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