7 men and 8 boys can do a piece of work in 2 days. 4 men and 12 boys can do 29/56 of the same work in 1 day, in how many days will 1 man do this work?

Options

A. 28 days
B. 26 days
C. 50 days
D. None of these

Answer

To solve the problem step by step, we will define the work done by men and boys, set up equations based on the information given, and then solve for the number of days it takes for one man to complete the work.

### Step 1: Define Variables
Let:
- \( M \) = work done by 1 man in 1 day
- \( B \) = work done by 1 boy in 1 day

### Step 2: Set Up the First Equation
From the problem, we know that 7 men and 8 boys can complete the work in 2 days. Therefore, the total work can be expressed as:
\[ 7M + 8B = \frac{1}{2} \quad \text{(since they complete 1 part in 2 days)} \]
This simplifies to:
\[ 7M + 8B = \frac{1}{2} \]

### Step 3: Set Up the Second Equation
Next, we know that 4 men and 12 boys can complete \( \frac{29}{56} \) of the work in 1 day. Thus, we can write:
\[ 4M + 12B = \frac{29}{56} \]

### Step 4: Solve the Equations
We now have two equations:
1. \( 7M + 8B = \frac{1}{2} \) (Equation 1)
2. \( 4M + 12B = \frac{29}{56} \) (Equation 2)

To eliminate one variable, we can multiply Equation 1 by 3 and Equation 2 by 2:
\[ 3(7M + 8B) = 3 \cdot \frac{1}{2} \implies 21M + 24B = \frac{3}{2} \quad \text{(Equation 3)} \]
\[ 2(4M + 12B) = 2 \cdot \frac{29}{56} \implies 8M + 24B = \frac{29}{28} \quad \text{(Equation 4)} \]

### Step 5: Subtract the Equations
Now, we subtract Equation 4 from Equation 3:
\[ (21M + 24B) - (8M + 24B) = \frac{3}{2} - \frac{29}{28} \]
This simplifies to:
\[ 13M = \frac{3}{2} - \frac{29}{28} \]

### Step 6: Find a Common Denominator
To subtract the fractions, we find a common denominator:
\[ \frac{3}{2} = \frac{42}{28} \quad \text{(converting to a common denominator)} \]
Thus:
\[ \frac{3}{2} - \frac{29}{28} = \frac{42}{28} - \frac{29}{28} = \frac{13}{28} \]

### Step 7: Solve for M
Now we have:
\[ 13M = \frac{13}{28} \]
Dividing both sides by 13 gives:
\[ M = \frac{1}{28} \]

### Step 8: Calculate the Days for 1 Man to Complete the Work
Since 1 man can do \( \frac{1}{28} \) of the work in one day, the number of days for 1 man to complete the whole work is:
\[ \text{Days} = \frac{1}{M} = \frac{1}{\frac{1}{28}} = 28 \text{ days} \]

### Final Answer
Thus, 1 man will take **28 days** to complete the work.

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