If x and y are positive integers and \((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\) , then what is the value of y-x? GMAT Problem Solving

Question: If x and y are positive integers and \((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\) , then what is the value of y-x?

1. -18
2. -17
3. 1
4. 17
5. 18

“If x and y are positive integers and \((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\) , then what is the value of y-x?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Answer

Approach Solution 1

\((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\)

Cross multiply the terms, we will get:

\((28)^{18y}=7^x*8^{12}\)

Factorize the following terms:

\((2)^{36y}*(7)^{18y}=7^x*2^{36}\)

Equate the power of 2 on both the sides, we will get:

36y = 36

y = 1

Now, equate the powers of 7 on both the sides, we will get:

18*1 = x

x = 18

Therefore, y – x = 1 – 18 = -17

Correct option: B

Approach Solution 2

\((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\)

Cross multiply the terms, we will get:

\((28)^{18y}=7^x*8^{12}\)------------ (1)

Firstly, we will solve LHS of the above equation, we will get:

\((28)^{18y}=(4*7)^{18y}\)

This can be rewritten as:

\((2^2*7)^{18y}=(2^2)^{18y}*7^{18y}\)

This will become:

\(2^{36y}*7^{18y}\)

Now compare this above equation with RHS of equation (1), we will get:

\(2^{36y}*7^{18y}=7^x*8^{12}\)

\(2^{36y}*7^{18y}=7^x*2^{36}\)

So, we will get:
36y = 36
y = 1
And
18y = x
18*1 = x
x = 18
Hence, y – x = 1 – 18 = -17

Correct option: B

Approach Solution 3

We are given that: \((\frac{1}{7})^x*(\frac{1}{8})^{12}=(\frac{1}{8})^{18y}\)

This can be rewritten as:

\(7^{-x}*8^{-12}=8^{-18y}\)
Factorize the terms in the above equation, we will get:

\(7^{-x}*2^{3*(-12)}=7^{-18y}*2^{2^*(-18y)}\)

\(7^{-x}*2^{-36}=7^{-18y}*2^{-36y}\)

From here we can see that

-x = -18y or x = 18y
And

-36 = -36y or 36 = 36y
y = 1

Therefore, x = 18
Hence, required answer is y – x = 1 – 18 = -17

Correct option: B

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