Most frequent Gmat problems. Part 1
GMAT MAF:28
Difficult Problems from the Math Section
1. The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then n=?
Sol: First term a=2, common difference d=2 since even number
therefore sum to first n numbers of Arithmetic progression would be
n/2(2a+(n-1)d)
= n/2(2*2+(n-1)*2)=n(n+1) and this is equal to 79*80
therefore n=79 which is odd...
2. The price of a bushel of corn is currently $3.20, and the price of a peck of wheat is $5.80. The price of corn is increasing at a constant rate of 5x cents per day while the price of wheat is decreasing at a constant rate of cents per day. What is the approximate price when a bushel of corn costs the same amount as a peck of wheat?
(A) $4.50
(B) $5.10
(C) $5.30
(D) $5.50
(E) $5.60
Soln: 320 + 5x = 580 - .41x; x = # of days after price is same;
solving for x; x is approximately 48, thus the required price is 320 + 5 * 48 = 560 cents = $5.6
3. How many randomly assembled people do u need to have a better than 50% prob. that at least 1 of them was born in a leap year?
Soln: Prob. of a randomly selected person to have NOT been born in a leap yr = 3/4
Take 2 people, probability that none of them was born in a leap = 3/4*3/4 = 9/16. The probability at least one born in leap = 1- 9/16 = 7/16 < 0.5
Take 3 people, probability that none born in leap year = 3/4*3/4*3/4 = 27/64.
The probability that at least one born = 1 - 27/64 = 37/64 > 0.5
Thus min 3 people are needed.
4. In a basketball contest, players must make 10 free throws. Assuming a player has 90% chance of making each of his shots, how likely is it that he will make all of his first 10 shots?
Ans: The probability of making all of his first 10 shots is given by
(9/10)* (9/10)* (9/10)* (9/10)* (9/10)* (9/10)* (9/10)* (9/10)* (9/10)* (9/10) = (9/10)^10 = 0.348 => 35%
5. AB + CD = AAA, where AB and CD are two-digit numbers and AAA is a three digit number; A, B, C, and D are distinct positive integers. In the addition problem above, what is the value of C?
(A) 1
(B) 3
(C) 7
(D) 9
(E) Cannot be determined
Ans: AB + CD = AAA
Since AB and CD are two digit numbers, then AAA must be 111
Therefore 1B + CD = 111
B can assume any value between 3 and 9
If B = 3, then CD = 111-13 = 98 and C = 9
If B = 9, then CD = 111-19 = 92 and C = 9
So for all B between 3 & 9, C = 9
Therefore the correct answer is D (C = 9)
6. A and B ran a race of 480 m. In the first heat, A gives B a head start of 48 m and beats him by 1/10th of a minute. In the second heat, A gives B a head start of 144 m and is beaten by 1/30th of a minute. What is B’s speed in m/s?
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
Ans: race 1 :- ta = tb-6 ( because A beats B by 6 sec)
race 2 :- Ta = tb+2 ( because A looses to B by 2 sec)
By the formula D= S * T
we get two equations
480/Sa = 432/Sb -6 ------------1)
480/Sa = 336/Sb +2------------2)
Equating these two equations we get Sb = 12
ta,Sa stand for time taken by A and speed of A resp.
7. A certain quantity of 40% solution is replaced with 25% solution such that the new concentration is 35%. What is the fraction of the solution that was replaced?
(A) 1/4
(B) 1/3
(C) 1/2
(D) 2/3
(E) ¾
Ans: Let X be the fraction of solution that is replaced.
Then X*25% + (1-X)*40% = 35%
Solving, you get X = 1/3
8. A person buys a share for $ 50 and sells it for $ 52 after a year. What is the total profit made by him from the share?
(I) A company pays annual dividend
(II) The rate of dividend is 25%
(A) Statement (I) ALONE is sufficient, but statement (II) alone is not sufficient
(B) Statement (II) ALONE is sufficient, but statement (I) is not sufficient
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement alone is sufficient
(D) Each statement ALONE is sufficient
(E) Statements (I) and (II) TOGETHER are NOT sufficient
9. A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?
(A) 2/27
(B) 1/9
(C) 1/3
(D) 4/27
(E) 2/9
Ans: Case I: Red ball first and then white ball
P1 = 3/9*2/9= 2/27
Case 2: White ball first and then red ball
P2 = 2/9*3/9 = 2/27
Therefore total probability: p1 + p2 = 4/27
10. What is the least possible distance between a point on the circle x^2 + y^2 = 1 and a point on the line y = 3/4*x - 3?
