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how is it not d? we can rewrite 4 ^ (x + y) to 2 ^ 2 * ( x + y ) and 8 ^ 10 to 2 ^ 30 2 ^ 2*(x+y) = 2^30 x+y=15 now statement 1 tells us x-y=9 -> therefore its solveable x=12 y=3; 2^2*15=2^30 same goes for statement 2.

how is it not d? we can rewrite 4 ^ (x + y) to 2 ^ 2 * ( x + y ) and 8 ^ 10 to 2 ^ 30 2 ^ 2*(x+y) = 2^30 x+y=15 now statement 1 tells us x-y=9 -> therefore its solveable x=12 y=3; 2^2*15=2^30 same goes for statement 2.

so shouldnt it be d?

It is NOT give that \(4^{x+y}=8^{10}\). The question asks specifically about this: "is \(4^{x+y}=8^{10}\) ?"
_________________

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.

The answer is C.
_________________

Scott Woodbury-Stewart Founder and CEO

GMAT Quant Self-Study Course 500+ lessons 3000+ practice problems 800+ HD solutions

Clearly not sufficient as we can have multiple values for x and y

Hence, (1) ===== is NOT SUFFICIENT

(2) \frac{y}{x = 1/4}

\(x = 4y\)

Again this is not sufficient as we can get multiple values of x and y

Hence, (2) ===== is NOT SUFFICIENT

Combining (1) & (2)

When we substitute \(x = 4y\) in equation \(x - y = 9\)

we get,

\(4y - y = 9\)

\(3y = 9\)

\(y = 3\)

\(x = 4y\)

\(x = 4 * 3\)

\(x = 12\)

\(x + y = 12 + 3 = 15\)

Hence, (1) & (2) ===== is SUFFICIENT

Hence, Answer is C

Did you like the answer? 1 Kudos Please _________________

"Nothing in this world can take the place of persistence. Talent will not: nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not: the world is full of educated derelicts. Persistence and determination alone are omnipotent."

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Target question:Is 4^(x + y) = 8^10? This is a good candidate for rephrasing the target question.

Given equation: 4^(x + y) = 8^10 Rewrite 4 and 8 as powers of 2 to get: (2²)^(x + y) = (2³)^10 Apply power of a power law to get: 2^(2x +2y) = 2^30 This means that: 2x + 2y = 30 Divide both sides by 2 to get: x + y = 15 In other words, asking whether 4^(x + y) = 8^10 is the SAME as asking whether x + y = 15 REPHRASED target question:Is x + y = 15?

Statement 1: x - y = 9 Is this enough information to answer the REPHRASED target question? No. Consider these two CONFLICTING cases: Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15 Case b: x = 10 and y = 1. In this case, x + y = 10 + 1 = 11. So, x + y does NOT equal 15 Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y/x = 1/4 Is this enough information to answer the REPHRASED target question? No. Consider these two CONFLICTING cases: Case a: x = 12 and y = 3. In this case, x + y = 12 + 3 = 15. So, x + y DOES equal 15 Case b: x = 8 and y = 2. In this case, x + y = 8 + 2 = 10. So, x + y does NOT equal 15 Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined Statement 1 tells us that x - y = 9 Statement 2 tells us that y/x = 1/4 Since we have 2 different linear equations with 2 variables, we COULD solve the system for the individual values of x and y, which means we COULD answer the REPHRASED target question with certainty. Of course, we wouldn't waste precious time performing such calculations, since our sole goal is to determine the sufficiency of the combined statements. Since we can answer the target question with certainty, the combined statements are SUFFICIENT

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