A. 35

B. 32

C. 30

D. 28

E. 25

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

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A. 20

B. 30

C. 90

D. 100

E. 120

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

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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

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A. 1

B. 5/3

C. 5

D. 10

E. 15

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

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(1) x^3 > 100y^7

(2) x > 0

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

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