If 5 men and 3 boys can reap 23 hectares in 4 days and if 3 men and 2 boys can reap 7 hectares in 2 days. How many boys must assist 7 men in order that they may reap 45 hectares in 6 days?

Options

A. 2 boys
B. 6 boys
C. 4 boys
D. 5 boys

Answer

To solve the problem step by step, we will first set up equations based on the information provided in the question.

### Step 1: Set Up the Equations
From the first part of the question, we know that:
- 5 men and 3 boys can reap 23 hectares in 4 days.

We can express this in terms of work done in hectares per day:
\[ \text{Work done} = \frac{23 \text{ hectares}}{4 \text{ days}} = 5.75 \text{ hectares/day} \]
Let the work done by 1 man in 1 day be \( m \) hectares and the work done by 1 boy in 1 day be \( b \) hectares. Therefore, we can write the equation:
\[ 5m + 3b = 5.75 \quad \text{(Equation 1)} \]

From the second part of the question, we know that:
- 3 men and 2 boys can reap 7 hectares in 2 days.

Again, we can express this in terms of work done in hectares per day:
\[ \text{Work done} = \frac{7 \text{ hectares}}{2 \text{ days}} = 3.5 \text{ hectares/day} \]
Thus, we can write the second equation:
\[ 3m + 2b = 3.5 \quad \text{(Equation 2)} \]

### Step 2: Solve the Equations
Now, we will solve these two equations simultaneously.

**Multiply Equation 2 by 3:**
\[ 3(3m + 2b) = 3(3.5) \]
This gives us:
\[ 9m + 6b = 10.5 \quad \text{(Equation 3)} \]

**Multiply Equation 1 by 2:**
\[ 2(5m + 3b) = 2(5.75) \]
This gives us:
\[ 10m + 6b = 11.5 \quad \text{(Equation 4)} \]

### Step 3: Eliminate \( b \)
Now, we will eliminate \( b \) by subtracting Equation 3 from Equation 4:
\[ (10m + 6b) - (9m + 6b) = 11.5 - 10.5 \]
This simplifies to:
\[ m = 1 \]

### Step 4: Substitute \( m \) back to find \( b \)
Now that we have \( m = 1 \), we will substitute this value back into Equation 1 to find \( b \):
\[ 5(1) + 3b = 5.75 \]
This simplifies to:
\[ 5 + 3b = 5.75 \]
Subtracting 5 from both sides gives:
\[ 3b = 0.75 \]
Dividing by 3 gives:
\[ b = 0.25 \]

### Step 5: Find the number of boys needed for 7 men to reap 45 hectares in 6 days
Now we need to find how many boys \( x \) are needed for 7 men to reap 45 hectares in 6 days.

The total work required is:
\[ \text{Total work} = \frac{45 \text{ hectares}}{6 \text{ days}} = 7.5 \text{ hectares/day} \]

We can set up the equation:
\[ 7m + xb = 7.5 \]
Substituting \( m = 1 \) and \( b = 0.25 \):
\[ 7(1) + x(0.25) = 7.5 \]
This simplifies to:
\[ 7 + 0.25x = 7.5 \]
Subtracting 7 from both sides gives:
\[ 0.25x = 0.5 \]
Dividing by 0.25 gives:
\[ x = 2 \]

### Final Answer
Thus, **2 boys** are needed to assist 7 men in order to reap 45 hectares in 6 days.

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