Efficiency and Time Calculation in Linear Work Problems

Introduction

In the context of linear work problems, the efficiency and time required to complete a task are two fundamental concepts. These types of problems often involve determining the time taken by an individual or a group to perform a task based on their relative efficiencies. In this article, we will explore these concepts in detail through a series of examples and explain how to solve such problems step-by-step.

Example 1: A's vs B's Work Efficiency

In the first example, we are given that A can complete a work in 20 days, and A's efficiency is 60% of B's efficiency. We need to calculate the time taken by B to complete the same work.

First, let's calculate A's work rate:

A can do the work in 20 days rarr; A's work rate 1/20 of the work per day.

We know that A's efficiency is 60% of B's efficiency. Therefore:

B's work rate (100/60) * (1/20) 1/12

To find out the time taken by B to complete the work:

B's work rate * Time 1 full work rarr; (1/12) * Time 1

Time 12 days

Example 2: A and B's Work Efficiency

In the second example, A can complete the work in 8 days and does half the work compared to B. We will use this information to find the time taken by B alone.

A's work rate is given as:

A's work rate 1/8

B's work rate based on the relation given:

B's work rate 2 * A's work rate 2 * (1/8) 1/4

To find the time taken by B to complete the work:

B's work rate * Time 1 full work rarr; (1/4) * Time 1

Time 4 days

Example 3: A's vs B's Work Efficiency

The third example involves a similar setup but with different numbers. A can complete a work in 20 days, and A's efficiency is 60% of B's efficiency. We will find out the time taken by B to complete the work using this information.

We know that A's work rate is:

A's work rate 1/20

Rewriting the given relation to find B's work rate:

A's work rate 60% of B's work rate rarr; A's work rate (60/100) * B's work rate

Solving for B's work rate:

B's work rate (100/60) * (1/20) 1/12

To find the time taken by B to complete the work:

B's work rate * Time 1 full work rarr; (1/12) * Time 1

Time 12 days

Conclusion

Understanding the relationship between work, time, and efficiency is crucial in solving linear work problems. By using the given data and applying basic algebraic principles, we can determine the time taken by an individual to complete a task. The key is to correctly interpret the given efficiency and relate it to the work rates.

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