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If x and y are positive integers, what is

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by jjjinapinch » Tue Aug 01, 2017 10:08 am

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If x and y are positive integers, what is the value of √x + √y?
(1) x + y = 15
(2) √(xy) = 6

Official Guide question
Answer: C
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Source: — Data Sufficiency |

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by Brent@GMATPrepNow » Tue Aug 01, 2017 10:32 am
jjjinapinch wrote:If x and y are positive integers, what is the value of √x + √y?
(1) x + y = 15
(2) √(xy) = 6

Official Guide question
Answer: C
Target question: What is the value of √x + √y?

Statement 1: x + y = 15
This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 14 and y = 1, in which case √x + √y = √14 + √1 = √14 + 1
Case b: x = 9 and y = 6, in which case √x + √y = √9 + √6 = 3 + √6
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Aside: For more on this idea of plugging in values when a statement doesn't feel sufficient, read my article: https://www.gmatprepnow.com/articles/dat ... lug-values

Statement 2: √(xy) = 6
In other words xy = 36
This statement doesn't FEEL sufficient either, so I'll TEST some values.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 1 and y = 36, in which case √x + √y = √1 + √36 = 1 + 6 = 7
Case b: x = 4 and y = 9, in which case √x + √y = √4 + √9 = 2 + 3 = 5
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x + y = 15
Statement 2 tells us that √(xy) = 6
Recognize that (√x + √y)² = x + 2√(xy) + y
Rearrange to get: (√x + √y)² = 15 + 2(6)
Evaluate: (√x + √y)² = 27
So, √x + √y = √27
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Jeff@TargetTestPrep » Wed Aug 09, 2017 11:10 am
jjjinapinch wrote:If x and y are positive integers, what is the value of √x + √y?
(1) x + y = 15
(2) √(xy) = 6

Official Guide question
Answer: C
We need to determine the value of √x + √y.

Statement One Alone:

x + y = 15

If x = 1 and y = 14, then √x + √y = 1 + √14. However, if x = 4 and y = 11, then √x + √y = 2 + √11. We see that we don't have enough information to determine a unique value of √x + √y.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

√(xy) = 6

If x = 6 and y = 6, then √x + √y = 2√6. However, if x = 4 and y = 9, then √x + √y = 5. We see that we don't have enough information to determine a unique value of √x + √y.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Notice that (√x + √y)^2 = x + y + 2√(xy). From the two statements, we are given that x + y = 15 and √(xy) = 6, and thus (√x + √y)^2 = 15 + 2(6) = 27. Now, if we take the square root of both sides of the equation (√x + √y)^2 = 27, we have √x + √y = √27 = 3√3.

Answer: C

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