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.

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).

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