If a and b are positive integers, and x = 2*3*7*a, and y = 2*2*8*b the

Question

If a and b are positive integers, and x = 2*3*7*a, and y = 2*2*8*b, then a − b =

(1) The values of both x and y lie between 120 and 130, not including the two.

(2) The difference between x and y equals two times the difference between a and b.

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BrentGMATPrepNow · Answer

Given : x = (2)(3)(7)(a), and y = (2)(2)(8)(b)

Target question :    What is the value of a - b?

Statement 1 : The values of both x and y lie between 120 and 130, not including the two.  
x = (2)(3)(7)(a), which means x =42a, which means 120 < 42a < 130. 
Since a is an INTEGER, we now know that  a = 3 
Also, y = (2)(2)(8)(b), which means y = 32b, which means 120 < 32b < 130
Since b is an INTEGER, we now know that  b = 4 
So, the answer to the target question is  a - b = 3 - 4 = -1 
Since we can answer the  target question  with certainty, statement 1 is SUFFICIENT

Statement 2 : The difference between x and y equals two times the difference between a and b. 
In other words: x - y = 2(a - b)
Replace x and x to get: 42a - 32b = 2a - 2b
Rearrange to get: 40a = 30b
Divide both sides by 10 to get: 4a = 3b 
At this point we can see that there are infinitely many values of a and b that satisfy the equation 4a = 3b. Here are two: 
 Case a : a = 3 and b = 4. In this case, the answer to the target question is  a - b = 3 - 4 = -1 
 Case b : a = 6 and b = 8. In this case, the answer to the target question is  a - b = 6 - 8 = -2 
Since we cannot answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

righthand · Answer

x = 42a ; y =32b

s1: given a, b positive intergers & x,y both are between 120 to 130 ---> x & y can be worked out ---> a-b can also be.
hence, sufficient.

s2: x-y = 2(a-b) ---> will give relation between a & b.. not the differnence. not sufficient.

Answer:A

IanStewart · Answer

We know x is a multiple of 42, and y is a multiple of 32. There will only be one possible value of x between 120 and 130, and only one possible value of y, so using Statement 1 we can work out exactly what x and y are, and thus find a and b and answer the question.

From Statement 2, we know

(2)(3)(7)(a) - (2)(2)(8)(b) = 2(a - b)
42a - 32b = 2a - 2b
40a = 30b
a/b = 3/4

So Statement 2 simply gives us the ratio of a to b. It's possible that a = 3 and b = 4 (those are the values you find using Statement 1 alone), and a - b = -1. Or it's possible that a = 6 and b = 8, and then a - b = -2, for example. So Statement 2 is not sufficient, and the answer is A.

The wording of Statement 2 is a bit ambiguous though, since it's not clear if it means to describe a-b (which turns out to be negative), or the positive difference |a - b|. I don't think I've ever seen that situation in a real GMAT question (and I'd be surprised if I did).

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