duahsolo wrote:
If n and p are positive integers, what is the ratio of n to p?
(1) np = 48
(2) p² - n² = 28
Target question: What is the ratio of n to p? Given: n and p are positive integers Statement 1: np = 48 This statement doesn't FEEL sufficient, so I'll TEST some values.
There are several values of n and p that satisfy statement 1. Here are two:
Case a: n = 1 and p = 48. In this case,
the ratio of n to p is 1 to 48Case b: n = 48 and p = 1. In this case,
the ratio of n to p is 48 to 1Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Aside: For more on this idea of testing values when a statement doesn't feel sufficient, read my article: http://www.gmatprepnow.com/articles/dat ... lug-values Statement 2: p² - n² = 28 Factor the left side to get: (p - n)(p + n) = 28
IMPORTANT: If p and n are integers, then p-n and p+n will also be integers.
Also, if p+n is ODD then p-n must also be ODD
Likewise, if p+n is EVEN then p-n must also be EVEN
Since (p - n)(p + n) = 28, we know that
p+n and p-n must BOTH be EVENWhy is this?
Well, the product of two ODD numbers cannot be EVEN (28).
When we examine pairs of INTEGERS that multiply to get 28, we see that there is only ONE pair such that both values are EVEN: 2 and 14
At this point, we can conclude that p - n = 2 and p + n = 14
Here we have two equations with 2 variables:
p - n = 2 p + n = 14When we solve the system, we get: p = 8 and n = 6
This means
the ratio of n to p is 6 to 8 (aka 3 to 4)Since we can answer the
target question with certainty, statement 2 is SUFFICIENT
Answer: B
Cheers,
Brent
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