A can do a work in 18 days, B in 9 days and C in 6 days. A and B start working together and after 2 days C joins them. What is the total number of days taken to finish the work ?

Options

A. `4.33`
B. `4.5`
C. `4.66`
D. None

Answer

To solve the problem step by step, we will determine how much work A, B, and C can do individually and then collectively to find out the total time taken to finish the work.

### Step 1: Determine the work done by A, B, and C in one day.

- A can complete the work in 18 days. Therefore, the work done by A in one day is:
\[ \text{Work done by A in one day} = \frac{1}{18} \]

- B can complete the work in 9 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{9} \]

- C can complete the work in 6 days. Therefore, the work done by C in one day is:
\[ \text{Work done by C in one day} = \frac{1}{6} \]

### Step 2: Calculate the work done by A and B together in 2 days.

- The combined work done by A and B in one day is:
\[ \text{Work done by A and B in one day} = \frac{1}{18} + \frac{1}{9} \]
To add these fractions, we need a common denominator. The least common multiple of 18 and 9 is 18.
\[ \frac{1}{9} = \frac{2}{18} \]
Thus,
\[ \text{Work done by A and B in one day} = \frac{1}{18} + \frac{2}{18} = \frac{3}{18} = \frac{1}{6} \]

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

### Step 3: Calculate the remaining work after 2 days.

- The total work is considered as 1 unit. After A and B have worked for 2 days, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Calculate the work done by A, B, and C together in one day.

- Now, we calculate the combined work done by A, B, and C in one day:
\[ \text{Work done by A, B, and C in one day} = \frac{1}{18} + \frac{1}{9} + \frac{1}{6} \]
Again, we need a common denominator. The least common multiple of 18, 9, and 6 is 18.
\[ \frac{1}{9} = \frac{2}{18}, \quad \frac{1}{6} = \frac{3}{18} \]
Thus,
\[ \text{Work done by A, B, and C in one day} = \frac{1}{18} + \frac{2}{18} + \frac{3}{18} = \frac{6}{18} = \frac{1}{3} \]

### Step 5: Calculate the number of days required to finish the remaining work.

- The remaining work is \(\frac{2}{3}\). The number of days required by A, B, and C to finish this remaining work is:
\[ \text{Days required} = \frac{\text{Remaining work}}{\text{Work done by A, B, and C in one day}} = \frac{\frac{2}{3}}{\frac{1}{3}} = 2 \text{ days} \]

### Step 6: Calculate the total number of days taken to finish the work.

- The total number of days taken to finish the work is:
\[ \text{Total days} = \text{Days worked by A and B} + \text{Days worked by A, B, and C} = 2 + 2 = 4 \text{ days} \]

### Final Answer:
The total number of days taken to finish the work is **4 days**.

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