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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