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?
Answer & explanation
Correct answer: option 4
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.