A can do a piece of work in 10 days and B can do the same work in 12 days. How long will they take to finish the work, if both work together?

Answer

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

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

- B can complete the work in 12 days. Therefore, the work done by B in one day is:
\[ \text{Work done by B in one day} = \frac{1}{12} \]

### Step 2: Calculate the combined work done by A and B in one day.
- When A and B work together, their combined work done in one day is:
\[ \text{Combined work in one day} = \frac{1}{10} + \frac{1}{12} \]

### Step 3: Find a common denominator to add the fractions.
- The least common multiple (LCM) of 10 and 12 is 60. We convert the fractions:
\[ \frac{1}{10} = \frac{6}{60} \quad \text{and} \quad \frac{1}{12} = \frac{5}{60} \]
- Now we can add the fractions:
\[ \text{Combined work in one day} = \frac{6}{60} + \frac{5}{60} = \frac{11}{60} \]

### Step 4: Calculate the total time taken to complete the work.
- If A and B together can do \(\frac{11}{60}\) of the work in one day, then the total number of days required to complete the entire work (1 unit of work) is the reciprocal of the combined work done in one day:
\[ \text{Total days required} = \frac{1}{\frac{11}{60}} = \frac{60}{11} \]

### Step 5: Simplify the answer.
- The total days required can be expressed as:
\[ \frac{60}{11} \approx 5.45 \text{ days} \]
- In fraction form, it is \(5 \frac{5}{11}\) days.

### Final Answer:
A and B will take \(\frac{60}{11}\) days or approximately \(5 \frac{5}{11}\) days to finish the work if they work together.

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