 bottles was refunded, what was the cost of the juice consumed at the conference?
• $90
• $100
• $190
• $380
• $570
Answer: $190
Taking x as the total number of bottles of juice bought,
¾ x = 150 or x = 150 (4/3) = 200
Number of bottles consumed = 200 − 150 = 50
∴ Cost of the juice consumed = 50 x $3.80 = $190


14. How many factors of 52 are divisible by 4?
• 1
• 2
• 3
• 4
• 5
Answer: 2
Prime factorization gives 52 = 2 x 2 x 13.
The factors of 52 are 1, 2, 4, 13, 26 and 52.
Of these, 4 and 52 are the factors divisible by 4.
∴ Number of factors divisible by 4 = 2.


15. How many two-digit numbers can be written using the digits 0 through 4, given that 0 cannot be the first digit and no digit can be repeated?
• 15
• 16
• 20
• 21
• 25
Answer: 16
There are 4 possibilities (1, 2, 3 and 4) for the first digit of the number, because 0 cannot be the first digit.
There are 4 possibilities for the second digit too, because no digit can be repeated.
∴ Total possibilities (two-digit numbers) = 4 x 4 = 16


16. If (p + 4)2 = 121, which of these could be the value of p − 4?
• −7
• −3
• 3
• 15
• 19
Answer: 3
If (p + 4)2 = 121, then p + 4 = ±√121 = ±11
Taking p + 4 = 11, p = 7 and p − 4 = 3
Taking p + 4 = −11, p = −15 and p − 4 = −19
∴ The correct answer is 3, because −19 is not a listed choice.


17. Two trains were traveling in the same direction. The first train left the station at 1:00 and the second train followed an hour later. Given that their speeds are 60 miles per hour and 80 miles per hour respectively, when will they meet each other?
• 4:00
• 5:00
• 6:00
• 7:00
• 8:00
Answer: 5:00
Taking the time taken by the first train as t hours, the distance covered by the first train is 60t miles.
Distance covered by the second train is 80(t − 1) miles.
For the two trains to meet, 60t = 80t − 80
20t = 80 or t = 4
∴ The two trains will meet at 5:00 (4 hours after the starting time of the first train, 1:00).


18. The ratio of apples to oranges to bananas in a fruit-basket is 4 : 5 : 7. If the fruits in the basket are worth $96 and were purchased at an average rate of $1.50 per fruit, how many bananas are there in the fruit-basket?
• 7
• 16
• 28
• 36
• 64
Answer: 28
Number of fruits in the basket = 96/1.50 = 64
Sum of the parts of the ratio = 4 + 5 + 7 = 16
∴ Number of bananas in the basket = (7/16) x 64 = 28


19. The length of a rectangle is 9 feet more than its breadth. Given that its perimeter is 66 feet, what is the breadth of the rectangle in feet?
• 9
• 12
• 18
• 21
• 33
Answer: 12
If the breadth of the rectangle as b, then its length is b + 9
Perimeter = 2 (b + b + 9) = 66
2b + 9 = 33 or b = 24/2 = 12


20. A farmer decides to enclose a rectangular field of area 900 square feet.If the length of the field is 45 feet and the cost of fencing is $5.50 per foot, what is the cost of enclosing the field?
• $495
• $715
• $990
• $1890
• $4950
Answer: $715
Area of rectangle = Length x Width
So, Width of rectangle = Area/Length = 900/45 = 20 feet
Perimeter of rectangle = 2 (Length + Width) = 2 (45 + 20) = 2 x 65 = 130 feet
∴ Cost of enclosing the field = 130 x $5.50 = $715


21. Tennis rackets were marked down 5% to 25%. If they were marked down another 10% the next day, what was the lowest cost possible for a tennis racket that was originally selling at $40?
• $24
• $26
• $30
• $34
• $36
Answer: $26
The maximum discount available on the first day was 25%.
Another 10% on the next day would make the total discount 35%.
∴ Lowest cost of tennis racket = 0.65 x $40 = $26


22. 14 masons take 50 hours to lay bricks for making a wall. How much time (in hours) would it take 10 masons to lay ½ of the bricks, given that they work at the same rate?
• 35
• 70
• 105
• 140
• 175
Answer: 35
Since fewer men will take more time, this is a problem of inverse proportionality.
Thus, Time α (1/Men) or Men x Time = Constant.
Constant = 14 x 50 = 700 man-hours.
Time = Constant/Men = 700/10 = 70 hours.
Since only ½ of the bricks have to be laid, time taken would be 35 hours.



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