3 men and 4 boys do a piece of work in 8 days, while 4 men and 4 boys finish it in 6 days. 2 men and 4 boys will finish it in…..

Options

A. 9 days
B. 10 days
C. 12 days
D. 14 days

Answer

To solve the problem step by step, we will first define our variables and then set up equations based on the information given.

### Step 1: Define Variables
Let:
- \( x \) = work done by 1 man in 1 day
- \( y \) = work done by 1 boy in 1 day

### Step 2: Set Up Equations
From the problem, we have two scenarios:

1. **3 men and 4 boys do the work in 8 days:**
The total work done can be expressed as:
\[ 8 \times (3x + 4y) = \text{Total Work} \]

2. **4 men and 4 boys finish it in 6 days:**
The total work done can also be expressed as:
\[ 6 \times (4x + 4y) = \text{Total Work} \]

Since both expressions equal the total work, we can set them equal to each other:
\[ 8(3x + 4y) = 6(4x + 4y) \]

### Step 3: Simplify the Equation
Expanding both sides:
\[ 24x + 32y = 24x + 24y \]

Now, subtract \( 24x \) from both sides:
\[ 32y = 24y \]

### Step 4: Solve for \( y \)
Subtract \( 24y \) from both sides:
\[ 32y - 24y = 0 \]
\[ 8y = 0 \implies y = 0 \]

### Step 5: Substitute \( y \) Back to Find \( x \)
Now we can substitute \( y = 0 \) back into one of the original equations. Let's use the first one:
\[ 8(3x + 4(0)) = \text{Total Work} \]
This simplifies to:
\[ 24x = \text{Total Work} \]

### Step 6: Find Total Work from the Second Equation
Using the second equation:
\[ 6(4x + 4(0)) = \text{Total Work} \]
This simplifies to:
\[ 24x = \text{Total Work} \]

### Step 7: Calculate Work Done by 2 Men and 4 Boys
Now we need to find out how long it will take for 2 men and 4 boys to finish the work:
\[ \text{Work done by 2 men and 4 boys in 1 day} = 2x + 4y = 2x + 4(0) = 2x \]

### Step 8: Calculate the Days Required
The total work is \( 24x \), and the work done in one day by 2 men is \( 2x \):
\[ \text{Days required} = \frac{\text{Total Work}}{\text{Work done in 1 day}} = \frac{24x}{2x} = 12 \text{ days} \]

### Final Answer
Thus, 2 men and 4 boys will finish the work in **12 days**.

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