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. The ratio of the daily work done by a man to that of a boy is
(a) 2:1(b) 3:1(c) 3:2(d) 5:4

To solve this problem, let's assume that the daily work done by one man is **m** and the daily work done by one boy is **b**.

### Step 1: Set up the equations based on the given information

1. 12 men and 16 boys can complete the work in 5 days. This means the total work done by 12 men and 16 boys in 5 days is equal to 1 unit of work. Therefore, we have:
$$5\times (12m+16b)=1$$
Simplifying:
$$12m+16b=\frac{1}{5}\text{(Equation 1)}$$

2. 13 men and 24 boys can complete the work in 4 days. This means the total work done by 13 men and 24 boys in 4 days is equal to 1 unit of work. Therefore, we have:
$$4\times (13m+24b)=1$$
Simplifying:
$$13m+24b=\frac{1}{4}\text{(Equation 2)}$$

### Step 2: Solve the system of equations

We now have the following system of equations:
1. $12m+16b=\frac{1}{5}$
2. $13m+24b=\frac{1}{4}$

To solve this, we'll first eliminate the fractions by multiplying both sides of each equation by 20 (the least common denominator of 5 and 4).

Multiplying Equation 1 by 20:
$$20\times (12m+16b)=20\times \frac{1}{5}$$
$$240m+320b=4\text{(Equation 3)}$$

Multiplying Equation 2 by 20:
$$20\times (13m+24b)=20\times \frac{1}{4}$$
$$260m+480b=5\text{(Equation 4)}$$

### Step 3: Eliminate one variable

Now subtract Equation 3 from Equation 4 to eliminate **m**:
$$(260m+480b)-(240m+320b)=5-4$$
$$260m-240m+480b-320b=1$$
$$20m+160b=1$$
Now divide through by 20:
$$m+8b=\frac{1}{20}\text{(Equation 5)}$$

### Step 4: Solve for m in terms of b

From Equation 5, solve for **m**:
$$m=\frac{1}{20}-8b$$

### Step 5: Substitute into one of the original equations

Substitute **m = \frac{1}{20} - 8b** into Equation 1:
$$12(\frac{1}{20}-8b)+16b=\frac{1}{5}$$
Simplify:
$$\frac{12}{20}-96b+16b=\frac{1}{5}$$
$$\frac{3}{5}-80b=\frac{1}{5}$$
Now subtract $\frac{3}{5}$ from both sides:
$$-80b=\frac{1}{5}-\frac{3}{5}$$
$$-80b=-\frac{2}{5}$$
Now solve for **b**:
$$b=\frac{2}{5}\times \frac{1}{80}=\frac{1}{200}$$

### Step 6: Solve for m

Now substitute **b = \frac{1}{200}** into **m = \frac{1}{20} - 8b**:
$$m=\frac{1}{20}-8\times \frac{1}{200}$$
$$m=\frac{1}{20}-\frac{1}{25}$$
Find a common denominator (LCM of 20 and 25 is 100):
$$m=\frac{5}{100}-\frac{4}{100}=\frac{1}{100}$$

### Step 7: Find the ratio of m to b

The ratio of the daily work done by a man to that of a boy is:
$$\frac{m}{b}=\frac{\frac{1}{100}}{\frac{1}{200}}=\frac{200}{100}=2$$

Therefore, the ratio of the daily work done by a man to that of a boy is **2:1**.

**Answer: (a) 2:1**