Question

If three men or five women or ten children can do a


piece of work in 90 days. In what time two men, 10

women and 30 children do the same job?

Answer 1

To solve this problem, we first need to determine the work done by each person (man, woman, child) based on the initial information provided.

### Step 1: Determine the total work

We know that:
- 3 men can complete the work in 90 days.
- 5 women can complete the same work in 90 days.
- 10 children can also complete the work in 90 days.

Let's represent the total work as \( W \).

From the data we have:

1. Work done by 3 men in 90 days:
\[
3M \times 90 = W \implies 3M = \frac{W}{90} \implies M = \frac{W}{270}
\]

2. Work done by 5 women in 90 days:
\[
5W \times 90 = W \implies 5W = \frac{W}{90} \implies W = \frac{W}{450}
\]

3. Work done by 10 children in 90 days:
\[
10C \times 90 = W \implies 10C = \frac{W}{90} \implies C = \frac{W}{900}
\]

Now we have the work rates of one man, one woman, and one child in terms of \( W \):

- \( M = \frac{W}{270} \)
- \( W = \frac{W}{450} \)
- \( C = \frac{W}{900} \)

### Step 2: Calculate work done by 2 men, 10 women, and 30 children

Next, we find the combined work rate for 2 men, 10 women, and 30 children.

- Work rate for 2 men:
\[
2M = 2 \times \frac{W}{270} = \frac{2W}{270} = \frac{W}{135}
\]

- Work rate for 10 women:
\[
10W = 10 \times \frac{W}{450} = \frac{10W}{450} = \frac{W}{45}
\]

- Work rate for 30 children:
\[
30C = 30 \times \frac{W}{900} = \frac{30W}{900} = \frac{W}{30}
\]

### Step 3: Combine the work rates

Now we sum the work rates of 2 men, 10 women, and 30 children:
\[
\text{Total work rate} = \frac{W}{135} + \frac{W}{45} + \frac{W}{30}
\]

To combine these fractions, we first find a common denominator. The least common multiple (LCM) of 135, 45, and 30 is 270.

Calculating each term:

1.
\[
\frac{W}{135} = \frac{2W}{270}
\]

2.
\[
\frac{W}{45} = \frac{6W}{270}
\]

3.
\[
\frac{W}{30} = \frac{9W}{270}
\]

Now we can add these together:
\[
\text{Total work rate} = \frac{2W + 6W + 9W}{270} = \frac{17W}{270}
\]

### Step 4: Calculate the time taken to complete the work

Now that we have the total work rate, we can find the time it takes to complete the work.

Let \( T \) be the time in days to complete the work \( W \):
\[
W = \text{Total work rate} \times T
\]
\[
W = \frac{17W}{270} \times T
\]

Dividing both sides by \( W \):
\[
1 = \frac{17}{270} \times T
\]
\[
T = \frac{270}{17} \approx 15.88 \text{ days}
\]

### Conclusion

Therefore, 2 men, 10 women, and 30 children can complete the job in approximately **15.88 days**.