A contractor took a construction work for Rs.400000 and employed 50 men to complete the work in 20 days. He decided to set aside 25% of the amount as profit for himself and distributed rest of the amounts among the workers as their wages. But at the end of `18^(th)` day, he noticed that the work is behind the schedule. So he employed some women along with the men and decided to pay a woman 80% of what was paid to a man per day. The work was completed 3 days later than the schedule. If the contractor finally had a profit of Rs.31000 only, then how many women were employed by him?

Options

A. A)25
B. B)20
C. C)15
D. D)10

Answer

To solve the problem step by step, let's break it down:

### Step 1: Calculate the total amount set aside for wages
The contractor took a total of Rs. 400,000 for the construction work. He decided to set aside 25% of this amount as profit for himself.

**Calculation:**
- Profit = 25% of 400,000 = \( \frac{25}{100} \times 400,000 = 100,000 \)
- Amount available for wages = Total amount - Profit = 400,000 - 100,000 = Rs. 300,000

**Hint:** Remember to calculate the percentage of the total amount to find out how much is allocated for wages.

### Step 2: Determine the daily wage cost for the men
The contractor employed 50 men to complete the work in 20 days.

**Calculation:**
- Total wage for 20 days = Rs. 300,000
- Daily wage cost for 50 men = \( \frac{300,000}{20} = 15,000 \)
- Daily wage per man = \( \frac{15,000}{50} = 300 \)

**Hint:** Divide the total wages by the number of days to find the daily wage, then divide by the number of workers to find the wage per worker.

### Step 3: Calculate the total work done in 18 days
In 18 days, the work done is less than expected. We need to calculate how much work was completed in this time.

**Calculation:**
- Work done in 18 days = Daily wage cost for 50 men × 18 days = 15,000 × 18 = Rs. 270,000

**Hint:** Multiply the daily wage cost by the number of days worked to find the total expenditure on labor.

### Step 4: Determine the remaining amount for the last 5 days
The contractor has 3 days left to complete the work after the 18th day, but he actually completed the work 3 days later, totaling 23 days.

**Calculation:**
- Total amount spent on labor after 23 days = Rs. 300,000 (total wages)
- Amount spent on labor for the first 18 days = Rs. 270,000
- Remaining amount for the last 5 days = Rs. 300,000 - Rs. 270,000 = Rs. 30,000

**Hint:** Subtract the amount already spent from the total wages to find out how much is left for the additional labor.

### Step 5: Calculate the cost of employing women
Let the number of women employed be W. Each woman is paid 80% of what a man earns.

**Calculation:**
- Daily wage per woman = 80% of Rs. 300 = Rs. 240
- Total cost for W women for 5 days = \( W \times 240 \times 5 = 1200W \)

**Hint:** Remember to calculate the wage for women based on the percentage of men's wages.

### Step 6: Set up the equation
The total expenditure for 23 days must equal the total wages available for workers.

**Equation:**
- Total expenditure = Expenditure for men for 23 days + Expenditure for women for 5 days
- \( 270,000 + 1200W = 300,000 \)

### Step 7: Solve for W
Rearranging the equation gives:
- \( 1200W = 300,000 - 270,000 \)
- \( 1200W = 30,000 \)
- \( W = \frac{30,000}{1200} = 25 \)

**Hint:** Isolate W by moving all other terms to the other side of the equation.

### Final Answer
The contractor employed **25 women** to help complete the work.