A and B can do a work in 15 days and 10 days respectively. They started working together but after 2 days for some reason B had to leave the work and A alone did the remaining work. In how many days total work was finished?
A तथा B किसी कार्य को पूरा करने में क्रमशः 15 दिन तथा 10 दिन का समय लेते है। उन्होनें मिलकर कार्य करना आरंभ किया किंतु 2 दिन पश्चात किसी कारणवश B को कार्य छोड़ना पड़ा तथा शेष कार्य अकेले A ने पूरा किया। संपूर्ण कार्य कितने समय में पूरा हुआ?

Options

A. 10 days/10 दिन
B. 8 days/8 दिन
C. 12 days/12 दिन
D. 15 days/15 दिन

Explanation

To solve the problem, we need to determine how long it takes for A and B to complete the work together and then how long A takes to finish the remaining work after B leaves.

### Step-by-step Solution:

1. **Determine the work rates of A and B:**
- A can complete the work in 15 days. Therefore, A's work rate is \( \frac{1}{15} \) of the work per day.
- B can complete the work in 10 days. Therefore, B's work rate is \( \frac{1}{10} \) of the work per day.

2. **Calculate the combined work rate of A and B:**
- Combined work rate of A and B = A's work rate + B's work rate
\[ \text{Combined work rate} = \frac{1}{15} + \frac{1}{10} \]
- To add these fractions, we need a common denominator. The least common multiple of 15 and 10 is 30.
\[ \frac{1}{15} = \frac{2}{30}, \quad \frac{1}{10} = \frac{3}{30} \]
- Therefore,
\[ \text{Combined work rate} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \]
- This means together, A and B can complete \( \frac{1}{6} \) of the work in one day.

3. **Calculate the work done in the first 2 days:**
- In 2 days, the amount of work completed by A and B together is:
\[ \text{Work done in 2 days} = 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \]

4. **Determine the remaining work:**
- Since \( \frac{1}{3} \) of the work is done, the remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

5. **Calculate how long it takes A to finish the remaining work:**
- A's work rate is \( \frac{1}{15} \) of the work per day. To find out how many days A needs to complete \( \frac{2}{3} \) of the work:
\[ \text{Days required by A} = \frac{\text{Remaining work}}{\text{A's work rate}} = \frac{\frac{2}{3}}{\frac{1}{15}} = \frac{2}{3} \times 15 = 10 \text{ days} \]

6. **Calculate the total time taken to complete the work:**
- Total time = Time worked together + Time taken by A alone
\[ \text{Total time} = 2 \text{ days} + 10 \text{ days} = 12 \text{ days} \]

### Final Answer:
The total work was finished in **12 days**.