Two men and three boys can do a piece of work in ten days; while three men and two boys can do the same work in eight days. In how many days can two men and one boy do the work?
Step-by-step Solution:
1. Set Up the Work Equations - Let 'm' be the amount of work one man does per day, and 'b' be the work one boy does per day.
- The total work done is the same in both scenarios. We can express this as: Total Work = Rate × Time.
- Scenario 1: Total Work = (2m + 3b) × 10 days.
- Scenario 2: Total Work = (3m + 2b) × 8 days.
2. Find the Relationship Between Man and Boy Rates - Since the total work is equal, we can set the two equations equal to each other:
- 10(2m + 3b) = 8(3m + 2b)
- 20m + 30b = 24m + 16b
- 14b = 4m
- This gives us the relationship: m = (7/2)b. (This means one man works as fast as 3.5 boys).
3. Calculate the Total Work - Now, substitute the relationship back into the first equation to find the total work in terms of a single unit (like 'boy-days'):
- Total Work = (2(7/2)b + 3b) × 10 = (7b + 3b) × 10 = 10b × 10 = 100b.
4. Calculate Time for the New Group - The new group is 2 men and 1 boy. Let's find their combined rate in terms of 'b':
- New Rate = 2m + 1b = 2(7/2)b + 1b = 7b + 1b = 8b.
- The time required is Total Work divided by the New Rate:
- Time = 100b / 8b = 12.5 days.
Final Answer: It will take the new group 12.5 days to complete the work.