Garima and her sister together can paint the walls of their house in 45 days. If Garima alone did the job, it would have taken her 81 days. If they both started painting together but Garima's sister had to leave 9 days before the completion of the work, how many days in all would it take the sisters to paint their house?

Options

A. 49
B. 48
C. 52
D. 50

Answer

To solve the problem step by step, let's break down the information given and calculate the required values.

### Step 1: Determine the Work Rates
1. **Combined Work Rate of Garima and her Sister**: They can paint the house together in 45 days.
- Work done in one day = \( \frac{1}{45} \) of the house.

2. **Garima's Work Rate**: Garima alone can paint the house in 81 days.
- Work done by Garima in one day = \( \frac{1}{81} \) of the house.

3. **Sister's Work Rate**: Let the sister's work rate be \( S \).
- Combined work rate equation:
\[ \frac{1}{45} = \frac{1}{81} + S \]
- Rearranging gives:
\[ S = \frac{1}{45} - \frac{1}{81} \]

### Step 2: Calculate Sister's Work Rate
To calculate \( S \), we need a common denominator:
- The least common multiple (LCM) of 45 and 81 is 405.

Now convert the fractions:
\[ \frac{1}{45} = \frac{9}{405}, \quad \frac{1}{81} = \frac{5}{405} \]
Thus,
\[ S = \frac{9}{405} - \frac{5}{405} = \frac{4}{405} \]
So, the sister's work rate is \( \frac{1}{101.25} \) of the house per day.

### Step 3: Set Up the Work Equation
Let \( x \) be the total number of days taken to complete the work.
- Garima works for \( x \) days.
- The sister works for \( x - 9 \) days (since she leaves 9 days before completion).

The total work done can be expressed as:
\[ \text{Work done by Garima} + \text{Work done by Sister} = 1 \text{ (whole house)} \]
This translates to:
\[ \frac{x}{81} + \frac{x - 9}{101.25} = 1 \]

### Step 4: Solve the Equation
To solve for \( x \), we first convert \( \frac{x - 9}{101.25} \) to a fraction with a common denominator:
\[ \frac{x}{81} + \frac{(x - 9) \cdot 405}{405 \cdot 101.25} = 1 \]
This simplifies to:
\[ \frac{x}{81} + \frac{4(x - 9)}{405} = 1 \]
Multiplying through by 405 to eliminate the denominators:
\[ \frac{405x}{81} + 4(x - 9) = 405 \]
Calculating \( \frac{405}{81} = 5 \):
\[ 5x + 4x - 36 = 405 \]
Combining like terms:
\[ 9x - 36 = 405 \]
Adding 36 to both sides:
\[ 9x = 441 \]
Dividing by 9:
\[ x = 49 \]

### Conclusion
Thus, the total time taken by Garima and her sister to paint the house is **49 days**.