A piece of work can be done by Rams and Sham in 12 days, by Sham and Hart in 15 days and by Hart and Rams in 20 days. Rams alone will complete the work in how many days?

A. 35
B. 32
C. 30
D. 28
E. 25

Solution:
Set R, S, and H as the respective rates.

Then we have R + S = 1/12,   S + H = 1/15,   H + R = 1/20.

We want to solve for R, we can eliminate S first by subtracting the first two equations with each other:

R - H = 1/12 - 1/15 = (15 - 12) / (12*15) = 3 / (12*15) = 1/60.

We also have R + H = 1/20.

Add these two equations to get: 2R = 1/60 + 1/20 = 1/60 + 3/60 = 4/60 = 1/15.

R = 1/30, so it would take R 30 days to complete the task alone.

Ans: C

How many factors of 2^3 * 3^4 * 5^5 are even numbers?

A. 20
B. 30
C. 90
D. 100
E. 120

Solution:
The easier question to answer is "how many factors of 2^3 * 3^4 * 5^5 are NOT even numbers."
Those factors will only have prime factors of 3 or 5. To construct a factor that is odd, there are 5 possible powers of 3 (0, 1, 2, 3, 4) and 6 possible powers of 5 to choose from (0, 1, 2, 3, 4, 5). Therefore there are 5 * 6 = 30 factors that are not even.

With the same theory, there are 4 * 5 * 6 = 120 factors in total. Subtracting the 30 odd factors, there are 90 even.

Ans: C

Two friends separated by a certain distance start walking towards each other. When they meet one of them has walked 30 meters more than the other. If the ratio of the distances that each has covered is 2 : 3, find the distance that originally separated them.

A. 30
B. 60
C. 90
D. 120
E. 150

Solution:
We may set their walking distances as 2x and 3x, then the difference is x which is also 30 meters since one has walked 20 meters more than the other. Then the walking distances are 60 and 90, originally they were separated by 60 + 90 = 150.

Ans: E

A function f(x) is defined as f(x+y)=f(x)+f(y) for all real value of x and y, and f(1)=5. What is the value of f(3)?

A. 1
B. 5/3
C. 5
D. 10
E. 15

Solution:

We are only given f(1) = 5. We need to work our way to change the input to x = 3 instead of x = 1. Observe that if we plug in y = x the property becomes f(x + x) = f(x) + f(x), or f(2x) = 2f(x). We can do this again but with y = 2x to get f(x + 2x) = f(x) + f(2x) = f(x) + 2f(x), and we get f(3x) = 3f(x). Now we can plug in x = 1 into this to change the input to 3, f(3*1) = 3*f(1) =3 * 5 = 15.

​Ans: E

Is x > y?

(1) x^3 > 100y^7

(2) x > 0

Solution:
Analyzing the question:
We have no information on x and y so we need to consider positive/negative values and fractional/whole number values.

Statement 1:
We can have x = 100 and y = 1 for x > y, or x = 1/100 and y = 1/99 for x < y. Insufficient.

Statement 2:
Insufficient.

Combined:
The key here is to see we cannot prove x > y with this information. As an example, the case x = 1/100 and y = 1/99 still gives x < y so combined information is still insufficient.

Ans: E