It takes Abby a hours to build a birdhouse. When Becky and Caroline he

As usual, I try to solve with a RatexTimexWork table as shown below:



Since you have 3 given variables and no fix value at all, you can plug in freely a value for a. I used a=2.

Remember for Rate x Time x Work problems, the combined rate (in this case 2/a) is given by rate A + rate B + rate C:

\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)= \(\frac{2}{a}\) |multiply by a

\(1\)+\(\frac{a}{b}\)+\(\frac{a}{c}\) = \(2\)

It follows that a/b + a/c = 1. Since we are not concerned on any specific value (a, b, or c), we can plug in b=4 and c=4, which then equals 2/4+2/4 since a=2.

Finally we have values for a=2, b=4, c=4.

All one has to do is plug in these values in the different answer choices. Answer choice C equals 0.

abby's 1 hour work = 1/a
becky's 1 hour work = 1/b
caroline's 1 hour work = 1/c
they together will take half of abby's time. So their 1 hour work will be 2/a

it is given that

\(\frac{1}{a}\) + \(\frac{1}{b}\)+\(\frac{1}{c}\) = \(\frac{2}{a}\)

or \(\frac{1}{b}\)+\(\frac{1}{c}\) = \(\frac{1}{a}\)

or \(\frac{1}{b}\)+\(\frac{1}{c}\)-\(\frac{1}{a}\) = 0

or \(\frac{(ca+ab-bc)}{abc}\) = 0

or \(\frac{(bc-ca-ab)}{abc}\) = 0

Answer:- C

MANHATTAN GMAT OFFICIAL SOLUTION:

The key to this problem is the second sentence. When Becky and Caroline help, the time it takes to build the birdhouse is cut in half. That means that Becky and Caroline, when working together, work at a rate equal to the rate Abby works at.

We can represent the rates that Abby, Becky and Caroline work as 1/a, 1/b and 1/c, respectively. Therefore:

1/a = 1/b + 1/c

All of the answer choices have an expression equal to 0, so combine fractions and move them all to the left side of the equation. The common denominator for all fractions is abc.

1/a = 1/b + 1/c

bc/(abc) = ac/(abc) + ab/(abc)

bc/(abc) - ac/(abc) - ab/(abc) = 0

(bc - ac - ab)/(abc) = 0

The correct answer is C.

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