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GMAT Test Prep : Quantitative Problem Solving Test I

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

1. Faucet A fills a tank in 3 hours, and faucet B drains the 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 empty?
• 1/4
• 1/3
• 1/2
• 2/3
• 3/4
Answer: 1/3
Faucet A fills 1/3 of the tank in 1 hour.
Faucet B drains 1/9 of the tank in 1 hour.
Fraction of tank filled in 1 hour = 1/3 − 1/9 = 3/9 − 1/9 = 2/9
∴ Fraction of tank filled in 1½ hours = 2/9 x 3/2 = 1/3


2. A sequence of real numbers is defined by the rule: a(n) = 4 − 3 a(2 − n).
What is the value of a(0)?
• 0
• 1
• 4/3
• 2
• 3
Answer: 1
Substitute n = 0 and n = 2 in the given equation (rule). This gives the following two equations:
a(2) = 4 − 3 a(0)
a(0) = 4 − 3 a(2)
Eliminating a(2), one obtains a(0) = 4 − 3 [4 − 3 a(0)] = −8 + 9 a(0).
Thus, the solution is: a(0) = 1; a(2) = 1


3. If ~X is the integer closest to the real number X, what is the value of the expression:
~4.3 + ~6.7 + ~(-0.9) ?
• 8
• 9
• 10
• 11
• 12
Answer: 10
From the definition of ~X,
~ 4.3 = 4 ; ~ 6.7 = 7 ; ~ (−0.9) = −1 ;
The value of the given expression is: 4 + 7 − 1 = 10.



4. If x2 < 9 and −2 < y < 1, then which of the following is true?
x < 3y
• 3x < 2y
• 2x > −3y
xy < 6
• −xy > 6
Answer: xy < 6
x2 < 9 implies −3 < x < 3. Also −2 < y < 1.
Clearly xy < 6 for all possible values of x and y. This product xy approaches 6 as x approaches −3 and y approaches −2.
All other options are false as can be seen by substituting x = y = 0.


5. If 4x − 3y = 24, then what is the value of 2y − 8(x / 3)?
• −16
• −12
• −8
• −6
• −4
Answer: −16
Multiply the given expression by −2/3.
Hence, −(2/3)(4x − 3y) = 2y 8(x / 3) = −(2/3)24 = −16.


6. A train travels at the rate of 10 miles/hr for the first hour of a trip, at 20 miles/hr for the second hour, at 30 miles/hr for the third hour and so on. How many hours will it take the train to complete a 450-mile journey? Assume that the train makes no intermediate stops.
• 8
• 8.5
• 9
• 9.5
• 10
Answer: 9
The speeds at which the train travels in successive hours are the terms of the arithmetic progression 10, 20, 30,... To find the time (say, t hours) taken by the train to complete the 450-mile journey, the sum of this sequence to the appropriate term is considered as follows.
Time x Speed = Distance = 450 miles.
1 hour x (10 + 20 + 30 + ... to t terms) = 450
Dividing by 10, one obtains 1 + 2 + 3 + ... + t = 45
Now, 1 + 2 + 3 + ... + t = t (t + 1)/2 (a well-known result worth remembering).
So, t (t + 1) = 90 giving t = 9.


7. [x] is the greatest integer less than or equal to the real number x. How many natural numbers n satisfy the equation [n1/2 ] = 17?
• 17
• 34
• 35
• 36
• 38
Answer: 35
Given that [n1/2] = 17, it is necessary that 17 < n1/2 < 18.
Squaring throughout, 289 < n < 324.
Clearly there are 35 values of n (natural numbers from 289 to 323) that satisfy [n1/2] = 17.


8. A watch costs $100. Anakin bought the watch at a discount of x%. He then sold the watch to Luke at a discount of y%. How many dollars did Luke pay Anakin for the watch?
• 100 − (x + y)
• 100 − (x + y + (xy/100))
• 100 − (x + y − (xy/100))
• 100 − (xy + (xy/100))
• 100 − (xy − (xy/100))
Answer: 100 − (x + y − (xy/100))
Since Anakin bought the watch at a discount of x%, he paid $100(1 − (x/100)) for the watch.
Luke bought the watch from Anakin at discount of y%.
So, Luke paid Anakin 100(1 − (x/100))(1 − (y/100)) = 100 − xy + (xy/100) = 100 − (x + y − (xy/100)).


9. The value of the recurring decimal number 0.6313131... is equal to
• 65/106
• 125/198
• 627/990
• 691/1124
• 737/1200
Answer: 125/198
0.6313131... = (6/10) + (31/1000) + (31/105) + (31/107) + ...
= (6/10) + (31/103) x (1 + (1/102) + (1/104) + ...)
= (6/10) + (31/103) x (1/(1 − (1/100))) on summing the infinite geometric series
= (6/10) + (31/990) = (594 + 31)/990 = 625/990 = 125/198.


10. The smallest whole number which is exactly divisible by 11/2, 21/4, 33/2, 42/5 is given by:
• 441
• 462
• 1089
• 1386
• 4158
Answer: 462
The given numbers are 11/2, 21/4, 33/2, and 42/5.
The required number that would be exactly divisible by all the above fractions would be the least common multiple (LCM) of 11, 21, 33 and 42.
The LCM may be obtained by prime factorization as follows:
21 = 3 x 7; 33 = 3 x 11; and 42 = 2 x 3 x 7 giving LCM = 2 x 3 x 7 x 11 = 462.


