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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. m − n < 0 • I only
• II only • III only
• I and III
• II and III
Answer: II onlyI 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 m − n = −2) or III may not be true (if m > n, e.g., m = −3 and n = −5 gives m − n = +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: 3/53/8 = 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 nonterminating decimal) 5/3 = 1.666... (recurring or nonterminating 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 64^{2n + 5} = 16^{5n + 2}? • −1
• 1
• 2¼
• 2¾ • 3¼
Answer: 2¾Converting to same base: (4^{3})^{2n + 5} = (4^{2})^{5n + 2} ∴ 4^{6n + 15} = 4^{10n + 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 a^{2} + a = 6? • a^{2} − 4
• a^{2} − 9
• a^{2} − 4a + 4
• a^{2} − 5a + 6
• a^{2} − a − 6
Answer: a^{2} − a − 6Solving given expression: a^{2} + a − 6 = 0 Factoring: (a + 3) (a − 2) = 0 Thus, a = −3 or a = 2. Substituting above values of a in each choice shows thata^{2} − a − 6 = 6 or −4 (not zero). Alternatively, factor each choice as follows. a^{2} − 4 = (a + 2) (a − 2) a^{2} − 9 = (a + 3) (a − 3) a^{2} − 4a + 4 = (a − 2) (a − 2) a^{2} − 5a + 6 = (a − 2) (a − 3) a^{2} − a − 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: $280Now, 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 + 4Let 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: 1110% 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 πr^{2}. If the new radius is 0.9r, the new area will be π(0.9r)^{2} = 0.81 πr^{2}. 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 minutesIn 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 rightangled triangle, the two unknown angles are in the ratio of 1:4. The smaller angle is • 18^{o} • 20^{o}
• 22^{o}
• 22.5^{o}
• 30^{o}
Answer: 18^{o}The angles of a triangle add up to 180^{o}. So the two unknown angles of the rightangled triangle add up to 90^{o}. If the smaller angle is x^{o}, then the other unknown angle is 4x^{o}. So, x + 4x = 90 or x = 90/5 = 18. 12. If x * y = 2x^{3} + 5y^{2} − 3, then 3 * 2 = • 35
• 68
• 70
• 71 • 74
Answer: 71Taking x as 3 and y as 2 in the given function, 3 * 2 = (2 x 3^{3}) + (5 x 2^{2}) − 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: $190Taking 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: 2Prime 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 twodigit 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: 16There 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 (twodigit 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: 3If (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: 5:00Taking 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 fruitbasket 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 fruitbasket? • 7
• 16
• 28 • 36
• 64
Answer: 28Number 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: 12If 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: $715Area 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: $26The 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: 35Since 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 manhours. Time = Constant/Men = 700/10 = 70 hours. Since only ½ of the bricks have to be laid, time taken would be 35 hours.
Try the Quiz : GMAT Test Prep : Quantitative Math Problem Solving Test IV
