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Is 4^(x + y) = 8^10?

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jjjinapinch Senior | Next Rank: 100 Posts Default Avatar
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Posted:
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Is 4^(x + y) = 8^10?

Post Thu Aug 03, 2017 12:09 pm
Is 4^(x + y) = 8^10?
(1) x - y = 9
(2) y/x = 1/4

Official Guide question
Answer: C

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Post Thu Aug 03, 2017 1:54 pm
jjjinapinch wrote:
Is 4^(x + y) = 8^10?
(1) x - y = 9
(2) y/x = 1/4

Official Guide question
Answer: C
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?

Aside: Here’s a video with tips on rephrasing the target question: http://www.gmatprepnow.com/module/gmat-data-sufficiency?id=1100

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

Answer: C

Cheers,
Brent

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