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:
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Work done by 3 men in 90 days: \[ 3M \times 90 = W \implies 3M = \frac{W}{90} \implies M = \frac{W}{270} \]
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Work done by 5 women in 90 days: \[ 5W \times 90 = W \implies 5W = \frac{W}{90} \implies W = \frac{W}{450} \]
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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.
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Work rate for 2 men: \[ 2M = 2 \times \frac{W}{270} = \frac{2W}{270} = \frac{W}{135} \]
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Work rate for 10 women: \[ 10W = 10 \times \frac{W}{450} = \frac{10W}{450} = \frac{W}{45} \]
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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:
\[ \frac{W}{135} = \frac{2W}{270} \]
\[ \frac{W}{45} = \frac{6W}{270} \]
\[ \frac{W}{30} = \frac{9W}{270} \]
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.