11. 8 professionals can paint a house in 12 days. However, it takes 12 amateurs 16 days
to paint the same house. 16 professional painters started painting the house. After 3
days, 10 of them were replaced by 4 amateurs. Now how many days will they take to
complete the remaining painting?
a. 8 days
b. 10 days
c. 12 days
d. 6 days
They will take 8 more days to complete the remaining painting, so the correct answer is (a) 8 days.
Time and Work Problem: Professionals and Amateurs
This question is a classic time and work problem with professionals and amateurs that tests your ability to compare individual work rates and handle changes in workforce over time. Such problems are common in competitive exams and require a clear understanding of how to convert days and workers into a single rate-based framework.
Step-by-Step Solution
1. Find individual work rates
Let the whole work (painting the house) be 1 unit of work.
- 8 professionals can paint the house in 12 days.
- Combined rate of 8 professionals = 1/12 work per day.
- So rate of 1 professional = 1/12 ÷ 8 = 1/96 work per day
- 12 amateurs can paint the house in 16 days.
- Combined rate of 12 amateurs = 1/16 work per day.
- So rate of 1 amateur = 1/16 ÷ 12 = 1/192 work per day
Summary:
- 1 professional = 1/96 work per day
- 1 amateur = 1/192 work per day
2. Work done in the first 3 days
16 professionals start the work.
- Rate of 16 professionals = 16 × 1/96 = 16/96 = 1/6 work per day
- Work done in 3 days = 3 × 1/6 = 3/6 = 1/2
So, half of the house is already painted in 3 days.
Remaining work = 1 – 1/2 = 1/2
3. New team’s daily work rate
After 3 days, 10 professionals are replaced by 4 amateurs.
- Initially: 16 professionals
- After change: Professionals left = 16 – 10 = 6 professionals
- Amateurs added = 4 amateurs
Now compute their combined rate:
- Rate of 6 professionals = 6 × 1/96 = 6/96 = 1/16
- Rate of 4 amateurs = 4 × 1/192 = 4/192 = 1/48
- Total rate of new team = 1/16 + 1/48 = 3/48 + 1/48 = 4/48 = 1/12 work per day
So the new team (6 professionals + 4 amateurs) can paint 1/12 of the house per day.
4. Time to complete remaining work
- Remaining work = 1/2
- Daily rate of new team = 1/12
- Time required = Work/Rate = (1/2) ÷ (1/12) = 1/2 × 12/1 = 6 days
So, it will take 6 days more to finish the remaining painting.
Total duration = 3 days (initial) + 6 days (after replacement) = 9 days, but the question asks only for the days needed to complete the remaining painting, which is 6 days.
Correct Answer
(d) 6 days
Explanation of All Options
Key Points for Exam Preparation
- Always convert group work into individual work rates first.
- Track work done and remaining work step by step when team composition changes.
- Use the formula Time = Work/Rate consistently.
- This structured approach helps in solving any time and work problem with professionals and amateurs efficiently in competitive exams.
