Quant - Work problem

mhoi wrote:Bea can paint a house three times faster than Alice can paint a house. If, working together, it takes Alice and Bea 24 hours to paint a house, then how many hours will it take Bea to paint a house alone?

A) 28
B) 30
C) 32
D) 36
E) 40

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Let B = number of hours it takes Bea to paint the house.
So, 3B = number of hours it takes Alice to paint the house [since Bea is three times faster than Alice]

Now examine what can be accomplished in 1 hour
If Bea takes B hours to paint the entire house, then after 1 hour, Bea will have painted 1/B of the house.
If Alice takes 3B hours to paint the entire house, then after 1 hour, Alice will have painted 1/3B of the house.
If Alice and Bea work together, then the fraction of the house painted in 1 hour = 1/B + 1/3B
= 3/3B + 1/3B
= 4/3B

Now consider the fact that we're told that Working together, it takes Alice and Bea 24 hours to paint a house
So, in one hour, Alice and Bea can paint 1/24 of a house

We can now write: 4/3B = 1/24
Cross-multiply to get: (4)(24) = (3B)(1)
Evaluate: 96 = 3B
Solve: B = 32

Answer: B

Cheers,
Brent

mhoi wrote:Bea can paint a house three times faster than Alice can paint a house. If, working together, it takes Alice and Bea 24 hours to paint a house, then how many hours will it take Bea to paint a house alone?

A) 28
B) 30
C) 32
D) 36
E) 40

Here's another approach:

Bea can paint a house three times faster than Alice can paint a house.
So, for every drop of paint that Alice applies, Bea applies 3 drops.
So, when they work together, we can say that, for every 4 drops of paint they apply, Bea applies 3 of those drops.
In other words, BEA DOES 3/4 OF THE WORK.

It takes Alice and Bea 24 hours to paint a house
If Bea does 3/4 of the works, we know that, in 24 hours, Bea has painted 3/4 of the house.
In other words, every 8 hours, Bea paints 1/4 of the house.
So, it will take Bea 32 hours to paint the entire house.

Answer: C

Cheers,
Brent