A can complete a work in 9 days, B in 10 days and C in 15 days. B and C together started the work but left the work after 2 days. The time taken to complete the remaining work by A will be

Options

A. 6 days
B. 8 days
C. 9 days
D. 5 days

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the work done by B and C together in one day.
- A can complete the work in 9 days, so A's work in one day = \( \frac{1}{9} \)
- B can complete the work in 10 days, so B's work in one day = \( \frac{1}{10} \)
- C can complete the work in 15 days, so C's work in one day = \( \frac{1}{15} \)

Now, we need to find the combined work of B and C in one day:
\[ \text{Work of B + Work of C} = \frac{1}{10} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 10 and 15 is 30.

Converting the fractions:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{15} = \frac{2}{30} \]
Now add them:
\[ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]
Thus, B and C together can complete \( \frac{1}{6} \) of the work in one day.

### Step 2: Calculate the work done by B and C in 2 days.
Since B and C work together for 2 days:
\[ \text{Work done in 2 days} = 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]
So, B and C complete \( \frac{1}{3} \) of the work in 2 days.

### Step 3: Calculate the remaining work.
The total work is considered as 1 (whole work). Therefore, the remaining work after B and C have worked for 2 days is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 4: Calculate the time taken by A to complete the remaining work.
A can complete the whole work in 9 days, which means A's work rate is \( \frac{1}{9} \) of the work in one day. To find out how many days A will take to complete \( \frac{2}{3} \) of the work, we set up the equation:
\[ \text{Time taken by A} = \text{Remaining work} \div \text{A's work rate} = \frac{2}{3} \div \frac{1}{9} \]
This can be simplified as:
\[ \frac{2}{3} \times 9 = \frac{2 \times 9}{3} = \frac{18}{3} = 6 \]
Thus, A will take 6 days to complete the remaining work.

### Final Answer:
A will take **6 days** to complete the remaining work.

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