Data Sufficiency - 42

Is a-3b an even number?

1). b=3a+3
2). b-a is an odd number

Answer: C

From statement (1): Given that b=3a+3
Thus a-3b=a-3(3a+3) = -8a-9 which may be even, odd, integer, non-integer, rational etc ... Hence insufficient

From statement (2): Given that b-a is an odd number implies b is of the form b=(2k+1)+a where k is an integer
Thus a-3b= a-3[(2k+1)+a] = -2a -6k-3 which may be even, odd, integer, non-integer, rational etc ..Hence insufficient

Taking statement (1) and (2) together: -8a-9=-2a-6k-3 for some integer k
or -6a=-6k+6=-6(k+1) implies a=k+1
Thus a is an integer, either odd or even

Now statement (2) tells us that b is also an integer and that exactly one of {a,b} is even
If a is even and b is odd, a-3b is odd
If b is even and a is odd a-3b is odd

Thus (1) and (2) combined tell us that a-3b is an odd number...hence sufficient

Problem Solving - 43

If the sum of four consecutive positive integers a three digit multiple of 50, the mean of the these integers must be one of x possible values, where x=

(A) 7
(B) 8
(C) 9
(D) 10
(E) more than 10

Answer: C

Suppose four integers are a, a+1, a+2 and a+3
Hence a + (a+1) + (a+2) + (a+3) = 4a+6
Now 4a+6 will be a multiple of 50 i.e 4a+6=50m where m can take any value from {2,3...,19}

But a = (50m-6)/4 = (25m-3)/2 must be an integer, so m must be odd.

Thus m can be any odd integer from 3 to 19
3=1+1*2
19=1+9*2

So there are 9 different values for m, a and 4a+6, as well as (4a+6)/4

Data Sufficiency - 41

Sania has a circular garden in her backyard. She puts poles A,B and C on the circumference of her garden. Then she ties ropes between these poles. Is length of one of the ropes is equal to the diameter of her garden?

1. Slope of line joining pole A and B is 3/4 and slope of line joining poles B and C is -4/3
2. Length of line joining pole A and B is 12 and length of line joining B and C is 5


Answer: A

From statement (1): Given that the slope of line AB is 3/4 and slope of line BC is -4/3. This implies that the product of slopes = -1. Hence AB perpendicular BC and B is a right angle. Thus AC is a diameter which implies ABC form a semi-circle.
Hence sufficient

From statement (2): Given that length of line AB is 12 and length of BC is 5. However this does not imply that ABC is a right angled triangle. We can draw number of different triangles with the same given two sides but with different third side.
Hence insufficient

Problem Solving - 42

E is a collection of four odd integers and the greatest difference between any two integers in E is 4. The standard deviation of E must be one of how many numbers?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 7


Answer: B

Suppose the integers are 1, 3 and 5. Therefore the four integers can be:

1, 5, 5, 5
1, 3, 5, 5
1, 3, 3, 5
1, 1, 5, 5
1, 1, 1, 5
1, 1, 3, 5

Here two pairs have the same standard deviation. thus in all we have four different standard deviations

Data Sufficiency - 40

In XY plane, does the line with equation y=3x+2 contain point (r,s)?

1) (3r + 2 - s)(4r + 9 - s) = 0
2) (4r - 6 - s)(3r + 2 - s) = 0

Answer: C

Given that y = 3x+2 implies that does 3x+2-y = 0 contains the point (r,s) implies is (3r+2-s) = 0 ?

From statement (1): (3r+2-s)(4r+9-s) = 0 implies either (3r+2-s) = 0 or (4r+9-s) = 0.
Now when (3r+2-s)...the line passes through (r,s)
When (4r+9-s) = 0 ...we cannot determine that whether the line passes through (r,s) or not.
Hence insufficient

From statement (2): (4r-6-s)(3r+2-s) = 0 implies either (4r-6-s) = 0 or (3r+2-s) = 0
Now when (4r-6-s) = 0 ... we cannot determine that whether the line passes through (r,s) or not
When (3r+2-s) = 0..the line passes through (r,s)
Hence insufficient

Taking statement (1) and (2) together: (3r+2-s)(4r+9-s) = 0 and (4r-6-s)(3r+2-s) =0... We cannot have both 4r+9-s=0 and 4r-6-s=0 so it is (3r+2-s) = 0 ... only this equation makes both the equations to be 0
Hence sufficient

Problem Solving - 41

If eleven consecutive integers are listed from least to greatest, what is the average (arithmetic mean) of the eleven integers?

(1) The average of the first nine integers is 7.
(2) The average of the last nine integers is 9.

