Mahesh and Suresh can complete a work in 16 days and 12 days, respectively. Starting
with Mahesh, they work on alternate basis. How many days will take to finalise the
complete work?
Ans
1. 


To determine how many days it will take Mahesh and Suresh to complete the work when they alternate working days, we need to find their combined work rate and then determine how many days it will take them together.

Given:
- Mahesh can complete the work in 16 days.
- Suresh can complete the work in 12 days.

First, we find the work rates:
- Mahesh's work rate is \(\frac{1}{16}\) of the work per day.
- Suresh's work rate is \(\frac{1}{12}\) of the work per day.

When they work alternately, let's analyze the work done in two days:
- Day 1: Mahesh works \(\frac{1}{16}\) of the work.
- Day 2: Suresh works \(\frac{1}{12}\) of the work.

The total work done in two days is:
\[ \frac{1}{16} + \frac{1}{12} \]

To combine these rates, we need a common denominator:
\[ \frac{1}{16} = \frac{3}{48} \]
\[ \frac{1}{12} = \frac{4}{48} \]

Adding these fractions:
\[ \frac{3}{48} + \frac{4}{48} = \frac{7}{48} \]

So, in two days, they complete \(\frac{7}{48}\) of the work.

Next, we need to determine how many full cycles (of two days each) are needed to get close to completing the work.

Since one cycle of two days completes \(\frac{7}{48}\) of the work, the number of cycles needed to get close to 1 (the entire work) can be found by dividing 1 by \(\frac{7}{48}\):
\[ \text{Number of cycles} = \frac{48}{7} \approx 6.857 \]

This means 6 full cycles will complete:
\[ 6 \times \frac{7}{48} = \frac{42}{48} = \frac{7}{8} \]

After 6 cycles (12 days), \( \frac{7}{8} \) of the work is done. The remaining work is:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]

On the 13th day, Mahesh will work and he completes \(\frac{1}{16}\) of the work per day. Therefore, to finish the remaining \(\frac{1}{8}\) of the work:

\[ \frac{1}{8} \div \frac{1}{16} = 2 \]

This means Mahesh will need 2 additional days to complete the remaining work. So, the total number of days taken to finalize the complete work is:

\[ 12 \text{ days (6 cycles)} + 1 \text{ day (13th day)} + 1 \text{ day (14th day)} = 14 \text{ days} \]

Thus, the correct answer is:

**14 days**

Would you like further details or have any questions?

Here are 5 questions you might want to ask:
1. How do you calculate work rates for multiple workers?
2. What is the concept of alternate work in shared tasks?
3. Can you provide more examples of work problems involving multiple workers?
4. How do you handle fractional work rates?
5. What are some tips for solving time and work problems efficiently?

Tip: When solving work problems involving alternating tasks, always consider the combined work rate and analyze how many cycles are needed to complete the task.

Learn how Mahesh and Suresh alternate work days to complete a task. Understand the concept of work rates and how to calculate total completion time using fractional work rates. Suitable for high school students studying work and time problems.

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