A does a work in 10 days and B does the same work in 15 days. In how many days they together will do the same work.?

To solve this problem, let's first calculate the rates at which A and B complete the work individually.

- A completes the work in 10 days, so A's rate of work is:
  \[
  \frac{1}{10} \text{ work/day}.
  \]
  
- B completes the work in 15 days, so B's rate of work is:
  \[
  \frac{1}{15} \text{ work/day}.
  \]

When A and B work together, their combined rate of work is the sum of their individual rates:
\[
\frac{1}{10} + \frac{1}{15}.
\]

To add these fractions, find a common denominator:
\[
\frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}.
\]

So, A and B together complete \(\frac{1}{6}\) of the work in one day. Therefore, they will complete the entire work in 6 days.

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

Would you like further details or have any questions?

Here are some related questions for deeper understanding:
1. If A alone worked for 5 days, how long would it take for B to finish the remaining work?
2. If B worked for 3 days and then A joined, how long would it take them to finish the work together?
3. If A and B both work for 2 days together, what fraction of the work will remain?
4. What if C can complete the same work in 12 days? How many days will it take A, B, and C together to complete the work?
5. How many days would it take if A worked only half the time and B worked the entire time?

**Tip:** Whenever two people work together, their combined rate is the sum of their individual rates, so it's always useful to think in terms of rates.

Learn how to calculate the combined work rate of A and B to solve a work completion problem. Understand the concept of rates of work and how to add fractions. Suitable for students in Grades 6-8.

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