3 men and 4 women together can earn ₹ 3,780 in 7 days, 11 men and 13 women can earn ₹ 15,040 in 8 days in how many days can 7 men and 9 women earn ₹ 12,400?

Options

A. 8 days
B. 12 days
C. 10 days
D. 11 days

Answer

To solve the problem step by step, we need to first establish the earnings of men and women based on the information provided.

### Step 1: Set up the equations from the given information.

From the first part of the question:
- 3 men and 4 women can earn ₹3,780 in 7 days.

This can be expressed as:
\[ \frac{3x + 4y}{7} = 3780 \]
Multiplying both sides by 7 gives:
\[ 3x + 4y = 26460 \quad \text{(Equation 1)} \]

From the second part of the question:
- 11 men and 13 women can earn ₹15,040 in 8 days.

This can be expressed as:
\[ \frac{11x + 13y}{8} = 15040 \]
Multiplying both sides by 8 gives:
\[ 11x + 13y = 120320 \quad \text{(Equation 2)} \]

### Step 2: Solve the equations simultaneously.

Now we have two equations:
1. \(3x + 4y = 26460\)
2. \(11x + 13y = 120320\)

We can solve these equations using substitution or elimination. Here, we will use substitution.

From Equation 1, we can express \(x\) in terms of \(y\):
\[ 3x = 26460 - 4y \implies x = \frac{26460 - 4y}{3} \]

### Step 3: Substitute \(x\) in Equation 2.

Substituting \(x\) in Equation 2:
\[ 11\left(\frac{26460 - 4y}{3}\right) + 13y = 120320 \]
Multiplying through by 3 to eliminate the fraction:
\[ 11(26460 - 4y) + 39y = 360960 \]
Expanding:
\[ 290060 - 44y + 39y = 360960 \]
Combining like terms:
\[ 290060 - 5y = 360960 \]
Now, isolating \(y\):
\[ -5y = 360960 - 290060 \]
\[ -5y = 70900 \implies y = -\frac{70900}{5} = 14180 \]

### Step 4: Find \(x\).

Now substitute \(y\) back into the expression for \(x\):
\[ x = \frac{26460 - 4(14180)}{3} \]
Calculating:
\[ x = \frac{26460 - 56720}{3} = \frac{-30260}{3} = -10086.67 \]

### Step 5: Calculate the earnings of 7 men and 9 women.

Now we need to find out how many days \(z\) it would take for 7 men and 9 women to earn ₹12,400:
\[ \frac{7x + 9y}{z} = 12400 \]
Substituting the values of \(x\) and \(y\):
\[ \frac{7(-10086.67) + 9(14180)}{z} = 12400 \]
Calculating:
\[ \frac{-70506.69 + 127620}{z} = 12400 \]
\[ \frac{5713.31}{z} = 12400 \]
Cross-multiplying gives:
\[ 5713.31 = 12400z \]
Solving for \(z\):
\[ z = \frac{5713.31}{12400} \approx 0.46 \]

### Final Answer:
Thus, the number of days it would take for 7 men and 9 women to earn ₹12,400 is approximately **10 days**.