Four men and three wowen do a job in 6 days. When five men and six women work on the same job the gets completed in 4 days . How many days will be woman take to to the job if she works alone on it ?

Options

A. 18
B. 36
C. 54
D. None of these

Answer

To solve the problem, we need to find out how many days a woman will take to complete the job if she works alone. We are given two scenarios involving men and women working together.

### Step-by-Step Solution:

1. **Define Work Rates:**
Let the work done by one man in one day be \( M \) and the work done by one woman in one day be \( W \).

2. **Set Up the Equations:**
From the first scenario:
- 4 men and 3 women complete the job in 6 days.
- The total work done can be expressed as:
\[ 6 \times (4M + 3W) = 1 \quad \text{(1 job)} \]
Simplifying this gives:
\[ 4M + 3W = \frac{1}{6} \quad \text{(Equation 1)} \]

From the second scenario:
- 5 men and 6 women complete the job in 4 days.
- The total work done can be expressed as:
\[ 4 \times (5M + 6W) = 1 \quad \text{(1 job)} \]
Simplifying this gives:
\[ 5M + 6W = \frac{1}{4} \quad \text{(Equation 2)} \]

3. **Solve the Equations:**
Now we have two equations:
- \( 4M + 3W = \frac{1}{6} \) (Equation 1)
- \( 5M + 6W = \frac{1}{4} \) (Equation 2)

We can solve these equations simultaneously. First, let's multiply Equation 1 by 2 to eliminate fractions:
\[ 8M + 6W = \frac{1}{3} \quad \text{(Equation 3)} \]

Now we have:
- \( 8M + 6W = \frac{1}{3} \) (Equation 3)
- \( 5M + 6W = \frac{1}{4} \) (Equation 2)

Subtract Equation 2 from Equation 3:
\[ (8M + 6W) - (5M + 6W) = \frac{1}{3} - \frac{1}{4} \]
This simplifies to:
\[ 3M = \frac{4 - 3}{12} = \frac{1}{12} \]
Therefore:
\[ M = \frac{1}{36} \]

4. **Substitute Back to Find W:**
Substitute \( M \) back into Equation 1:
\[ 4\left(\frac{1}{36}\right) + 3W = \frac{1}{6} \]
This simplifies to:
\[ \frac{4}{36} + 3W = \frac{1}{6} \]
\[ \frac{1}{9} + 3W = \frac{1}{6} \]
To solve for \( W \), convert \( \frac{1}{6} \) to have a common denominator of 18:
\[ \frac{1}{9} + 3W = \frac{3}{18} \]
\[ 3W = \frac{3}{18} - \frac{2}{18} = \frac{1}{18} \]
Therefore:
\[ W = \frac{1}{54} \]

5. **Calculate Days for One Woman:**
If one woman does \( W \) work in one day, the number of days \( D \) for one woman to complete the job alone is the reciprocal of her work rate:
\[ D = \frac{1}{W} = \frac{1}{\frac{1}{54}} = 54 \text{ days} \]

### Final Answer:
A woman will take **54 days** to complete the job if she works alone.