A can do a piece of work in 10 days and B can do it in 12 days. They work together for 3 days. Then B leaves and A alone continues. 2 days after that C joins and the work is completed in 2 days more. In how many days can C do it, if he works alone ?

Options

A. 40 days
B. 50 days
C. 40 days
D. 60 days

Answer

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

### Step 1: Determine the work efficiencies of A and B
- A can complete the work in 10 days, so A's efficiency is:
\[ \text{Efficiency of A} = \frac{1 \text{ work}}{10 \text{ days}} = \frac{1}{10} \text{ work/day} \]
- B can complete the work in 12 days, so B's efficiency is:
\[ \text{Efficiency of B} = \frac{1 \text{ work}}{12 \text{ days}} = \frac{1}{12} \text{ work/day} \]

### Step 2: Calculate the combined efficiency of A and B
- The combined efficiency of A and B is:
\[ \text{Combined Efficiency} = \frac{1}{10} + \frac{1}{12} \]
- To add these fractions, find a common denominator (which is 60):
\[ \text{Combined Efficiency} = \frac{6}{60} + \frac{5}{60} = \frac{11}{60} \text{ work/day} \]

### Step 3: Calculate the work done by A and B in 3 days
- The amount of work done by A and B together in 3 days is:
\[ \text{Work done in 3 days} = 3 \times \frac{11}{60} = \frac{33}{60} = \frac{11}{20} \text{ of the work} \]

### Step 4: Determine the remaining work
- The total work is 1 (or 60 units if we consider the total work as 60 units). Thus, the remaining work after 3 days is:
\[ \text{Remaining Work} = 1 - \frac{11}{20} = \frac{9}{20} \text{ of the work} \]

### Step 5: Calculate the work done by A alone in the next 2 days
- A continues to work alone for 2 days. The work done by A in 2 days is:
\[ \text{Work done by A in 2 days} = 2 \times \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \text{ of the work} \]

### Step 6: Determine the remaining work after A's 2 days
- The remaining work after A works for 2 days is:
\[ \text{Remaining Work} = \frac{9}{20} - \frac{1}{5} = \frac{9}{20} - \frac{4}{20} = \frac{5}{20} = \frac{1}{4} \text{ of the work} \]

### Step 7: Calculate the time taken by A and C together to finish the remaining work
- A's efficiency is \(\frac{1}{10}\) and let C's efficiency be \(c\). Together, their efficiency is:
\[ \text{Efficiency of A and C} = \frac{1}{10} + c \]
- They complete the remaining \(\frac{1}{4}\) of the work in 2 days:
\[ 2 \left(\frac{1}{10} + c\right) = \frac{1}{4} \]
Simplifying gives:
\[ \frac{1}{10} + c = \frac{1}{8} \]
\[ c = \frac{1}{8} - \frac{1}{10} \]
Finding a common denominator (40):
\[ c = \frac{5}{40} - \frac{4}{40} = \frac{1}{40} \text{ work/day} \]

### Step 8: Calculate the number of days C would take to complete the work alone
- If C's efficiency is \(\frac{1}{40}\), then the time taken by C to complete the entire work alone is:
\[ \text{Time taken by C} = \frac{1 \text{ work}}{\frac{1}{40} \text{ work/day}} = 40 \text{ days} \]

Thus, C can complete the work alone in **40 days**.

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