Math Problem Statement

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.

Solution

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 116\frac{1}{16} of the work per day.
  • Suresh's work rate is 112\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 116\frac{1}{16} of the work.
  • Day 2: Suresh works 112\frac{1}{12} of the work.

The total work done in two days is: 116+112\frac{1}{16} + \frac{1}{12}

To combine these rates, we need a common denominator: 116=348\frac{1}{16} = \frac{3}{48} 112=448\frac{1}{12} = \frac{4}{48}

Adding these fractions: 348+448=748\frac{3}{48} + \frac{4}{48} = \frac{7}{48}

So, in two days, they complete 748\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 748\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 748\frac{7}{48}: Number of cycles=4876.857\text{Number of cycles} = \frac{48}{7} \approx 6.857

This means 6 full cycles will complete: 6×748=4248=786 \times \frac{7}{48} = \frac{42}{48} = \frac{7}{8}

After 6 cycles (12 days), 78\frac{7}{8} of the work is done. The remaining work is: 178=181 - \frac{7}{8} = \frac{1}{8}

On the 13th day, Mahesh will work and he completes 116\frac{1}{16} of the work per day. Therefore, to finish the remaining 18\frac{1}{8} of the work:

18÷116=2\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 days (6 cycles)+1 day (13th day)+1 day (14th day)=14 days12 \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.

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Math Problem Analysis

Mathematical Concepts

Work and Time Problems
Alternate Work
Fractional Work Rates

Formulas

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Theorems

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Suitable Grade Level

High School