Kareena can do a piece of work in 9 days and Karishma can do the same work in 18 days. They started the work. After 3 days Shahid joined them, who can complete alone the same whole work in 3 days. What is the total number of days in which they had completed the work?

Options

A. 12
B. 8
C. 4
D. 6

Answer

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

### Step 1: Determine the work efficiency of each person.
- **Kareena's efficiency**: She can complete the work in 9 days. Therefore, her efficiency is:
\[ \text{Efficiency of Kareena} = \frac{1 \text{ work}}{9 \text{ days}} = \frac{1}{9} \]

- **Karishma's efficiency**: She can complete the work in 18 days. Therefore, her efficiency is:
\[ \text{Efficiency of Karishma} = \frac{1 \text{ work}}{18 \text{ days}} = \frac{1}{18} \]

- **Shahid's efficiency**: He can complete the work in 3 days. Therefore, his efficiency is:
\[ \text{Efficiency of Shahid} = \frac{1 \text{ work}}{3 \text{ days}} = \frac{1}{3} \]

### Step 2: Calculate the total work done by Kareena and Karishma in the first 3 days.
- Together, their combined efficiency for one day is:
\[ \text{Combined efficiency} = \frac{1}{9} + \frac{1}{18} \]
To add these fractions, we find a common denominator (which is 18):
\[ \frac{1}{9} = \frac{2}{18} \]
So,
\[ \text{Combined efficiency} = \frac{2}{18} + \frac{1}{18} = \frac{3}{18} = \frac{1}{6} \]

- In 3 days, the work done is:
\[ \text{Work done in 3 days} = 3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \]

### Step 3: Calculate the remaining work.
- After 3 days, half of the work is completed. Therefore, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 4: Calculate the combined efficiency when Shahid joins.
- Now, all three (Kareena, Karishma, and Shahid) are working together. Their combined efficiency is:
\[ \text{Total efficiency} = \frac{1}{9} + \frac{1}{18} + \frac{1}{3} \]
Again, finding a common denominator (which is 18):
\[ \frac{1}{9} = \frac{2}{18}, \quad \frac{1}{3} = \frac{6}{18} \]
So,
\[ \text{Total efficiency} = \frac{2}{18} + \frac{1}{18} + \frac{6}{18} = \frac{9}{18} = \frac{1}{2} \]

### Step 5: Calculate the time taken to complete the remaining work.
- The time taken to complete the remaining work of \(\frac{1}{2}\) at a rate of \(\frac{1}{2}\) work per day is:
\[ \text{Time taken} = \frac{\text{Remaining work}}{\text{Total efficiency}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1 \text{ day} \]

### Step 6: Calculate the total time taken to complete the work.
- The total time taken is the sum of the time worked by Kareena and Karishma alone and the time worked together with Shahid:
\[ \text{Total time} = 3 \text{ days} + 1 \text{ day} = 4 \text{ days} \]

### Final Answer:
The total number of days in which they completed the work is **4 days**.

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