2 men and 4 boys can complete a work in 10 days where as 4 men and 5 boys can complete it in 6 days. These men and boys get their wages according to their work. According, if per day wages of a man is ₹ 40, then find the ratio of per day wages of 1 man and 1 boy?

Options

A. `7:4`
B. `7:3`
C. `5:3`
D. `5:2`

Answer

To solve the problem step by step, let's denote the work done by a man in one day as \( M \) and the work done by a boy in one day as \( B \).

### Step 1: Establish the equations based on the given information
From the problem, we have two scenarios:

1. **Scenario 1**: 2 men and 4 boys can complete the work in 10 days.
- The total work can be expressed as:
\[ \text{Total Work} = \text{(Work done per day)} \times \text{(Number of days)} \]
\[ \text{Total Work} = (2M + 4B) \times 10 \]
\[ \text{Total Work} = 20M + 40B \]

2. **Scenario 2**: 4 men and 5 boys can complete the work in 6 days.
- Similarly, we can express the total work as:
\[ \text{Total Work} = (4M + 5B) \times 6 \]
\[ \text{Total Work} = 24M + 30B \]

### Step 2: Set the two expressions for total work equal to each other
Since both expressions represent the same total work, we can set them equal:
\[ 20M + 40B = 24M + 30B \]

### Step 3: Rearrange the equation to isolate terms
Rearranging the equation gives:
\[ 20M + 40B - 24M - 30B = 0 \]
\[ -4M + 10B = 0 \]
\[ 10B = 4M \]
\[ B = \frac{4}{10}M = \frac{2}{5}M \]

### Step 4: Find the ratio of wages
Given that the per day wage of a man is ₹ 40, we can find the wage of a boy:
- If \( M \) corresponds to ₹ 40, then:
\[ B = \frac{2}{5}M = \frac{2}{5} \times 40 = ₹ 16 \]

### Step 5: Calculate the ratio of wages
Now we can find the ratio of the wages of a man to a boy:
\[ \text{Ratio of wages} = \frac{M}{B} = \frac{40}{16} = \frac{5}{2} \]

### Final Answer
Thus, the ratio of per day wages of 1 man to 1 boy is \( 5:2 \).

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