Manhattan GMAT Challenge Problem of the Week - 5 Oct 2011

by Manhattan Prep, Oct 5, 2011

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Question

If abc 0 and the sum of the reciprocals of a, b, and c equals the reciprocal of the product of a, b, and c, then a =

A. (1 + bc)/(b + c)

B. (1 bc)/(b + c)

C. (1 + b + c)/(bc)

D. (1 b c)/(bc)

E. (1 b c)/(b + c)

Answer

First, translate the given information into an equation. Go phrase by phrase. The sum of the reciprocals of a, b, and c is 1/a + 1/b + 1/c. Notice that you first take reciprocals, then you add the reciprocals together.

Now, set that equal to the reciprocal of the product of a, b, and c, which is 1/(abc). Notice that we first take the product of a, b, and c (which is abc), and then take the reciprocal of that product.

The equation is this:

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

Now rearrange to isolate a on one side. Make a common denominator on the left side (abc), so that you can add the fractions:

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

Since the right side of the original equation is 1/(abc), which happens to have the same denominator, you can set the numerators equal:

bc + ac + ab = 1

Now solve for a:

ac + ab = 1 bc

a(c + b) = 1 bc

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

Theoretically, you can solve this problem by plugging numbers for the variables, but finding three consistent values of a, b, and c (to satisfy the complicated condition) is rather difficult. A pure algebraic approach is faster and more secure in this case.

The correct answer is B.

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