A) 1.4
B) sqrt (2)
C) 1.7
D) sqrt (3)
E) 2.0
Ans: The equation of the line will be 3x - 4y - 12 = 0.
This crosses the x and y axis at (0,-3) and (4,0)
The circle has the origin at the center and has a radius of 1 unit.
So it is closest to the given line when, a perpendicular is drawn to the line, which passes through the origin.
This distance of the line from the origin is 12 / sqrt (9 + 16) which is 2.4
[Length of perpendicular from origin to line ax +by + c = 0 is
mod (c / sqrt (a^2 + b^2))]
The radius is 1 unit.
So the shortest distance is 2.4 - 1 unit = 1.4 units
11.
In the square above, 12w = 3x = 4y. What fractional part of the square is shaded?
A) 2/3
B) 14/25
C) 5/9
D) 11/25
E) 3/7
Sol: Since 12w=3x=4y,
w:x=3:12=1:4 and x:y=4:3
so, w = 1
x = 4
y = 3
the fractional part of the square is shaded:
{(w+x)^2 - [(1/2)wx + (1/2)wx +(1/2)xy + (1/2)w(2w)]}/(w+x)^2
= {(w+x)^2 - [wx + (1/2)xy + w^2)]}/[(w+x)^2]
=[(5^2) -(4+6+1)]/(5^2)
= (25 - 11)/25
= 14/25
12. The average of temperatures at noontime from Monday to Friday is 50; the lowest one is 45, what is the possible maximum range of the temperatures?
20 25 40 45 75
Ans: The answer 25 doesn't refer to a temperature, but rather to a range of temperatures.
The average of the 5 temps is: (a + b + c + d + e) / 5 = 50
One of these temps is 45: (a + b + c + d + 45) / 5 = 50
Solving for the variables: a + b + c + d = 205
In order to find the greatest range of temps, we minimize all temps but one. Remember, though, that 45 is the lowest temp possible, so: 45 + 45 + 45 + d = 205
Solving for the variable: d = 70
70 - 45 = 25
13. If n is an integer from 1 to 96, what is the probability for n*(n+1)*(n+2) being divisible by 8? 25% 50% 62.5% 72.5% 75%
Soln: E = n*(n+1)*(n+2)
E is divisible by 8, if n is even.
No of even numbers (between 1 and 96) = 48
E is divisible by 8, also when n = 8k - 1 (k = 1,2,3,.....)
Such numbers total = 12(7,15,....)
Favorable cases = 48+12 = 60.
Total cases = 96
P = 60/96 = 62.5
Method 2:
From 1 to 10, there are 5 sets, which are divisible by 8.
(2*3*4) (4*7*6) (6*7*8)(7*8*9)(8*9*10)
So till 96, there will be 12 * 5 such sets = 60 sets
so probability will be 60/96 = 62.5
14. Kurt, a painter, has 9 jars of paint:
4 are yellow
2 are red
rest are brown
Kurt will combine 3 jars of paint into a new container to make a new color, which he will name accordingly to the following conditions:
Brun Y if the paint contains 2 jars of brown paint and no yellow
Brun X if the paint contains 3 jars of brown paint
Jaune X if the paint contains at least 2 jars of yellow
Jaune Y if the paint contains exactly 1 jar of yellow
What is the probability that the new color will be Jaune
a) 5/42
b) 37/42
c) 1/21
d) 4/9
e) 5/9
Sol: 1. This has at least 2 yellow meaning..
a> there can be all three Y => 4c3
OR
b> 2 Y and 1 out of 2 R and 3 B => 4c2 x 5c1
Total 34
2.This has exactly 1 Y and remaining 2 out of 5 = > 4c1 x 5c2
Total 40
Total possibilities = (9!/3!6!) = 84
Adding the two probabilities: probability = 74/84 = 37/42
15. TWO couples and a single person are to be seated on 5 chairs such that no couple is seated next to each other. What is the probability of the above??
Soln:
Ways in which the first couple can sit together = 2*4! (1 couple is considered one unit)
Ways for second couple = 2*4!
These cases include an extra case of both couples sitting together
Ways in which both couple are seated together = 2*2*3! = 4! (2 couples considered as 2 units- so each couple can be arrange between themselves in 2 ways and the 3 units in 3! Ways)
Thus total ways in which at least one couple is seated together = 2*4! + 2*4! - 4! = 3*4!
Total ways to arrange the 5 ppl = 5!
Thus, prob of at least one couple seated together = 3*4! / 5! = 3/5
Thus prob of none seated together = 1 - 3/5 = 2/5
