Problem Solving - 55

Which of the following fractions has a decimal equivalent that is a terminating decimal?

A. 10/189
B. 15/196
C. 16/225
D. 25/144
E. 39/ 128

Answer: E

RULE: If denominator of a fraction has just the prime factors of 2 or 5 or both it is terminating otherwise not

128 = 2*2*2*2*2*2*2 => has only factors of 2..hence the ans

Problem Solving - 54

This year Henry will save a certain amount of his income, and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1 + r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he was available to spend will be equal to half the amount that he spends this year?

(A) 1/(r+2)
(B) 1/2(r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)


Answer: E

Let total income = x
Let saved = y
=> spent = x-y

Total dollars available that he can spend next year = y(1+r)

Given y(1+r) = (x-y)/2
=> y = x/(2r+3)

Fraction (y/x) = 1/(2r+3)

Hence E

Problem Solving - 53

A student worked 20 days. For each of the amount shown (see attached table) in the first row of the table, second row gives the number of days the student earned that amount. Median amount of money earned per day for 20 days is?

A) 96
B) 84
C) 80
D) 70
E) 48

Answer: B

Median day = 20+1)/2 = 10.5 th -- money earned was 84
= Average value of 10th and 11th day in the sequence = Median amount of money
Average value of 10th day = 84
Average value of 11th day = 84
Average value of 10th and 11th day = 84 ans

Data Sufficiency - 50

What is the median number of employees assigned per project for the projects at Company Z?

(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.

Answer: OA - C

From Statement 1): It is given that 25 percent of the projects at Company Z have 4 or more employees assigned to each project - but we donot know the percentage of projects who have employees less than 4 or in other words we do not have any information about the rest 75% projects ---- hence insufficient

From Statement 2): It is given that 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project - but we donot know the percentage of projects who have employees more than 2 or in other words we do not have any information about the rest 65% projects---- hence insufficient

Taking both the statements together:
25 percent of the projects at Company Z have employees 4 , 5, 6..
35 percent of the projects at Company Z have employees 2, 1, 0
=> 40% of projects have 3 employees = > median value is 3

(1-35)employees -- (36-75)employees -- (76-100)employees
2 or less than 2 ---------3, 3, 3, ------------ 4 or more than 4

Problem Solving - 52

A boat traveled up stream a distance of 90 miles at an average speed of (v-3) mph and then traveled the same distance downstream at an average speed of (v+3) mph. If the trip upstream took half an hour longer than the trip downstream, how many hours did it take the boat to travel downstream?

(A) 2.5
(B) 2.4
(C) 2.3
(D) 2.2
(E) 2.1

Answer: A

Total upstream time taken by boat to travel = 90/ (v-3) hrs
Total downstream time taken by boat to travel = 90/ (v+3) hrs

It is given that upstream took half an hour longer than the trip downstream:

=> 90/(v-3) - 90/(v+3) = 1/2
=> [90*(v+3) - 90(v-3)] / [(v^2) -9)] = 1/2
=> 90*[(v+3)-(v-3)] / [(v^2) -9] = 1/2
=> 90*6/ [(v^2) -9] = 1/2
=> v^2 - 9 = 2*90*6
=> v^2 = 2*90*6 + 9 = 2*9*10*6 + 9 = 9(2*10*6+1) = 9(121) = 9*11*11
=> v = 3*11 = 33

We are suppose to calculate = 90/(v+3) = 90/36 = 30/12 = 5/2 = 2.5 ans

Problem Solving - 51

Each of the 10 machines works at the same constant rate of doing certain job. The amount of time needed by 10 machines, working together to complete the job is 16 hrs. How many hours are needed if only 8 machines working together were to complete the job?

A. 18
B. 20
C. 22
D. 24
E. 26

Answer: B

10 machines, working at same constant rate, take time to complete a job = 16 hrs
thus 1 machine takes time to complete a job = 10 * 16 = 160 hrs
=> 8 machines will take = 160/8 = 20 hrs to complete the job.

Problem Solving - 50

If x = -|x|, then which one of the following statements could be true?

I. x=0
II. x < 0
III. x > 0

A). none
B). I only
C). III only
D). I and II
E). II and III

Answer: D

Given x = - |x|
We know that |x| >= 0
=> -|x| = x <=0

Thus inequalities represented by I and II

Data Sufficiency - 49

Does the curve (x-a)^2 + (y-b)^2=16 intersect the y-axis ?

