A can do a piece of work in 10 days while B alone can do it in 15 days . In how many days can both finish the same work?

Answer

To solve the problem of how many days A and B can finish the work together, we can follow these steps:

### Step 1: Determine the work done by A and B in one day.
- A can complete the work in 10 days. Therefore, in one day, A does:
\[ \text{Work done by A in one day} = \frac{1}{10} \]
- B can complete the work in 15 days. Therefore, in one day, B does:
\[ \text{Work done by B in one day} = \frac{1}{15} \]

### Step 2: Add the work done by A and B in one day.
- To find the total work done by A and B together in one day, we add the fractions:
\[ \text{Total work in one day} = \frac{1}{10} + \frac{1}{15} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple (LCM) of 10 and 15 is 30. We convert the fractions:
\[ \frac{1}{10} = \frac{3}{30} \quad \text{and} \quad \frac{1}{15} = \frac{2}{30} \]
- Now we can add them:
\[ \frac{3}{30} + \frac{2}{30} = \frac{5}{30} \]

### Step 4: Simplify the total work done in one day.
- Simplifying \(\frac{5}{30}\) gives:
\[ \frac{5}{30} = \frac{1}{6} \]
- This means together, A and B can complete \(\frac{1}{6}\) of the work in one day.

### Step 5: Calculate the total days required to complete the work.
- If A and B together complete \(\frac{1}{6}\) of the work in one day, then the total number of days required to complete the entire work is:
\[ \text{Total days} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

### Final Answer:
A and B together can finish the work in **6 days**.

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