What is the numerical value of 1/a + 1/b + 1/c ?

Question

(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?

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)?

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)?

mainhoon · Answer

yes if no solution is possible you would select E

In this case you can see that the sum is ab+bc+ca/(abc) neither of the options tell you what this sum would be , hence E

IanStewart · Answer

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.

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.

milanezr · Answer

Please, could someone help me understand why solving like this is wrong?

1/a+1/b+1/c = (a+b+c)/(abc)

(1) a+b+c = 1
(2) abc=1

(1) and (2) gives us = (a+b+c)/(abc)=1/1

amanvermagmat · Answer

1/a + 1/b + 1/c = (bc + ac + ab) / (abc)

and NOT the way you have solved.

NunuBot · Answer

Given:
1/a + 1/b + 1/c

Combine to a single fraction, multiply by abc/abc to every single term:
1/a + 1/b + 1/c = (bc + ac+ ab) / abc

There is no way to derive bc / ac / ab from any of the below options:
(1) a + b + c = 1
(2) abc = 1

Answer: E, Neither

satyadev123 · Answer

AM*HM = GM^2

If we apply this formula, the answer should be C. Since option A is 3*AM of a,b,c and option B is GM^3 of numbers a,b,c.

IanStewart · Answer

Unfortunately you can't use anything like that here, because any relationships you might have learned about geometric means are only true for sets of positive numbers, and here, two of the unknowns must be negative. The answer is genuinely E to this question. If you imagine first that a=5, then you have these equations: b + c = -4 bc = 1/5 and the solutions for b and c are the same as the solutions to the quadratic x^2 + 4x + (1/5). You can immediately see this will have two negative solutions (the discriminant is positive), and if you solve for them, or just think about what they would need to approximately equal, they are roughly -0.05 and -3.95. In this case, if we add 1/a + 1/b + 1/c, we get something close to 1/5 - 1/4 - 1/(0.05) which is close in value to -20. Instead if you imagine a = 101, then we find b and c by solving x^2 + 100x + (1/101). The solutions here are roughly -100 and -1/10,000. Now if we add the three required reciprocals, the sum is very close to -10,000. As I said in my post above from several years ago (

), the question is really beyond the algebraic scope of the GMAT, so test takers shouldn't be concerned about it.

sujoykrdatta · Answer

But, it does not say that a, b, c are integers. We can always have a = b = c = 1/3  (from statement 1)

IanStewart · Answer

I do not understand why you have quoted me in your reply -- I obviously haven't assumed a, b and c are integers when my solutions are not integers. Using both Statements, there are obviously no integer solutions for a, b and c. Perhaps you meant to quote a different post?

sujoykrdatta · Answer

We have: a + b + c = 1 ... (i) abc = 1 ... (ii) Simple logic: There are 3 unknowns and 2 equations => We cannot solve for a, b and c => We cannot determine the value of (1/a + 1/b + 1/c) Counter logic: Often, we don't really need to determine the values of a, b, c and still can determine the value of an expression. For example, had the question been to determine the value of (1/ab + 1/bc + 1/ac), we can easily get the answer by combining the statements. Hence, we need to do some analysis: Assuming a, b, c are positive and using AM ≥ GM for (i): (a + b + c)/3 ≥ (a * b * c)^(1/3) => (a + b + c)/3 ≥ (a * b * c)^(1/3) => (a * b * c)^(1/3) ≤ 1/3 => abc ≤ 1/27 Clearly, this contradicts with the second statement: abc = 1 This leads to 2 possibilities: Two of the terms of a, b, c are negative (note - we cannot have only one negative) OR a, b, c are complex numbers (beyond the scope of GMAT) Considering the first case: We can at best simplify as follows: a = 1 - b - c Thus, abc = 1 implies: (1 - b - c) * bc = 1 => (b^2) * c + b * (c^2 - c) + 1 = 0 Which is a quadratic in 'b' and CAN have multiple real solutions => there are many such values of a, b, c possible. A couple of such values I obtained: Say a = 2 => b + c = 1 - a = -1 and bc = 1/a = 1/2 => 1/b + 1/c = (b + c)/bc = -2 => 1/a + 1/b + 1/c = 1/2 - 2 = -3/2 Say a = 3 => b + c = 1 - a = -2 and bc = 1/a = 1/3 => 1/b + 1/c = (b + c)/bc = -6 => 1/a + 1/b + 1/c = 1/3 - 6 = -17/3 Thus, the value of 1/a + 1/b + 1/c is NOT unique - Not sufficient Answer E Note: This is NOT going to be asked in the GMAT - don't lose your sleep over it

hemantbafna · Answer

What is the numerical value of 1/a + 1/b + 1/c ?

(1) a + b + c = 1
(2) abc = 1

bc+ac+ab/abc

Is this an illogical way to solve the equation stated in the question?

sujoykrdatta · Answer

Oh, yes ... I think I tagged a different post than the one I wanted to

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What is the numerical value of 1/a + 1/b + 1/c ?

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