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

Options

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

Answer

To solve the problem step by step, we will follow these instructions:

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

### Step 2: Set Up the Equations
From the problem, we have two scenarios:
1. 12 men and 16 boys can complete the work in 5 days.
\[ 12x + 16y = \frac{1}{5} \quad \text{(Equation 1)} \]

2. 13 men and 24 boys can complete the work in 4 days.
\[ 13x + 24y = \frac{1}{4} \quad \text{(Equation 2)} \]

### Step 3: Eliminate Fractions
To eliminate the fractions, we can multiply both equations by the least common multiple of the denominators:
- For Equation 1, multiply by 20:
\[ 20(12x + 16y) = 20 \cdot \frac{1}{5} \implies 240x + 320y = 4 \quad \text{(Equation 3)} \]

- For Equation 2, multiply by 4:
\[ 4(13x + 24y) = 4 \cdot \frac{1}{4} \implies 52x + 96y = 1 \quad \text{(Equation 4)} \]

### Step 4: Solve the System of Equations
Now we have:
- Equation 3: \( 240x + 320y = 4 \)
- Equation 4: \( 52x + 96y = 1 \)

To eliminate \( y \), we can multiply Equation 4 by 10:
\[ 520x + 960y = 10 \quad \text{(Equation 5)} \]

Now we can subtract Equation 3 from Equation 5:
\[ (520x + 960y) - (240x + 320y) = 10 - 4 \]
This simplifies to:
\[ 280x + 640y = 6 \]

### Step 5: Isolate \( y \)
Now we can express \( y \) in terms of \( x \). Rearranging gives:
\[ 640y = 6 - 280x \]
\[ y = \frac{6 - 280x}{640} \]

### Step 6: Substitute Back to Find \( x \)
We can substitute \( y \) back into one of the original equations to find \( x \). Let's use Equation 1:
\[ 12x + 16\left(\frac{6 - 280x}{640}\right) = \frac{1}{5} \]

### Step 7: Solve for \( x \)
Multiply through by 640 to eliminate the fraction:
\[ 640(12x) + 16(6 - 280x) = 128 \]
This simplifies to:
\[ 7680x + 96 - 4480x = 128 \]
Combine like terms:
\[ 3200x + 96 = 128 \]
Subtract 96 from both sides:
\[ 3200x = 32 \]
Divide by 3200:
\[ x = \frac{32}{3200} = \frac{1}{100} \]

### Step 8: Find \( y \)
Now substitute \( x \) back into one of the equations to find \( y \):
Using Equation 1:
\[ 12\left(\frac{1}{100}\right) + 16y = \frac{1}{5} \]
This simplifies to:
\[ \frac{12}{100} + 16y = \frac{20}{100} \]
Subtract \( \frac{12}{100} \):
\[ 16y = \frac{20}{100} - \frac{12}{100} = \frac{8}{100} \]
Divide by 16:
\[ y = \frac{8}{1600} = \frac{1}{200} \]

### Step 9: Find the Ratio of \( x \) to \( y \)
Now we have:
- \( x = \frac{1}{100} \)
- \( y = \frac{1}{200} \)

The ratio of the daily work done by a man to that of a boy is:
\[ \frac{x}{y} = \frac{\frac{1}{100}}{\frac{1}{200}} = \frac{200}{100} = 2 \]

### Final Answer
The ratio of the daily work done by a man to that of a boy is \( 2:1 \).

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