Last Updated on May 6, 2023
GMAT OFFICIAL GUIDE DS
Solution:
We need to determine whether 4^(x+y) = 8^10. We start by breaking down our two bases into prime factors.
4^(x+y) = (2^2)^(x+y) = 2^(2x+2y)
8^10 = (2^3)^10 = 2^30
We can now rephrase the question as:
Is 2^(2x+2y) = 2^30 ?
Because the bases are the same, we can drop them and set the exponents equal to each other. The question becomes:
Is 2x+2y = 30 ?
Is x + y = 15 ?
After simplifying the equation, we see that we need to determine whether the sum of x and y is equal to 15.
Statement One Alone:
x – y = 9
Knowing the difference of x and y is not the same as knowing the sum of x and y; thus, statement one is not sufficient to answer the question. We can eliminate answer choices A and D.
Statement Two Alone:
y/x = ¼
When we cross multiply obtain:
4y = x
4y = x is not enough information to determine the value of x + y. Statement two alone is not sufficient. We can eliminate answer choice B.
Statements One and Two Together:
Using statements one and two we know the following:
x – y = 9 and 4y = x
Since 4y = x, we can substitute 4y for x into the equation x – y = 9 and we have:
4y – y = 9
3y = 9
y = 3
Since y = 3, x = 4(3) = 12.
Thus, x + y = 12 + 3 = 15. We can answer yes to the question. Both statements together are sufficient.
Answer: C