11. A natural number is divided into two positive unequal parts such that the ratio of the original number to the larger (divided) part is equal to the ratio of the larger part to the smaller part. What is the value of this ratio?
• (51/2 − 1)
• (51/2 + 1) / 2
• (51/2 + 1) / 4
• (51/2 + 1) / (51/2 − 1)
• (51/2 + 3) / (51/2 − 1)
Answer: (51/2 + 1) / 2
Let N be the natural number and L be the larger of the two individual parts.
Then, the given relation is N / L = L / (NL).
N 2LN = L2.
Let the required ratio be R = N / L. Dividing the above equation by L2 gives
R2R − 1 = 0.
The solution to the above quadratic equation isR = (1 + 51/2) / 2.
Since N is a natural number, only the positive root is considered for R.


12. If a and b are the nonzero roots of the quadratic equation x2 + ax + b = 0, then
a = 1 and b = −4
a = 1 and b = −2
a = −1 and b = −4
a = −1 and b = 2
a = −1 and b = 4
Answer: a = 1 and b = −2
Since a and b are the roots, x = a or x = b.
So, (xa) (xb) = 0.
x2 − (a + b) x + ab = 0.
On comparing with given quadratic equation, − (a + b) = a and ab = b.
Since a and b are nonzero, the solution is a = 1 and b = −2.


13. If a2 + 10b2 + c2 = 2b(3a + c), then which of the following is true?
I. a = 3b
II. a = b / 3
III. c = b
IV. c = 3b
V. c = a
VI. c = a / 3
• Only I
• Only II
• Only III
• I, III and VI
• II, IV and V
Answer: I, III and VI
Rearranging the given expression as follows:
(a2 + 9b2 − 6ab + (b2 + c2 − 2bc) = 0
Thus, (a − 3b)2 + (bc)2 = 0
Since the square of a quantity is always non-negative, the only possibility is that a = 3b and b = c.


14. If a = b = c / 3 and abc = 384, then a =
• 2 (21/3)
• 4 (21/3)
• 2
• 4
• 8
Answer: 4 (21/3)
On substituting b = a and c = 3a in abc = 384, one obtains (a) (a) (3a) = 384 or 3a3 = 384.
Thus, a3 = 128 or a = 1281/3= (641/3) (21/3) = 4 (21/3).


15. How many of the integers between 170 and 180 are prime numbers?
• 0
• 1
• 2
• 3
• 4
Answer: 2
Clearly, 172, 174, 176 and 178 are divisible by 2 and hence not prime.
175 is divisible by 5.
171 and 177 are divisible by 3 (because the sum of their digits is divisible by 3).
The numbers that remain are 173 and 179, which are prime.


16. If 1/x < 1/y, then
x < y
x > y
x < y if xy < 0
x < y if xy > 0
x > y if xy < 0
Answer: x < y if xy < 0
Multiply both sides of the inequality by xy; so, left-hand side = y and right-hand side = x.
If xy > 0, then the inequality is unchanged; so, y < x.
But, if xy < 0, then the inequality is reversed; so, y > x.


17. If the radius of a circle increases by 50%, by what percentage does the ratio of its perimeter to area decrease?
• 25
• 33.33
• 50
• 66.67
• 75
Answer: 33.33
Let the original radius be R. Then, Original perimeter = 2πR; Original area = πR2; Original ratio = 2/R.
Now, new radius = 1.5R; New perimeter = 3πR; New area = 2.25πR2; New ratio = 2/(1.5R) = 4/(3R).
Fractional change = (4/(3R) − 2/R) / (2/R) = −1/3.
Percentage decrease = 1/3 x 100% = 33.33%.


18. If a, b and m are integers and a = b − 5 + 2m, then which of the following integers must be odd?
a
b
a + 2b
ab
ab − 1
Answer: ab − 1
a, b and m are integers.
Now, 2m − 5 is odd, which implies that ab is odd.
So, if a is odd, then bis even or vice versa. This eliminates options "a" and "b".
The option "a + 2b" is eliminated since it may be odd or even depending on whether a is odd or even.
ab must be even because either a or b has to be even.
Hence, (ab − 1) must be odd.


19. If 21 is less than seven times the integer N, which in turn is less than 64, what is the sum of all possible values of N?
• 30
• 33
• 39
• 42
• 48
Answer: 39
The problem statement can be mathematically written as 21 < 7N < 64.
∴ 3 < N < 9.143
The integers that satisfy this inequality are 4, 5, 6, 7, 8 and 9. Their sum is 39.


20. Lines p and q are perpendicular to each other. The slope of line p is 1/8. What is the sum of the slopes of the two lines?
• −63/8
• −1
• 0
• 1/4
• 65/8
Answer: −63/8
When two lines are perpendicular to each other, the product of their slopes is −1.
Since the slope of line p is 1/8, the slope of line q must be −8.
Hence, the sum of the slopes of the two lines is (−8) + (1/8) = −63/8.


21. Hans Solo, a navy admiral, has to choose 3 ships out of his fleet of 8 destroyer ships. In how many ways can this be achieved, assuming that he is not biased toward choosing any particular ship?
• 24
• 48
• 56
• 120
• 336
Answer: 56
This problem requires calculation of the number of ways in which 8 different objects can be combined into groups of 3.
The number of ways in which r objects can be chosen from n objects is given by the expression:
nCr = n! / (r!(nr)!), where ! stands for the factorial notation.
Since n = 8 and r = 3, the required number of ways is 8! / (3! 5!) = 8 x 7 x 6 / (3 x 2 x 1) = 56.
Note that 8! = 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1; 3! = 3 x 2 x 1; and 5! = 5 x 4 x 3 x 2 x 1.


22. If 3x + 2y = −7 and 2x + 3y = 12, then x + y =
• 1
• 2
• 3
• 5
• None of these
Answer: 1
Adding the given equations, 5x + 5y = 5
Dividing by 5, x + y = 1.



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