14 men can complete a task in 9 days and 16 women can complete the same task in 7 days. In how many days can 6 men and 8 women complete the task?

Correct option is B

Given:

14 men can complete the task in 9 days.

16 women can complete the task in 7 days.

We are asked to find how many days 6 men and 8 women can complete the same task.

Concept Used:

The rate of work is the reciprocal of the time taken to complete the task.

Solution:

Rate of 1 man =$$\frac{1}{14 \times 9} = \frac{1}{126} \text{ tasks/day}$$​

Rate of 6 men =$$6 \times \frac{1}{126} = \frac{6}{126} = \frac{1}{21} \text{ tasks/day}$$​

Rate of 1 woman = $$\frac{1}{16 \times 7} = \frac{1}{112} \text{ tasks/day}$$​

Rate of 8 women =$$8 \times \frac{1}{112} = \frac{8}{112} = \frac{1}{14} \text{ tasks/day}$$​

Combined rate of 6 men and 8 women:

Combined rate =$$\frac{1}{21} + \frac{1}{14} = \frac{2}{42} + \frac{3}{42} = \frac{5}{42} \text{ tasks/day}$$​

Time taken to complete 1 task:

Time taken = $$\frac{1}{\text{Combined rate}} = \frac{1}{\frac{5}{42}} = \frac{42}{5} = 8.4 \text{ days}$$​

Thus, 6 men and 8 women can complete the task in 8.4 days.

Alternate Method: 

$$\frac{M_1 D_1H_1}{W_1}=\frac{M_2 D_2H_2}{W_2}$$ 

M = Man, D = Days , H = Hours,   W = Work

14M $$\times 9 =$$ 16W $$\times 7$$

$$\frac{M}{W}=\frac 89$$

Total Work = 14 $$\times$$ 8 $$\times$$ 9 = 1008

Now,

6M + 8W= 6 $$\times$$ 8 + 8 $$\times$$ 9

= 48 + 72 = 120

Time taken = $$\frac{1008}{120}$$ = 8.4 days​


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