18 women complete a work in 24 days and 24 men complete the same work in 15 days. 16 women worked for 3 days and then they left. 20 men worked for the next 2 days, and then they are joined by 16 women. In how many days will they finish the remaining work?

$8\frac{2}{5}$

The correct answer is Option (3) → $8\frac{2}{5}$

1. Find the Efficiency Ratio

Let $W$ be the work rate of one woman and $M$ be the work rate of one man. According to the problem:

• $18 \text{ women} \times 24 \text{ days} = \text{Total Work}$
• $24 \text{ men} \times 15 \text{ days} = \text{Total Work}$

Equating the two:

$18 \times 24 \times W = 24 \times 15 \times M$

$18 \times W = 15 \times M$

Dividing both sides by 3:

$6W = 5M ⇒\frac{W}{M} = \frac{5}{6}$

Thus, we can assume a woman's efficiency is 5 units/day and a man's efficiency is 6 units/day.

2. Calculate Total Work

$\text{Total Work} = 18 \times 5 \text{ (efficiency)} \times 24 \text{ (days)} = 2160 \text{ units}$

3. Calculate Work Done in Stages

• Stage 1 (16 women for 3 days):

$\text{Work done} = 16 \times 5 \times 3 = 240 \text{ units}$

• Stage 2 (20 men for 2 days):

$\text{Work done} = 20 \times 6 \times 2 = 240 \text{ units}$

• Total Work Done So Far:

$240 + 240 = 480 \text{ units}$$

4. Calculate Remaining Work

$\text{Remaining Work} = 2160 - 480 = 1680 \text{ units}$

5. Calculate Time for Final Group

The remaining work is done by 20 men and 16 women together.

$\text{Combined Daily Rate} = (20 \times 6) + (16 \times 5) = 120 + 80 = 200 \text{ units/day}$

$\text{Days Required} = \frac{1680}{200} = \frac{168}{20} = \frac{42}{5} = 8\frac{2}{5} \text{ days}$