Answer: D

Let the numbers be a, b, c, d, e, f, g, h , i, j, k

i) For odd number of consecutive integers median = mean
ii)We also know that the median is the "middle" number in a group (when arranged in ascending or descending order) consisting of an odd number of numbers

We have to find f

From statement (1): It is given that average of first nine numbers = 7
Hence this implies e = 7 ..since it is given numbers are consecutive hence f = 8
Thus sufficient

From statement (2): It is given that average of last nine numbers = 9
Hence this implies g = 9..since it is given numbers are consecutive hence f = 8
Thus sufficient

Data Sufficiency - 39

If *triangle* denotes one of the four arithmetic operations addition, subtraction, multiplication and division, what is the value of 1 *triangle* 2 ?

(1) n *triangle* 0 = n for all integers n.
(2) n *triangle* n = 0 for all integers n.

Answer: B

From statement (1): *triangle* can be both positive or negative as

n-0 = n
n+0 = n

Hence insufficient

From statement (2): *triangle* can only be negative in this case as

n-n = 0

Hence sufficient

Problem Solving - 40

5 pieces of wood have an average (arithmetic mean) length of 124 centimeters and a median length of 140 centimeters. What is the maximum possible length in centimeters of the shortest piece of wood?

A. 90

B. 100

C. 110

D. 130

E. 140

Answer: B

Shortest to Longest length ---- L1, L2, L3 = 140, L4, L5
L1 + L2 + 140 + L4 + L5 = 5 * 124
For L1 to be the maximum, L4 and L5 should be minimum

L1=L2
Thus, L1 + L2 = (5*124) - (140*3) = 620 - 420 = 200

Hence L1 = L2 = 100

Problem Solving - 39

On a certain day, Tim invest $1,000 at 10 percent annual interest, compound annually, and Lana invested $ 2,000 at 5 percent annual interest, compound annually. The total amount of interest earned by Tim's investment in the first 2 years was how much greater than the total amount of interest earned by Lana's investment in the first 2 years?

A. 5
B. 15
C. 50
D. 100
E. 105

Answer: A

Amount = P[1+(r/100)] ^ t
Tim's investment Amount = 1000(1+10%)^2=1210
Interest earned by Tim = Amount - P = 1210 - 1000 = 210
Lana's investment Amount = 2000(1+5%)^2=2205
Interest earned by Lana = Amount - P = 2205 - 2000 = 205
Hence 210-205=5

Problem Solving - 38

6 machines, each working at the same constant rate, together can complete a certain job in 12 days, How many additional machines, each working at the same constant rate, will be needed to complete the job in 8 days?

A. 2

B. 3

C. 4

D. 6

E. 8

Answer: B

6 machines take = 12 days.
Therefore 1 machine = 12*6 days.
8 days will take = (12*6)/8 = 9 machines.

No of additional machines required = 9-6 = 3

Problem Solving - 37

Of the 800 companies in Company X, 70% have been with the company for at least 10 years. If y of these "long-term" members were to retire, and no other employee changes were to occur, what value of y would reduce the percent of "long-term" employees in the company to 60%.

A) 200
B) 160
C) 112
D) 80
E) 56

Answer: A

Assume y=x
The number of people working more than 10 years = 70% of 800 = 560
Hence (560-x)/(800-x)=60%
Thus x=200

OR

The number of long term workers = 70% of 800 = 560
Now if y of the long term workers retired, then long term workers left are 560-y and the number of total employees = 800-y

Thus (560-y)/(800-y) * 100 = 60
560-y = 480 - 0.6y
80 = 0.4y
y = 200 Ans

Problem Solving - 36

Equal amounts of water were poured into two empty jars of different capacities, which made one jar 1/4 full and the other jar 1/3 full. If the water in the jar with lesser capacity is then poured into the jar with the greater capacity, what fraction of the larger jar will be filled with water?

A. 1/7

B. 2/7

C. 1/2

D. 7/12

E. 2/3

Answer: C

The jar that is 1/3 full is smaller
water in jar 1 = water in jar 2
jar 1 is now twice as full (1/4)*2 = 1/2

or

1/4 + 1/4 = 1/2

Problem Solving - 35

The rate of a certain chemical reaction is directly proportional to the square of the concentration of chemical A present and inversely proportional to the concentration of chemical B present. If the concentration of chemical B is increased by 100 percent, which of the following is closest to the percent change in the concentration of chemical A required to keep the reaction rate unchanged?

A. 100% decrease
B. 50% decrease
C. 40% decrease
D. 40% increase
E. 50% increase

Answer: D

rate = k*(A1^2)/B
If concentration of chemical B is increased by 100 percent then
rate = k*(A2^2)/2B

(A2/A1) ^ 2 = 2
A2/A1 = Square root (2)
(A2/A1) - 1 = Square root (2)-1 = 0.414
(A2-A1)/A1 = 0.414 = 40% approximately

Problem Solving - 34

A certain library assesses fines for overdue books as follows. On the first day that a book is overdue, the total fine is $0.10. For each additional day that the book is overdue the total fine is either increased by $0.30 or double, whichever results in the lesser amount. What is the total fine for a book on the fourth day it is overdue?

