Amar can complete a work in 15 days, and Prem, in 12 days. Starting with Amar, they work on alternate days. In how many days will the work be completed?  

To solve this problem, let's first determine how much work Amar and Prem can complete in a single day:

- **Amar** can complete the entire work in 15 days. Therefore, in 1 day, Amar completes:
  \[
  \text{Amar's 1-day work} = \frac{1}{15}
  \]
  
- **Prem** can complete the entire work in 12 days. Therefore, in 1 day, Prem completes:
  \[
  \text{Prem's 1-day work} = \frac{1}{12}
  \]

Now, since they work on alternate days, the pattern will be as follows:
- **Day 1**: Amar works, completing \(\frac{1}{15}\) of the work.
- **Day 2**: Prem works, completing \(\frac{1}{12}\) of the work.

The total work done in 2 days is the sum of Amar's and Prem's contributions:
\[
\text{Total work in 2 days} = \frac{1}{15} + \frac{1}{12}
\]
To add these fractions, we find a common denominator:
\[
\frac{1}{15} + \frac{1}{12} = \frac{4}{60} + \frac{5}{60} = \frac{9}{60} = \frac{3}{20}
\]
Thus, in 2 days, they complete \(\frac{3}{20}\) of the work.

Now, to find out how many full cycles of 2 days are required to finish the work, we need to calculate how many cycles will make the work complete.

Let \(x\) represent the number of 2-day cycles. After \(x\) such cycles, the total work done will be:
\[
\frac{3}{20} \times x
\]
Setting this equal to 1 (since the total work to be completed is 1):
\[
\frac{3}{20} \times x = 1
\]
Solving for \(x\):
\[
x = \frac{20}{3} \approx 6.67
\]
This means after 6 full cycles (12 days), they will have completed:
\[
6 \times \frac{3}{20} = \frac{18}{20} = \frac{9}{10} \text{ of the work.}
\]
At this point, \( \frac{1}{10} \) of the work remains.

On Day 13, it will be Amar’s turn to work, and he can complete \(\frac{1}{15}\) of the work. Since the remaining work is \(\frac{1}{10}\), and Amar can do \(\frac{1}{15}\), let’s compare the two:

\[
\frac{1}{10} > \frac{1}{15}
\]
Thus, Amar will complete \(\frac{1}{15}\) of the work on Day 13, leaving:
\[
\frac{1}{10} - \frac{1}{15} = \frac{3}{30} - \frac{2}{30} = \frac{1}{30}
\]
Now, only \(\frac{1}{30}\) of the work remains.

On Day 14, it will be Prem's turn to work, and he can complete \(\frac{1}{12}\) of the work, which is more than \(\frac{1}{30}\). So Prem will finish the remaining work on Day 14.

Thus, the total number of days required to complete the work is **14 days**.

Would you like further details on this solution?

Here are 5 related questions for practice:
1. If Amar takes 18 days and Prem takes 15 days to complete a work, how long will it take if they work on alternate days, starting with Prem?
2. If both Amar and Prem work together every day, in how many days will the work be completed?
3. What if Amar starts on the first day and Prem only works every third day, alternating with Amar?
4. If Amar works for 2 days consecutively followed by Prem for 1 day, in how many days will they complete the work?
5. How much work would be completed in 10 days if Amar and Prem work on alternate days, starting with Amar?

**Tip**: When solving alternate day work problems, look for patterns in 2-day or multi-day cycles to simplify calculations.

Amar can complete a work in 15 days, and Prem, in 12 days. Starting with Amar, they work on alternate days. In how many days will the work be completed?

Learn how Amar and Prem complete a task in 14 days by working on alternate days. Amar finishes work in 15 days and Prem in 12 days, using key concepts of work and time, rates, and fractions. Suitable for Grades 9-10.

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