P completed $\frac{2}{5}$ part of certain work in 12 days. Further, Q completed the rest of the work in 18 days. In how many days could P and Q complete the work, if they worked together?

15

The correct answer is Option (4) → 15

1. Find P's total time

We are told P completes $\frac{2}{5}$ of the work in 12 days.

• To find the time for the full work ($1$ unit), we set up the equation: $\frac{2}{5} \times \text{Total Time}_P = 12$
• $\text{Total Time}_P = 12 \times \frac{5}{2} = 30$ days.
• P's efficiency (work per day): $\frac{1}{30}$

2. Find Q's total time

Q completes the rest of the work.

• Rest of the work = $1 - \frac{2}{5} = \frac{3}{5}$
• Q completes $\frac{3}{5}$ of the work in 18 days.
• To find the time for the full work: $\frac{3}{5} \times \text{Total Time}_Q = 18$
• $\text{Total Time}_Q = 18 \times \frac{5}{3} = 30$ days.
• Q's efficiency (work per day): $\frac{1}{30}$

3. Calculate time working together

When P and Q work together, their daily efficiencies are added:

$\text{Combined Efficiency} = \frac{1}{30} + \frac{1}{30} = \frac{2}{30} = \frac{1}{15}$

The total time taken is the reciprocal of the combined efficiency:

$\text{Total Days} = 15 \text{ days}$

Conclusion

P and Q together can complete the work in 15 days.