1 man, 3 women and 4 children can complete a work in 96 hours, whereas 2 man and 8 children can complete that work in 80 hours. 2 men and 3 women can complete the same work in 120 hours, then 10 men and 5 women can complete the same work in how many hours?

Options

A. 36 hours
B. 33 hours
C. 45 hours
D. 54 hours

Answer

To solve the problem step by step, we need to determine the efficiencies of men, women, and children based on the information provided. Let's break it down:

### Step 1: Set Up the Equations
Let:
- Efficiency of 1 man = M
- Efficiency of 1 woman = W
- Efficiency of 1 child = C

From the problem, we have the following scenarios:

1. **1 man, 3 women, and 4 children can complete the work in 96 hours:**
\[ 1M + 3W + 4C = \frac{1}{96} \]

2. **2 men and 8 children can complete the work in 80 hours:**
\[ 2M + 8C = \frac{1}{80} \]

3. **2 men and 3 women can complete the work in 120 hours:**
\[ 2M + 3W = \frac{1}{120} \]

### Step 2: Convert Equations
We can convert these equations into a more manageable form by multiplying through by the denominators:

1. From \(1M + 3W + 4C = \frac{1}{96}\):
\[ 96M + 288W + 384C = 1 \quad \text{(Equation 1)} \]

2. From \(2M + 8C = \frac{1}{80}\):
\[ 160M + 640C = 1 \quad \text{(Equation 2)} \]

3. From \(2M + 3W = \frac{1}{120}\):
\[ 240M + 360W = 1 \quad \text{(Equation 3)} \]

### Step 3: Solve the Equations
Now we will solve these equations to find the values of M, W, and C.

**From Equation 2:**
\[ 160M + 640C = 1 \implies M + 4C = \frac{1}{160} \quad \text{(Equation 4)} \]

**From Equation 3:**
\[ 240M + 360W = 1 \implies 4M + 6W = \frac{1}{60} \quad \text{(Equation 5)} \]

### Step 4: Substitute and Solve
We can express C in terms of M from Equation 4:
\[ C = \frac{1}{640} - \frac{M}{4} \]

Substituting C into Equation 1:
\[ 96M + 288W + 384\left(\frac{1}{640} - \frac{M}{4}\right) = 1 \]
Simplifying this will give us a relationship between M and W.

### Step 5: Find Efficiency of 10 Men and 5 Women
Once we have the values for M and W, we can find the total efficiency of 10 men and 5 women:
\[ 10M + 5W \]

### Step 6: Calculate Time Taken
Finally, we can use the total work done (1 unit) and the efficiency to find the time taken:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Efficiency}} = \frac{1}{10M + 5W} \]

### Conclusion
After solving the equations, we will find the time taken for 10 men and 5 women to complete the work.

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