What is the numerical value of 1/a + 1/b + 1/c ?
(1) a + b + c = 1
(2) abc = 1
See notes below
This question comes from the Kaplan GMAT Advanced
2009 - 2010 edition, question #3 on page 350. Kaplan
's explanation is not very thorough - it says that there are 3 unknowns and 2 equations, thus multiple solutions are possible. It took me 20 minutes of trial and error to understand the answer. Can anyone offer a methodical explanation?
Any solution here that is simply based on counting equations and unknowns is mathematical nonsense. We can't count equations and unknowns here for three reasons:
* one of the equations here is non-linear. There are no simple rules that tell us how many solutions we have to systems of nonlinear equations;
* we have three unknowns. As soon as you have more than two unknowns, you can't simply count equations and unknowns, since it is not clear when your equations will be independent. For example, if you have the three equations:
a + b + c = 5
2a + b + c = 4
3a + 2b + 2c = 9
it might appear that you have three distinct equations, three unknowns, but you can only solve for a here (a = -1). The most you can learn about b and c is that b + c = 6. The issue here is that the three equations are not independent; we can get the third equation by adding the first two equations. With only two equations/unknowns, it's easy to tell when linear equations are not independent: one equation is a multiple of the other. With three equations/unknowns, it is not at all easy, in some cases, to tell if you have independent equations.
* the question doesn't ask us to solve for any of the unknowns anyway, so it's not clear what counting equations tells us. Perhaps we can work out what 1/a + 1/b + 1/c is without knowing the value of any one of our unknowns. If one of the statements said something like ab + ac + bc = abc, then that one equation alone would be enough (since we can rewrite it to get 1/a + 1/b + 1/c = 1).
So counting equations and unknowns tells you almost nothing in this question.
I agree that the answer is E, but I don't think it's all that straightforward to prove that. It's clear two of the unknowns need to be negative, and the other positive. If you pick a value like 5 for a, then by substitution you can get a quadratic which lets you find b and c (using the quadratic formula, which you'd never need on a real GMAT question). Still, even if you can find various solutions for a, b and c, you're still left with the task of proving that these different solutions give you different values for 1/a + 1/b + 1/c. Unless I'm missing a trick, this is just not a well-designed question, since it's far too time-consuming, and requires math beyond what the GMAT requires, if you want to actually prove
the answer is E. If you want to just count equations and, from that, guess
the answer is E, I suppose you can do the question quickly, but if you use that strategy on your real test, you'll get a lot of wrong answers. No real GMAT question ever requires you to use some kind of guessing strategy; it's always possible using simple math to prove the answer is what it is.
Also, does the GMAT ever provide a set of equations to which there is no solution? What would the answer be if that were the case, (E)?
No, this never happens on real GMAT DS questions. There wouldn't be a single logically correct answer if that could happen -- if we have enough information using both statements to be certain there are no solutions, is the answer C or is it E? There's no good way to answer that question, so they can never ask a DS question with no solutions - your task is to decide if there is exactly one solution, or if there is more than one solution.
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