8 men and 10 women can do a work in 15 days while 10 men and 18 women can do that work in 10 days. 4 men and 5 women started the work and after 10 days all men left the work then find how many more women would be required to complete the whole work in 21 days?

Options

A. 12
B. 15
C. 16
D. 10

Answer

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

### Step 1: Determine the total work done in terms of men and women.
Given:
- 8 men and 10 women can complete the work in 15 days.
- 10 men and 18 women can complete the work in 10 days.

**Total work (W)** can be calculated using the formula:
\[ W = \text{(Number of workers)} \times \text{(Days)} \]

From the first condition:
\[ W = (8M + 10W) \times 15 \]
From the second condition:
\[ W = (10M + 18W) \times 10 \]

### Step 2: Set the equations equal to each other.
Since both expressions equal the total work, we can set them equal:
\[ (8M + 10W) \times 15 = (10M + 18W) \times 10 \]

### Step 3: Expand and simplify the equation.
Expanding both sides:
\[ 120M + 150W = 100M + 180W \]

Rearranging gives:
\[ 120M - 100M = 180W - 150W \]
\[ 20M = 30W \]

### Step 4: Find the ratio of men to women.
Dividing both sides by 10:
\[ 2M = 3W \]
Thus, the ratio of men to women is:
\[ \frac{M}{W} = \frac{3}{2} \]

### Step 5: Calculate the total work in terms of one variable.
Let's express the total work in terms of women (W):
Using the first equation:
\[ W = (8M + 10W) \times 15 \]
Substituting \( M = \frac{2}{3}W \):
\[ W = \left(8 \times \frac{2}{3}W + 10W\right) \times 15 \]
\[ W = \left(\frac{16}{3}W + 10W\right) \times 15 \]
\[ W = \left(\frac{16 + 30}{3}W\right) \times 15 \]
\[ W = \left(\frac{46}{3}W\right) \times 15 \]
\[ W = 230W \]

### Step 6: Calculate the work done by 4 men and 5 women in 10 days.
Using the ratio of work done:
- Work done by 4 men in 10 days:
\[ \text{Work by 4 men} = 4M \times 10 = 4 \times \frac{2}{3}W \times 10 = \frac{80}{3}W \]
- Work done by 5 women in 10 days:
\[ \text{Work by 5 women} = 5W \times 10 = 50W \]

Total work done in 10 days:
\[ \text{Total Work Done} = \frac{80}{3}W + 50W = \frac{80 + 150}{3}W = \frac{230}{3}W \]

### Step 7: Calculate remaining work.
Total work is \( 230W \), and work done is \( \frac{230}{3}W \):
Remaining work:
\[ \text{Remaining Work} = 230W - \frac{230}{3}W = \frac{690 - 230}{3}W = \frac{460}{3}W \]

### Step 8: Determine how many days remain.
Total days = 21, days already worked = 10, so remaining days = 11.

### Step 9: Calculate the efficiency required to complete the remaining work.
Let \( x \) be the number of women required to complete the remaining work:
\[ xW \times 11 = \frac{460}{3}W \]
Dividing both sides by \( W \):
\[ 11x = \frac{460}{3} \]
\[ x = \frac{460}{33} \approx 13.94 \]

### Step 10: Determine how many more women are needed.
Initially, there were 5 women working:
\[ \text{More women needed} = x - 5 = 13.94 - 5 \approx 8.94 \]

Since we need whole women, we round up to 9.

### Conclusion
To complete the work in 21 days, **9 more women** are required.