15 men take 42 days of 4 hours each to do piece of work. How many days of 6 hours each would 21 women take if 3 women do as much work as 2 men?

Options

A. 15
B. 22
C. 25
D. 30

Answer

To solve the problem step by step, let's break down the information given and find the solution.

### Step 1: Calculate the total work done by men
We know that:
- 15 men take 42 days, working 4 hours each day.

First, we calculate the total hours worked by 15 men:
\[ \text{Total hours} = \text{Number of men} \times \text{Number of days} \times \text{Hours per day} \]
\[ \text{Total hours} = 15 \times 42 \times 4 \]
\[ \text{Total hours} = 2520 \text{ hours} \]

### Step 2: Calculate the work done in terms of efficiency
Let’s define the work done as 1 unit. The total work done by 15 men in 2520 hours can be expressed in terms of their efficiency:
\[ \text{Efficiency of 15 men} = \frac{1 \text{ unit}}{2520 \text{ hours}} = \frac{1}{2520} \text{ units per hour} \]

### Step 3: Calculate the efficiency of one man
The efficiency of one man can be calculated as:
\[ \text{Efficiency of 1 man} = \frac{1}{15} \times \frac{1}{2520} = \frac{1}{37800} \text{ units per hour} \]

### Step 4: Relate women’s efficiency to men’s efficiency
We are given that 3 women do as much work as 2 men. Therefore, the efficiency ratio of women to men is:
\[ \frac{\text{Efficiency of women}}{\text{Efficiency of men}} = \frac{2}{3} \]

Let the efficiency of one man be \(3k\) and the efficiency of one woman be \(2k\).

### Step 5: Calculate the total efficiency of 21 women
The total efficiency of 21 women is:
\[ \text{Total efficiency of 21 women} = 21 \times 2k = 42k \]

### Step 6: Set up the equation for work done
Now we can set up the equation for the work done:
\[ \text{Total work done by 15 men} = \text{Total work done by 21 women} \]
\[ 2520 \text{ hours} = \frac{1}{42k} \times \text{Total hours worked by women} \]

### Step 7: Calculate the total hours worked by women
Let \(x\) be the number of days taken by 21 women working 6 hours each day:
\[ \text{Total hours worked by women} = 6x \]
So we have:
\[ 2520 = 42k \times 6x \]

### Step 8: Solve for \(x\)
We can simplify:
\[ 2520 = 252k \times x \]
Dividing both sides by \(252k\):
\[ x = \frac{2520}{252} = 10 \text{ days} \]

### Step 9: Convert days to hours
Since the women work 6 hours each day, we need to find out how many days it takes:
\[ \text{Total days} = \frac{Total hours}{6} = \frac{2520}{6} = 420 \text{ hours} \]

### Final Calculation
Now we need to find out how many days of 6 hours each would be required:
\[ \text{Days} = \frac{420}{6} = 70 \text{ days} \]

### Conclusion
Thus, 21 women would take **70 days of 6 hours each** to complete the work.

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