Rates Practice - Word Problem

This aptitude practice question is a Work Time problem solving question. Concept tested: A word problem, where the crux of solving the question is framing a pair of equations and finding a solution.

Question

Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?

1. 18 days
2. 36 days
3. 54 days
4. None of these

  Correct Answer      Choice (3). A man and woman will take 54 days to complete the job.

Explanatory Answer

Let the amount of work done by a man in a day be 'm' and the amount of work done by a woman in a day be 'w'.

Therefore, 4 men and 3 women will do 4m + 3w amount of work in a day.
If 4 men and 3 women complete the entire work in 6 days, they will complete \\frac {1} {6}^{th}) of the work in a day.
Hence, 4m + 3w = \\frac{1}{6}) ----- eqn (1)

From statement (2), we know 5 men and 6 women take 4 days to complete the job.
i.e., 5 men and 4 women working together will complete \\frac{1}{4})th of the job in a day.
So, 5m + 6w = \\frac{1}{4}) ----- eqn (2)

2 * eqn(1) - eqn (2):
8m + 6w - (5m + 6w) = 2 * \\frac{1}{6}) - \\frac{1}{4})
3m = = \\frac{1}{12}) or m = \\frac{1}{36}).
i.e., a man does \\frac {1}{36}^{th}) of the work in a day.

Substitute the value of m in eqn (1), we get 4 x \\frac{1}{36}) + 3w = \\frac{1}{6})
=> 3w = \\frac{1}{6}) - \\frac{1}{9}) = \\frac{3 - 2}{18}) = \\frac{1}{18})
Or w = \\frac{1}{54}).
i.e. a woman does \\frac {1}{54}^{th}) of the work in a day.
Hence, she will take 54 days to do the entire work.

  The correct answer is Choice (3).