DS Question in Advanced Quant Guide

[sgyoung12345]

Hi,
The following question is on pg. 124, or problem set 4 (#14) of the Advanced Quant guide.

Is [(a-k)/(b-k)] > [(a+k)/(b+k)]?

1) a>b>k
2) k>0

The book has somewhat of a lengthy approach in terms of multiplying across the inequalities, and I wanted to see if the method I used just happened or work of if it is correct conceptually.

Instead of cross-multiplying I just looked at the two equations as is. For statement 1, I just picked arbitrary numbers for a,b and played around with k (ex. a=5, b=4). So if k is positive, the improper fraction on the left side will be greater than the one on the right (since if you add the same constant to top and bottom the improper fraction decreases). I reasoned that if k is negative, it must be the opposite case, so thus the statement is insufficient.

After seeing statement 2 clearly didn't say anything about a or b, it was clear to me that 1+2 meant the fraction on the left is greater than the one on the right.

Was this a correct alternative thought process? I was just concerned because I could have just gotten lucky in this case with my reasoning, esp. since I just used integers for a,b (although I don't think using fractions would have changed the matter). So does this logic seem sound?

[RonPurewal]

that's a fantastic, and highly intuitive, solution. well done!

(you can probably understand why we wouldn't put something like this in an answer key, though -- it's too intuitive. if someone is struggling with the problem and needs help seeing a solution, then that solution should be something that's at least fairly methodical.)