(1) a^2+b^2>16
(2) a=|b|+5

Answer: B

The given curve will intersect the y-axis when x=0
Thus we get a^2 + (y-b)^2 = 16
<=> a^2 + y^2 + b^2 - 2yb = 16
<=> y^2 - 2yb + a^2 + b^2 -16 = 0

In order to have real roots b^2 - 4ac >= 0

=> 4b^2 - 4(1)(a^2 + b^2 -16) >=0
=> a^2 <=16

From statement (1): Given that a^2 + b^2 > 16
No information about a^2 <=16 --- hence insufficient

From statement (2): Given a = |b|+5
=> |b| is positive or zero.
=> a is atleast equal to 5 and value of a^2 is atleast 25.
But we know that a^2 <=16 ====> does not intersect the y axis ---- hence sufficient

Hence the answer B

Problem Solving - 49

Pumps A, B, and C operate at their respective constant rates. Pumps A and B, operating simultaneously, can fill a certain tank in 6/5 hours, pumps A and C, operating simultaneously, can fill the tank in 3/2 hours; and pumps B and C, operating simultaneously, can fill the tank in 2 hours. How many hours does it take pumps A, B, and C, operating simultaneously, to fill the tank?

A) 1/3
B) 1/2
C) 2/3
D) 5/6
E) 1

Answer: E

1/A + 1/B = 5/6
1/B +1/C = 2/3
1/C + 1/A =1/2

Adding  all three equations above we get:

1/A + 1/B + 1/B +1/C + 1/C + 1/A = 5/6 + 2/3 + 1/2
2(1/A + 1/B + 1/C) = (5 + 4 + 3)/ 6 = 12/6
1/A + 1/B + 1/C = 12/6*2 = 1

=> Pumps A, B, and C, operating simultaneously will fill the tank in 1 hr

Set Theory - Formulas

Formulas for three-component set problems:
u = union
n = intersection

1. For 3 sets A, B, and C: P(AuBuC) = P(A) + P(B) + P(C) – P(AnB) – P(AnC) – P(BnC) + P(AnBnC)

2. No of persons in exactly one set:
P(A) + P(B) + P(C) – 2P(AnB) – 2P(AnC) – 2P(BnC) + 3P(AnBnC)

3. No of persons in exactly two of the sets: P(AnB) + P(AnC) + P(BnC) – 3P(AnBnC)

4. No of persons in exactly three of the sets: P(AnBnC)

5. No of persons in two or more sets: P(AnB) + P(AnC) + P(BnC) – 2P(AnBnC)

6. No of persons in atleast one set:
P(A) + P(B) + P(C) - P(AnB) - P(AnC) - P(BnC) + 2 P(AnBnC)

1. For three sets A, B, and C, P(AuBuC): (A+B+C+X+Y+Z+O)
2. Number of people in exactly one set: ( A+B+C)
3. Number of people in exactly two of the sets: (X+Y+Z)
   
4. Number of people in exactly three of the sets: O
   
5. Number of people in two or more sets: ( X+Y+Z+O)
   
6. Number of people only in set A: A
7. P(A): A+X+Y+O
8. P( AnB): X+O

Problem Solving - 48

What is the angle formed by the diagonals of two adjacent surfaces of a cube?

A) 45
B) 60
C) 90
D) 30
E) None of these


Answer: B

Draw from two adjacent surfaces of a cube the diagonals.

Now see that it forms an equilateral triangle if we draw a third diagonal i.e diagonal in green color.
Thus angles of the triangle will be 60 degrees; hence the answer.

Data Sufficiency - 48

In the number line, are x and y on different sides of zero point?

1). The distance from x to zero is equal to the distance from y to 1
2). The sum of the distance from x to zero and the distance from y to 1 is less than 1

Answer: E

From statement (1): |x-0|=|y-1|
x = y-1
x = 1-y ....hence insufficient

From statement (2): |x-0|+|y-1| less than 1
= x+y-1 less than 1
= x+1-y less than 1
= -x+y-1 less than 1
= -x+1-y less than 1 ....hence insufficient

Taking statements (1) and (2) together: still insufficient

Data Sufficiency - 47

If Line K in the XY-Plane has equation y=mx+b, where m and b are constants, what is the slope of K?

1. K is parallel to the line with equation y=(1-m)x+(b+1)
2. K intersects the line with equation y=2x+3 at the point (2,7)

Answer: A

From statement (1): y=(1-m)x+(b+1) has the same slope as y=mx+b. (Parallel lines have same slope)
Thus 1-m = m
implies Slope of K=m=1/2 ---- Hence sufficient

From statement (2): just says line y=2x+3 is not parallel to K, these two lines can have any angle between them other than 0, 180, 360 degrees ---- hence insufficient

Hence answer A

Data Sufficiency - 46

What is the greatest common divisor of positive integers a and b?

(1) a and b share exactly one common factor
(2) a and b are both prime numbers


Answer: A

From statement (1): we know that a and b have only one common factor, and we also know that all positive integers share the common factor 1 only, so we know it must be 1...hence sufficient

From Statement (2): we know that a and b are both prime, this implies the greatest common factor will have to be 1 or if a = b could be the same prime number then the GCF would be a (=b). ...hence insufficient

NOTE: You cannot assume that a and b are different integers if the question stem does not states the same

Problem Solving - 47

If n is a positive integer and the product of all the integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?

