Bunuel wrote:
If x and y are positive integers, what is the value of \(\sqrt{x} + \sqrt{y}\)?
(1) x + y = 15
(2) \(\sqrt{xy}= 6\)
Kudos for a correct solution.
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 + 1Case b: x = 9 and y = 6, in which case
√x + √y = √9 + √6 = 3 + √6Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
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 = 7Case b: x = 4 and y = 9, in which case
√x + √y = √4 + √9 = 2 + 3 = 5Since 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 = 15Statement 2 tells us that
√(xy) = 6Recognize 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:
But, isn't √x + √y = ± √27 which would not result in a single solution for the question?