Time and Work - Mathematics (SSC CGL CHSL)

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Using Ratio Method
Problems Based on LCM Method

In many problems involving multiple workers (or machines, pipes, etc.) working together, we can use the **LCM method** to solve them efficiently.

What is the LCM Method? The **LCM method** simplifies calculations when we have different workers with different time durations for completing the same work. This method helps us find a common time unit (LCM) in which the workers will all complete the work together.

Key Steps Using the LCM Method:

  1. Step 1: Find the time taken by each worker (or group) to complete the work.
  2. Step 2: Find the **LCM** of these time values. This represents the common time in which all workers can complete the work simultaneously.
  3. Step 3: Calculate the efficiency (rate) of each worker, and sum their efficiencies.
  4. Step 4: Use the formula: Time = Total Work / Combined Efficiency

Example 1: Simple LCM Method

Q: 3 men can complete a work in 12 days, while 4 women can complete it in 15 days. How long will 3 men and 4 women take to complete the same work together?

Solution:

Step 1: Find the time taken by each group.

  • 3 men take 12 days to complete the work. Therefore, 1 man will take 12 × 3 = 36 days.
  • 4 women take 15 days. So, 1 woman will take 15 × 4 = 60 days.

Step 2: Find the LCM of 36 and 60.

The LCM of 36 and 60 is 180.

Step 3: Calculate efficiencies

  • Efficiency of 1 man = 1/36
  • Efficiency of 1 woman = 1/60

Step 4: Find combined efficiency of 3 men and 4 women working together.

  • Efficiency of 3 men = 3 × (1/36) = 1/12
  • Efficiency of 4 women = 4 × (1/60) = 1/15
  • Combined efficiency = 1/12 + 1/15

To add the fractions, find the LCM of 12 and 15. The LCM of 12 and 15 is 60.

Step 5: Combine the fractions

\[ \frac{1}{12} + \frac{1}{15} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \]

Step 6: Calculate total time

\[ \text{Time} = \frac{1}{\text{Combined Efficiency}} = \frac{1}{\frac{3}{20}} = \frac{20}{3} \text{ days} \approx 6.67 \text{ days} \]

Answer: 3 men and 4 women together will complete the work in approximately 6.67 days.

Example 2: More Complex LCM Method with Machines and Workers

Q: A machine can complete a work in 10 days, while a worker can complete the same work in 30 days. How long will it take if the machine and 2 workers work together?

Solution:

Step 1: Find time taken by machine and workers

  • The machine takes 10 days to complete the work.
  • The worker takes 30 days to complete the work.

Step 2: Find the LCM of 10 and 30.

The LCM of 10 and 30 is 30.

Step 3: Calculate efficiencies

  • Efficiency of machine = 1/10
  • Efficiency of 1 worker = 1/30
  • Efficiency of 2 workers = 2 × (1/30) = 2/30 = 1/15

Step 4: Combined efficiency of machine + 2 workers

\[ \text{Combined Efficiency} = \frac{1}{10} + \frac{1}{15} \]

LCM of 10 and 15 is 30, so:

\[ \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} \]

Step 5: Calculate total time

\[ \text{Time} = \frac{1}{\text{Combined Efficiency}} = \frac{1}{\frac{1}{6}} = 6 \text{ days} \]

Answer: The machine and 2 workers together will complete the work in 6 days.

Example 3: Worker Efficiency Varies Over Time

Q: 3 men, 4 women, and 6 children can complete a work in 20 days. After 10 days, 1 man and 1 woman leave. How many more days will it take to finish the work?

Solution:

Step 1: Find total work in man-days

Let’s calculate total work in terms of child-days, assuming the following efficiency ratios: Man : Woman : Child = 6 : 4 : 2 (i.e., 1 man = 6 children, 1 woman = 4 children)

  • 3 men = 3 × 6 = 18 children
  • 4 women = 4 × 4 = 16 children
  • 6 children = 6 children

Total efficiency in terms of children = 18 + 16 + 6 = 40 children/day

Total work = 40 × 20 = 800 child-days

Step 2: Work done in first 10 days

In 10 days, the team completes 40 × 10 = 400 child-days of work. Remaining work = 800 − 400 = 400 child-days.

Step 3: New team efficiency after 1 man and 1 woman leave

  • Remaining team: 2 men, 3 women, and 6 children
  • 2 men = 2 × 6 = 12 children
  • 3 women = 3 × 4 = 12 children
  • 6 children = 6 children

New efficiency = 12 + 12 + 6 = 30 children/day

Step 4: Calculate remaining time

\[ \text{Time} = \frac{400}{30} = \frac{40}{3} \approx 13.33 \text{ days} \]

Answer: It will take approximately 13.33 more days to complete the remaining work.

Key Takeaways

  • LCM helps in solving problems where workers have different time durations for completing the same work.
  • The total work done can be found by calculating the combined efficiency of all workers.
  • The formula for work is Time = Total Work / Combined Efficiency.
  • The LCM of worker times gives the common period during which work will be completed.