A. $0.60
B. $0.70
C. $0.80
D. $0.90
E. $1.00

Answer: B

Day 1 = 0.10
Day 2 = 0.20
Day 3 = 0.20*2 = 0.40

Thus on 4th day the total fine is: 0.10 + 0.20 + 0.40 = 0.70

Data Sufficiency - 38

If q is a integer, is q^4 a multiple of 64?

(1) q^4 is not a multiple of 128.
(2) q^2 has 27 factors, 7 of which are less than or equal to 10

Answer: A

OE:
From statement (1): Given q is an integer, thus q can be written as product of distinct prime factors, where the power of 2 must be a whole number(a non-negative integer). This implies that the power of 2 in the prime factorization of q^4 must be a multiple of 4. As 2^7 is not a factor of q^4, the highest power of 2 that could be a factor of q^4 is 2^4.
Hence q^4 is not a multiple of 64=2^6......sufficient

From statement (2): Given that q^2 has 27 factors. Thus q^2 could be of the form
(i) a^26, or
(ii)b^2*c^8 or
(iii) a^2*b^2*c^2 where a b and c are distinct prime numbers.

Because q^2 has 7 factors less than 11, (i) is impossible.

As for (ii) 2^8*3^2 has exactly 7 factors under 11 (1,2,3,4,6,8,9), in which case q^4 would be a multiple of 64. 3^8*2*2 is not a possibility for q^2 (it only has 6 factors under 1: 1,2,3,4,6,9)

Regarding (iii) if q^2=2^2*3^2*5^2, q^2 would have 8 factors under 11- 1,2,3,4,5,6,9,10 and q^2=2^2*3^2*7^2 would have 7 factors under 11: 1,2,3,4,6,7,9) In this case, q^4 would not be a multiple of 64. Thus (2) is insufficient

hence the answer A

Problem Solving - 33

A hiker walking at a constant rate of 4 miles per hour is passed by a cyclist traveling in the same direction along the same path at a constant rate of 20 miles per hour. The cyclist stops to wait for the hiker 5 minutes after passing her, while the hiker continue to walk at her constant rate. How many minutes must the cyclist wait until the hiker catches up?

A. 20/3
B. 15
C. 20
D. 25
E. 80/3

Answer: C

Hiker's relative speed = 16 m/h
Hiker traveled in 5 min: 16 * 5/60 = 4/3 miles.
Time taken by hiker to cover 4/3 miles: 4/(4/3) = 1/3 hrs = 20 minutes

Data Sufficiency - 37

At least 100 students at a certain high school study Japanese. If 4 percent of the students who study French also study Japanese, do more students at the school study French than Japanese?

1). 16 students at the school study both French and Japanese.
2). 10 percent of the students at the school who study Japanese also study French.

Answer: B

From statement (1):16 students study both French and Japanese, so 16/0.04=400 students study French. But we don't know how many the total number of students are so the number Japanese student can be at least 100 or more than 400....insufficient

From statement (2): 10% Japanese studying students = 4% French studying students
10/100 study Japanese = 4/100 study French
25 study Japanese = 10 study French
Hence study French > study Japanese ... thus more students at the school study French than study Japanese....sufficient

Hence B

Problem Solving - 32

List K consists of 12 consecutive integers, if -4 is the least integer in list K, what is the range of the positive integers in the list K?

A. 5
B. 6
C. 7
D. 11
E. 12

Answer: B

The least number in the list is -4, thus the list is: -4,-3,-2,-1, 0, 1, 2, 3, 4, 5, 6, 7
Positive integers in the above list: 1, 2, 3, 4, 5, 6, 7
Therefore the range of the positive integers is 7-1 = 6

Data Sufficiency - 36

If set S consist of the numbers 1, 5, -2, 8, and n, is 0 less than n less than 7 ?

1). the median of the numbers in S is less than 5.
2). the median of the numbers in S is greater than 1

Answer: C

From statement (1): Median will be less than 5 only if n is located below 5
Thus the median will either be 1 if n less than 1 or n if 1 less than n less than 5
Hence in both cases n is less than 5 but it can also be n less than 0 ....insufficient

From statement (2): Median will be greater than 1 only if n is located above 1
Thus median will either be 5 if n greater than 5 or n if 1 less than n less than 5
Hence in both cases n greater than 1 but it can also be n greater than 7 ....insufficient

Taking statements (1) and (2) together: 1 less than n less than 5 which lies within the given interval 0 less than n less than 7 ...thus possible values for n are be 2, 3, or 4

Hence the answer C

Data Sufficiency - 35

Data Sufficiency - 34

In the xy-plane, at what two points does the graph of y=(x+a)(x+b) intersect the x-axis?

1). a+b= -1
2). The graph intersects the y-axis at (0,-6)

Answer: C

From Statement (1) -- a+b = -1...no information about a and b ...hence insufficient

From Statement (2) -- If x = 0 the y = -6 thus ab = 6....insufficient

Taking statements (1) and (2) together: (x+a)*(x+b)=0

x^2+(a+b)x+ab=0

Hence x=-3, x=2

Thus the answer C