12 men and 16 boys can do a piece of work in 5 days, 13 men and 24 boys can do it in 4 days. Then the ratio of daily work done by a man to that of a boy is
A)`2:1`
B)`3:1`
C)`3:2`
D)`5:4`

Options

A. `2:1`
B. `3:1`
C. `3:2`
D. `5:4`

Answer

To solve the problem, we need to determine the ratio of the daily work done by a man to that of a boy based on the information given.

### Step-by-Step Solution:

1. **Define Variables**:
Let the daily work done by one man be \( M \) and the daily work done by one boy be \( B \).

2. **Set Up the Equations**:
From the problem, we have two scenarios:
- **Scenario 1**: 12 men and 16 boys can complete the work in 5 days.
- **Scenario 2**: 13 men and 24 boys can complete the work in 4 days.

The total work can be expressed in terms of daily work:
- For Scenario 1:
\[ \text{Total Work} = (12M + 16B) \times 5 \]
- For Scenario 2:
\[ \text{Total Work} = (13M + 24B) \times 4 \]

3. **Equate the Total Work**:
Since both expressions represent the same total work, we can set them equal to each other:
\[ (12M + 16B) \times 5 = (13M + 24B) \times 4 \]

4. **Expand the Equations**:
Expanding both sides gives:
\[ 60M + 80B = 52M + 96B \]

5. **Rearrange the Equation**:
Now, rearranging the equation to isolate terms involving \( M \) and \( B \):
\[ 60M - 52M = 96B - 80B \]
\[ 8M = 16B \]

6. **Simplify the Ratio**:
Dividing both sides by 8 gives:
\[ M = 2B \]

7. **Find the Ratio**:
The ratio of the daily work done by a man to that of a boy is:
\[ \frac{M}{B} = \frac{2B}{B} = 2:1 \]

### Conclusion:
Thus, the ratio of daily work done by a man to that of a boy is \( 2:1 \).