5 men can prepare 150 toys in 5 days working 6 hours a day. In how many hours 10 boys can prepare 200 toys in 10 days, if a man works thrice as fast as a boy?

Answer

To solve the problem step by step, we will use the relationship between men, boys, hours, days, and work done.

### Step 1: Calculate the total work done by men
First, we need to find out how much work is done by the 5 men in the given time.

- **Total work done by 5 men in 5 days:**
\[ \text{Total work} = \text{Number of men} \times \text{Number of days} \times \text{Number of hours per day} \]
\[ = 5 \text{ men} \times 5 \text{ days} \times 6 \text{ hours/day} = 150 \text{ man-hours} \]

### Step 2: Determine the work done per toy
Now, we know that 150 toys are prepared in 150 man-hours.

- **Work done per toy:**
\[ \text{Work per toy} = \frac{\text{Total work}}{\text{Number of toys}} = \frac{150 \text{ man-hours}}{150 \text{ toys}} = 1 \text{ man-hour/toy} \]

### Step 3: Calculate the total work needed for 200 toys
Next, we need to find out how much work is required to prepare 200 toys.

- **Total work for 200 toys:**
\[ \text{Total work for 200 toys} = 200 \text{ toys} \times 1 \text{ man-hour/toy} = 200 \text{ man-hours} \]

### Step 4: Convert man-hours to boy-hours
Since a man works thrice as fast as a boy, we can find out how many boy-hours are equivalent to 200 man-hours.

- **Boy-hours equivalent:**
\[ \text{Boy-hours} = \frac{\text{Man-hours}}{3} = \frac{200 \text{ man-hours}}{3} \approx 66.67 \text{ boy-hours} \]

### Step 5: Calculate the total boy-hours available in 10 days
Now, we need to find out how many hours 10 boys can work in 10 days.

- **Total boy-hours available:**
\[ \text{Total boy-hours} = \text{Number of boys} \times \text{Number of days} \times \text{Number of hours per day} \]
Let \( H \) be the number of hours worked per day by each boy.
\[ = 10 \text{ boys} \times 10 \text{ days} \times H = 100H \text{ boy-hours} \]

### Step 6: Set up the equation to find H
We know that the total boy-hours must equal the boy-hours needed to prepare the toys.

\[ 100H = 66.67 \]

### Step 7: Solve for H
Now, we can solve for \( H \):

\[ H = \frac{66.67}{100} \approx 0.6667 \text{ hours} \]

### Conclusion
Thus, the number of hours each boy needs to work per day to prepare 200 toys in 10 days is approximately 0.67 hours, or about 40 minutes.

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