A alone can do a piece of work in 10 days and B alone can do it in 15 days. In how many days will A and B together do the same work?

Options

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

Answer

To solve the problem step by step, we will determine how long A and B will take to complete the work together.

### Step-by-step Solution:

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

2. **Add the work rates of A and B together:**
- To find the combined work rate of A and B, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{15} \]

3. **Find a common denominator:**
- The least common multiple (LCM) of 10 and 15 is 30. We will convert both fractions to have a denominator of 30:
\[ \frac{1}{10} = \frac{3}{30} \quad \text{(multiply numerator and denominator by 3)} \]
\[ \frac{1}{15} = \frac{2}{30} \quad \text{(multiply numerator and denominator by 2)} \]

4. **Add the fractions:**
- Now we can add the two fractions:
\[ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} \]

5. **Simplify the combined work rate:**
- Simplifying \(\frac{5}{30}\):
\[ \frac{5}{30} = \frac{1}{6} \]
- This means A and B together can complete \(\frac{1}{6}\) of the work in one day.

6. **Calculate the total time taken to complete the work:**
- If A and B together complete \(\frac{1}{6}\) of the work in one day, then the total time taken to complete the entire work is the reciprocal of \(\frac{1}{6}\):
\[ \text{Total time} = 6 \text{ days} \]

### Final Answer:
A and B together will complete the work in **6 days**.

---