Advanced GMAT Quant Chap 4 13 (a-k)/(b-k)

[JoaoM763]

Hello,

My doubt regards the DS question 13 of Chap 4:

Is (a-k)/(b-k) > (a+k)/(b+k) ?
(1) a>b>k
(2) k>0

It took me a lot of time and I solved by testing numbers.
I couldn´t get the explanation clearly and was unable to came up with the trigger to do the rephrasing by cross-multiplying the denominators.

Tks.

[RonPurewal]

hi,

I couldn´t get the explanation clearly

regarding this ^^, please tell us what, exactly, was unclear in the explanation.
without something specific to go on, we can't really give you a meaningful response (beyond basically just regurgitating what's already in the answer key).

thanks.

[RonPurewal]

in any case, some observations about this one:

1/
testing numbers shouldn't take that long, if you're sufficiently aware of your GOALS while picking the numbers.

specifically, as for any other "yes/no" problem, there are two goals:
– get the "yes" possibility (left-hand side > right-hand side)
– get the "no" possibility (left-hand side ≤ right-hand side).

for instance, here's a sensible approach to statement 1.

• pick a random set of numbers first-- say a = 3, b = 2, k = 1.
in this case, the left-hand side is 2/1 = 2, and the right-hand side is 4/3. since 2 > 4/3, that takes care of the "yes" case above.

• NOW, this is where efficiency is the point. if you're just testing random values, you will waste tons of time and effort!

• the point is that you need to make the "≤" case happen. so, you need to choose numbers deliberately to make that happen.

• one straightforward way to make "≤" happen is just to let k = 0 (with any two different positive numbers for "a" and "b"). in that case, the left- and right-hand sides are both a/b, so they are equal.

once you've made both things happen, you have "not sufficient". then proceed in the same way with further testing.

[RonPurewal]

2/
there's no need to think in terms of "cross multiplying". just consider basic actions that you'd take with normal fractions.

in fact, i actively DIScourage students from "cross multiplying", because "cross multiplying" is ... not really a thing. it's a shortcut that works only in certain circumstances -- specifically, when both denominators are known to have the same sign, or when you're dealing with an equality rather than an inequality -- and NOT in others.

instead, you should understand what's actually happening when you "cross multiply" -- namely, you're just making a common denominator.

e.g.,
"Is 13/30 greater than 3/7?"
to make this comparison, you'd probably just create a common denominator (= 210). then, you have "Is 91/210 greater than 90/210?", to which the answer is clearly yes.

the shortcut of "cross multiply" just turns this into "Is 91 greater than 90?", which is the same thing you'd get by doing the above (standard) operation and then just multiplying both sides by 210.**
in other words, the essence is exactly the same -- there's little, if any, increase in efficiency -- but the underlying understanding is gone. it's basically just a random operation that someone once told you you could do.

unless you FULLY understand why "cross multiply" works, it's a shaky foundation on which to build these processes... especially because DS in particular WILL test the exceptions to common "rules of "thumb". pretty much every time.

--

**notice that i said "multiplying both sides by 210", NOT "cancelling the 210's"... because "cancelling" is not actually a thing, either. (like "cross multiply", it's a shortcut that sometimes works, but that just as often doesn't.)

[RonPurewal]

...so, when you get to "both statements together", you can just do what you're already accustomed to:
• make the common denominator -- in this case, (b – k)(b + k)
• multiply both sides by that denominator (which gets rid of it).

note that the second step here -- "multiply both sides by the denominator" -- CANNOT be done for either individual statement, because we don't know that the denominators are positive until we have both statements.
(if you refer to that step as "cancelling", you may mistakenly think it's possible for the individual statements, a gaffe that would cause you to think the answer is A.)

[KakakkdeK400]

Thank you for good communication.

[RonPurewal]

you're welcome.