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If n and p are positive integers, what is the ratio of n to p? (1) n  [#permalink]

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If n and p are positive integers, what is the ratio of n to p?

(1) np = 48
(2) p^2 - n^2 = 28

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Re: If n and p are positive integers, what is the ratio of n to p? (1) n  [#permalink]

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If n and p are positive integers, what is the ratio of n to p?

(1) np = 48. Clearly insufficient. Consider n = 48 and p = 1 AND n = 1 and p = 48.

(2) p^2 - n^2 = 28

(p + n)(p - n) = 28. Notice here that p must be greater than n, so p + n must be greater than p - n.

The product of two positive integers is 28. 28 = 28*1 = 14*2 = 7*4.

p + n = 28 and p - n = 1 --> no integer solution.
p + n = 14 and p - n = 2 --> p = 8 and n = 6.
p + n = 7 and p - n = 4 --> no integer solution.

Only one possible set: p = 8 and n = 6. Ratio = n/p = 6/8. Sufficient.

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Re: If n and p are positive integers, what is the ratio of n to p? (1) n  [#permalink]

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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 48
Case b: n = 48 and p = 1. In this case, the ratio of n to p is 48 to 1
Since 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 EVEN
Why 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 = 14

When 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

Cheers,
Brent
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Re: If n and p are positive integers, what is the ratio of n to p? (1) n  [#permalink]

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