Solving Time & Work Problems with Multiple Entities

1. The Core Concept: The LCM Method

The fundamental relationship in this topic is:

From this, we can derive that Rate = Work / Time.

When multiple people work together, their individual rates add up. For example, if Person A works at a rate of $R_A$ and Person B at a rate of $R_B$, their combined rate is $R_A+R_B$.

The traditional method involves working with fractions (e.g., if a person takes 10 days, their rate is 1/10 of the job per day). This can be slow and prone to errors. Instead, we will use a more intuitive approach called the LCM Method.

Here is the process:

1. Assume Total Work: Instead of assuming the total work is '1 job', we assume it's a convenient number of 'units'. The best number to choose is the Least Common Multiple (LCM) of all the individual times given in the problem.
2. Calculate Individual Rates: For each worker, calculate their work rate in units/day (or units/hour, etc.) using the formula: Rate = Total Work (in units) / Time.
3. Calculate Combined Rate: Add the individual rates to find the rate at which they work together.
4. Find Total Time: Calculate the time it takes to complete the job together using: Time = Total Work (in units) / Combined Rate.

Let's see how this works in practice.

As you saw, this method converts the problem into simple arithmetic with whole numbers, which is much faster and more reliable under exam pressure.

2. Application: Pipes and Cisterns

"Pipes and Cisterns" problems are a classic variation of Time & Work. The logic is identical, with one small twist:

• Inlet pipes fill a tank, so they do positive work.
• Outlet/Drainage pipes empty a tank, so they do negative work.

When calculating the combined rate, you simply add the rates of inlet pipes and subtract the rates of outlet pipes. The 'Work' is the capacity of the cistern or tank.

Let's see how the LCM method is applied here.

The image below shows another example of this concept, visually breaking down the steps of the LCM method for a typical pipes and cisterns problem.

3. A Common Variation: Workers Joining or Leaving

CAT problems often add a layer of complexity where workers do not work for the entire duration. They might start together and one leaves midway, or one might leave a few days before the work is completed.

The LCM method is still our foundation. The strategy is to set up an equation based on the total work. Let the total time taken be $T$ days.

For each worker, their contribution to the total work is (Their Rate) × (Time they actually worked). The sum of all contributions must equal the total work (in units).

Let's examine a problem that uses this approach.

This algebraic approach is very powerful. By defining the total time as a variable ($T$ or $X$), you can express the time each person worked in terms of $T$ and solve for it, even when the situation is complex.

Conclusion

Today you've learned to master the fundamental Time & Work problems using a systematic and efficient method. This approach leverages the same logic as our previous lessons on Distance = Speed × Time, reinforcing the core concept of rate-based calculations.

Key Takeaways:

• The core formula is Work = Rate × Time.
• The LCM Method is a powerful strategy to avoid fractions and simplify calculations. You assume the total work is the LCM of the individual times.
• The combined rate of workers is the sum of their individual rates.
• Pipes and Cisterns problems are a direct application, where outlet pipes have a negative work rate.
• For problems where workers join or leave, you can set up an equation where the sum of individual work contributions equals the total work.

In our next lesson, we will delve deeper into more complex Time & Work scenarios, including those with workers of different efficiencies, alternate working days, and other variations frequently seen in the CAT exam. The foundation you've built today is the key to mastering them.