3 men and 4 boys can complete a certain amount of work in 28 days, whereas 4 men and 6 boys can complete the same work in 20 days. Find :
The number of days required by 7 men and 6 boys to complete the same work.

Answer

To solve the problem step by step, we will use the information given about the work done by men and boys to find out how many days 7 men and 6 boys would take to complete the same work.

### Step 1: Establish the Work Equations
We know that:
- 3 men and 4 boys can complete the work in 28 days.
- 4 men and 6 boys can complete the same work in 20 days.

Using the formula:
\[ \text{Work} = \text{Efficiency} \times \text{Time} \]

We can set up the equations based on the information provided.

For the first scenario:
\[ (3M + 4B) \times 28 = W \]
For the second scenario:
\[ (4M + 6B) \times 20 = W \]

### Step 2: Equate the Work Done
Since both expressions equal the same work \( W \), we can set them equal to each other:
\[ (3M + 4B) \times 28 = (4M + 6B) \times 20 \]

### Step 3: Expand and Simplify
Expanding both sides:
\[ 84M + 112B = 80M + 120B \]

Now, rearranging the equation:
\[ 84M - 80M = 120B - 112B \]
\[ 4M = 8B \]

### Step 4: Find the Relationship Between Men and Boys
From the equation \( 4M = 8B \), we can simplify it to find the relationship:
\[ M = 2B \]

This means 1 man is equivalent to 2 boys.

### Step 5: Substitute the Relationship
Now, we need to find the efficiency of 7 men and 6 boys. We can express the boys in terms of men:
\[ 6B = 3M \]
So, the total efficiency of 7 men and 6 boys can be expressed as:
\[ 7M + 3M = 10M \]

### Step 6: Set Up the Work Equation for 7 Men and 6 Boys
Now we can use the efficiency to find out how many days \( x \) it will take for 7 men and 6 boys to complete the work:
\[ 10M \times x = W \]

### Step 7: Use the Work from One of the Previous Equations
We can use the work done by 3 men and 4 boys:
\[ W = (3M + 4B) \times 28 \]
Substituting \( B = \frac{M}{2} \):
\[ W = (3M + 4 \times \frac{M}{2}) \times 28 \]
\[ W = (3M + 2M) \times 28 \]
\[ W = 5M \times 28 \]
\[ W = 140M \]

### Step 8: Set the Equations Equal
Now we can set the two expressions for work equal:
\[ 10M \times x = 140M \]

### Step 9: Solve for \( x \)
Dividing both sides by \( 10M \):
\[ x = \frac{140M}{10M} \]
\[ x = 14 \]

### Final Answer
The number of days required by 7 men and 6 boys to complete the same work is **14 days**.

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