A) 10
B) 11
C) 12
D) 13
E) 14

Answer: B

990 is a multiple of n! implies it must contain all the prime factors of 990
Largest prime factor of 990 is 11 implies n! must have 11 as a factor

Now since n! = 990x where x is integer it implies it can have prime factors more than 11 but not less than 11

Thus least possible value of n is thus 11

Problem Solving - 46

A certain restaurant offers 6 kinds of cheese and 2 kinds of fruit for its dessert platter. If each dessert platter contains an equal number of kinds of cheese and kinds of fruit, how many different dessert platters could the restaurant offer?

A) 8
B) 12
C) 15
D) 21
E) 27

Answer: E

Kinds of platter:

1 cheese + 1 fruit
Total = 6 * 2 = 12 types of platters

2 cheese + 2 fruit
Total = 6C2 * 2C2 = 15 * 1 = 15 types

Total: 12 + 15 = 27

Data Sufficiency - 45

p^a * q^b * r^c * s^d = x, where x is a perfect square. If p, q, r, and s are prime integers, are they distinct?

(1) 18 is a factor of ab and cd
(2) 4 is not a factor of ab and cd

Answer: B

OE: When a perfect square is broken down into its prime factors, those prime factors always come in "pairs." For example, the perfect square 225 (which is 15 squared) can be broken down into the prime factors 5 * 5 * 3 * 3. Notice that 225 is composed of a pair of 5's and a pair of 3's.

The problem states that x is a perfect square. The prime factors that build x are p, q, r, and s. In order for x to be a perfect square, these prime factors must come in pairs. This is possible if either of the following two cases hold:

Case One: The exponents a, b, c, and d are even. In the example 3^2 5^4 7^2 11^6, all the exponents are even so all the prime factors come in pairs.

Case Two: Any odd exponents are complemented by other odd exponents of the same prime. In the example 3^1 5^4 3^3 11^6, notice that 3^1 and 3^3 have odd exponents but they complement each other to create an even exponent (3^4), or "pairs" of 3's. Notice that this second case can only occur when p, q, r, and s are NOT distinct. (In this example, both p and r equal 3.)

Statement (1) tells us that 18 is a factor of both ab and cd. This does not give us any information about whether the exponents a, b, c, and d are even or not.

Statement (2) tells us that 4 is not a factor of ab and cd. This means that neither ab nor cd has two 2's as prime factors. From this, we can conclude that at least two of the exponents (a, b, c, and d) must be odd. As we know from Case 2 above, if paqbrcsd is a perfect square but the exponents are not all even, then the primes p, q, r and s must NOT be distinct.

The correct answer is B: Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Problem Solving - 45

If p is the smallest positive integer such that (p^3)/3920 is also an integer, what is the sum of the digits of p?

(A) 5
(B) 7
(C) 9
(D) 11
(E) 13

Answer: A

m = p*p*p/ 3920
m = p*p*p/ 2*2*2*2*5*7*7
m = p*p*p/ [(4^2)*5*(7^2)]

Hence p must be equal to 4*5*7
thus p = 140
Hence sum of digits = 1+4+0 = 5

Problem Solving - 44

Samar tried to type his new 7-digit phone number on a form, but what appeared on the form was 39269, since the '4' key on his computer no longer works. His secretary has decided to make a list of all of the numbers that could be Samar's new number. How many numbers will there be on the list?

(A) 21
(B) 24
(C) 25
(D) 30
(E) 36

Answer: A

It is clear from the question that there are two 4's missing as it is a seven digit number.
Total number of ways of choosing places for these two missing 4's in these 7 digits is 7C2 = 21 Once you fix the place for these two 4's rest all numbers will occupy remaining places.

Data Sufficiency - 44

In triangle ABC, AB has a length of 10 and D is the midpoint of AB. What is the length of line segment DC?

(1) Angle C= 90
(2) Angle B= 45

Answer: A

From statement (1): it is given that angle C = 90 degrees ...this implies that ABC is a right angle triangle with AB as the hypotenuse and DC as the median. We know that --- In all right triangles, the median on the hypotenuse is the half of the hypotenuse. Hence DC=5

Data Sufficiency - 43

In the figure shown, what is the value of x?

(1) The length of line segment of QR is equal to the length of line segment RS
(2) The length of line segment of ST is equal to the length of line segment TU

Answer: C

From statement (1): Length of line segment of QR is equal to the length of line segment RS ..this implies angle RQS = angle RSQ = p(say)
From statement (2): Length of line segment of ST is equal to the length of line segment TU .. this implies angle TUS = angle TSU = q(say)

Hence p+p+angle QRS = 180 --- eq(1) and q+q+angle UTS = 180 --- eq(2)
Thus, p+q+x = 180

Now because angle RPT = 90, QRS+UTS= 90
adding eq(1) and eq(2) we get:
2p+2q+QRS+UTS = 360
2p+2q+90=360
p+q = 270/2 = 135

Now x = 180-p-q..hence the answer C
x = 180 - (p+q) = 180 - 135 = 45

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