Time and Work - Mathematics (SSC CGL CHSL)

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Work Done by One Person
Introduction to Time and Work

1. What is Work?

In competitive exams like SSC CGL and CHSL, "work" refers to a task or job to be completed. To simplify calculations, we assume the total work = 1 unit.

Let us understand this idea clearly:

If a person completes a work in 5 days, then the part of the work done in 1 day is:

\[ \text{Work done in 1 day} = \frac{1}{5} \]

This concept of one day's work is the foundation of this entire chapter.

2. Fundamental Concepts

  • Work: The task to be completed. Often considered as 1 unit.
  • Time: Duration required to complete the work.
  • Efficiency: Amount of work done in one unit of time.

These three elements are closely related. Let’s see how.

3. Relationship Between Work, Time, and Efficiency

The basic formula connecting these concepts is:

\[ \text{Work} = \text{Efficiency} \times \text{Time} \]

From this, we can derive:

  • \[ \text{Time} = \frac{\text{Work}}{\text{Efficiency}} \]
  • \[ \text{Efficiency} = \frac{\text{Work}}{\text{Time}} \]

These formulas allow us to solve a variety of problems once we know any two of the three quantities.

4. One Day's Work

Let’s understand this step-by-step:

  1. If a person completes a work in x days, then in 1 day, the work done is: \[ \frac{1}{x} \]
  2. If two people can complete a job in x and y days respectively, their combined work in one day is: \[ \frac{1}{x} + \frac{1}{y} \]

This principle will be very helpful when we deal with questions involving two or more people working together.

5. Inverse Relationship Between Time and Efficiency

Time and efficiency are inversely proportional. This means:

\[ \text{Efficiency} \propto \frac{1}{\text{Time}} \]

In simpler words, if a person is more efficient, they will take less time to complete the same work.

Let’s understand this through an example:

Suppose A can do a work in 6 days, and B can do the same work in 12 days.

  • A’s 1-day work = \( \frac{1}{6} \)
  • B’s 1-day work = \( \frac{1}{12} \)

Therefore, A is twice as efficient as B.

6. Important Formulae and Concepts

  • If a person finishes a work in \( x \) days, then one day’s work = \( \frac{1}{x} \)
  • If one day’s work = \( \frac{1}{x} \), then the full work is completed in \( x \) days
  • \( \text{Work} = \text{Efficiency} \times \text{Time} \)
  • \( \text{Time} = \frac{\text{Work}}{\text{Efficiency}} \)
  • If A is \( k \) times as efficient as B, then A will take \( \frac{1}{k} \) times the time B takes

7. Summary

  • Total work is assumed to be 1 unit for simplicity.
  • Use “work done in 1 day” to solve most problems easily.
  • Efficiency and time are inversely related — more efficient people take less time.
  • Combine one-day work rates to calculate team performance.

In the next part, we will apply these ideas to real problems involving one person doing the work, and we’ll also introduce the popular LCM method used to speed up calculations in exams.