3m + 4b can earn ₹ 756 in 7 days. 11m + 13b can earn ₹ 3,008 in 8 days. in how many days, will 7 men + 9 boy earn ₹ 2,480?

Options

A. 10
B. 12
C. 8
D. 15

Answer

To solve the problem step by step, we need to derive the daily earnings of men and boys from the given information and then find out how many days it will take for 7 men and 9 boys to earn ₹2,480.

### Step 1: Calculate Daily Earnings from the First Equation
Given that 3 men and 4 boys can earn ₹756 in 7 days, we first find out how much they earn in one day.

\[ \text{Daily Earnings} = \frac{756}{7} = 108 \text{ (₹)} \]

Let’s denote the daily earnings of 1 man as \( m \) and 1 boy as \( b \). Thus, we can write the equation:

\[ 3m + 4b = 108 \quad \text{(Equation 1)} \]

### Step 2: Calculate Daily Earnings from the Second Equation
Next, for 11 men and 13 boys who earn ₹3,008 in 8 days, we find their daily earnings:

\[ \text{Daily Earnings} = \frac{3008}{8} = 376 \text{ (₹)} \]

This gives us the second equation:

\[ 11m + 13b = 376 \quad \text{(Equation 2)} \]

### Step 3: Solve the System of Equations
Now we have two equations:

1. \( 3m + 4b = 108 \)
2. \( 11m + 13b = 376 \)

To solve these equations, we can eliminate one variable. We can multiply Equation 1 by 11 and Equation 2 by 3 to align the coefficients of \( m \):

\[ 11(3m + 4b) = 11(108) \implies 33m + 44b = 1188 \quad \text{(Equation 3)} \]

\[ 3(11m + 13b) = 3(376) \implies 33m + 39b = 1128 \quad \text{(Equation 4)} \]

### Step 4: Subtract the Equations
Now, we subtract Equation 4 from Equation 3:

\[ (33m + 44b) - (33m + 39b) = 1188 - 1128 \]

This simplifies to:

\[ 5b = 60 \implies b = 12 \]

### Step 5: Substitute Back to Find \( m \)
Now that we have \( b \), we can substitute it back into Equation 1 to find \( m \):

\[ 3m + 4(12) = 108 \]

\[ 3m + 48 = 108 \implies 3m = 60 \implies m = 20 \]

### Step 6: Calculate Earnings of 7 Men and 9 Boys
Now we need to find out how much 7 men and 9 boys can earn in one day:

\[ \text{Daily Earnings} = 7m + 9b = 7(20) + 9(12) = 140 + 108 = 248 \text{ (₹)} \]

### Step 7: Calculate Days to Earn ₹2,480
Finally, we need to find out how many days it will take for them to earn ₹2,480:

\[ \text{Number of Days} = \frac{2480}{248} = 10 \]

### Conclusion
Thus, it will take **10 days** for 7 men and 9 boys to earn ₹2,480.

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