Work Efficiency Analysis: How Many Days Does A Take to Complete the Work Alone?

Work efficiency problems are commonly encountered in mathematics, particularly in the field of time and work. This article explores a typical problem where the individual work rates of individuals A and B are given, and we need to determine how long A alone would take to complete a task. We will dissect the problem, providing detailed explanations and the step-by-step solution to help you understand and apply the concepts effectively.

Understanding the Problem

The problem states that A can complete a work in 3 days, while B can do the same task in 5 days. We aim to find out how long A can complete the task alone. To solve this, we will analyze the work rates of A and B and use them to determine A's individual work rate and hence the number of days required for A to complete the work alone.

Solving Using Combined Work Rates

The first approach involves using combined work rates to find the individual rate:

Step 1: Determine Combined Work Ratios

The work rates for A and B are given as follows:

A can do the work in 3 days, so A's work rate is 1/3. B can do the work in 5 days, so B's work rate is 1/5.

Together, they can complete the work in 20/(1/3 1/5) days.

Step 2: Utilizing Combined Work and Individual Work Rates

We can use the formula for combined work rates to verify:

Let A alone take x days.

In 3 days, A does 3/8 of the work, and in 5 days, B does 3/8 of the work.

Thus, together they do 3/8 3/8 6/8 3/4 of the work in 8 days.

Hence, A alone takes 8/3 * 5 40/3 13.33 days.

Using Simultaneous Equations and Efficiencies

Another approach involves using the ratio of time taken to find the individual efficiencies.

Step 1: Define the Variables

Let the one-day work of A, B, and C be represented by a, b, and c respectively. We know that:

3a 5b, implying b (3a)/5. 2b 3c, implying c (2b)/3.

All together can complete a certain piece of work in 20 days.

Step 2: Establish the Combined Rate

The combined rate can be expressed as:

20abc 1.

Substitute the values of b and c in terms of a:

20a * (3a/5) * (2b/3) 1.

20a * (3a/5) * (2 * 3a/5 * 3/2) 1.

Solving for a, we get:

a 1/40.

The part of work done by A in one day is 1/40, meaning A can complete the work in 40 days.

Conclusion and Final Answer

In conclusion, both methods confirm that A alone can complete the work in 40 days. This confirms that the correct answer is 40 days.

Related Keywords

work efficiency time and work problems work and time ratios

This comprehensive analysis and solution will help you tackle similar time and work problems effectively, understanding the underlying principles and methods.