2 men or 5 women or 3 boys can do a work in 10 days. Then in how many days will 1 man, 2 women and 4 boys do the work?

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by 2 men, 5 women, and 3 boys in one day.
Given that 2 men, 5 women, or 3 boys can complete the work in 10 days, we can find the work done per day.

- Work done by 2 men in 10 days = 1 work unit
- Therefore, work done by 2 men in 1 day = 1/10 work unit
- Work done by 1 man in 1 day = (1/10) / 2 = 1/20 work unit

### Step 2: Calculate the work done by women and boys.
Using the same logic:

- Work done by 5 women in 10 days = 1 work unit
- Therefore, work done by 5 women in 1 day = 1/10 work unit
- Work done by 1 woman in 1 day = (1/10) / 5 = 1/50 work unit

- Work done by 3 boys in 10 days = 1 work unit
- Therefore, work done by 3 boys in 1 day = 1/10 work unit
- Work done by 1 boy in 1 day = (1/10) / 3 = 1/30 work unit

### Step 3: Find the efficiency of 1 man, 2 women, and 4 boys.
Now, we can calculate the total work done in one day by 1 man, 2 women, and 4 boys.

- Work done by 1 man in 1 day = 1/20 work unit
- Work done by 2 women in 1 day = 2 * (1/50) = 2/50 = 1/25 work unit
- Work done by 4 boys in 1 day = 4 * (1/30) = 4/30 = 2/15 work unit

### Step 4: Combine the efficiencies.
Now, let's add these efficiencies together:

1. Work done by 1 man = 1/20
2. Work done by 2 women = 1/25
3. Work done by 4 boys = 2/15

To add these fractions, we need a common denominator. The least common multiple of 20, 25, and 15 is 300.

- Convert each fraction:
- 1/20 = 15/300
- 1/25 = 12/300
- 2/15 = 40/300

Now, add them:

\[ \text{Total work done in one day} = \frac{15}{300} + \frac{12}{300} + \frac{40}{300} = \frac{67}{300} \text{ work units} \]

### Step 5: Calculate the total time to complete the work.
Since the total work is 1 work unit, we can find the number of days required:

\[ \text{Days required} = \frac{\text{Total work}}{\text{Work done in one day}} = \frac{1}{\frac{67}{300}} = \frac{300}{67} \approx 4.48 \text{ days} \]

Thus, 1 man, 2 women, and 4 boys will complete the work in approximately **4.48 days**.

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