How Long Does It Take for A to Complete the Work Alone?

In this article, we'll explore a mathematical problem that involves determining the individual working efficiency of a person based on their combined work rate. We'll break down the problem into smaller steps, understand the underlying principles, and provide a detailed solution. This will help you improve your SEO for related search queries, making this content attractive to Google.

Problem Setup and Definitions

Let's assume A and B together can finish a particular job in 30 days. They start working together for 20 days, after which B leaves, and A continues alone for another 20 days, completing the entire job. Our goal is to determine how long it would take for A to finish the job on their own.

Mathematical Analysis

We'll represent the work rate of A as A and the work rate of B as B. Given that A and B together can complete the work in 30 days, we can express their combined work rate as:

A B 1/30

What this means is that working together, A and B can finish 1/30 of the work in a day.

Work Done Together in 20 Days

Over the first 20 days, A and B together would have completed:

20 * (1/30) 2/3 of the work

This translates to 2/3 of the total work, leaving 1/3 of the work remaining for A to complete alone.

Work Done by A Alone in 20 Days

Let's denote the work rate of A as A. Since A completes the remaining 1/3 of the work in 20 days, we can set up the following equation:

20A 1/3

Solving for A, we get:

A 1/60

This means that A completes 1/60 of the work in a day, or it would take A 60 days to complete the entire work by themselves.

Verification and Alternative Solution

To verify this result, we can set up the initial equation as follows:

16A 44B 1 (since A works for 16 days and B works for 44 days per unit work)

From the combined work rate, we already know that:

A B 1/30

Substituting B from the second equation into the first, we get:

16A 44(1/30 - A) 1

Simplifying this equation, we find that:

16A 44/30 - 44A 1

-28A 1 - 44/30

-28A 30/30 - 44/30 -14/30

A 1/60

Therefore, A alone can complete the work in 60 days.

Conclusion and Application

The process of solving this problem involves understanding combined work rates and how they can be broken down into individual work rates. This concept is crucial for managing projects, scheduling tasks, and understanding team dynamics in various industries, from construction to software development.

Related Keywords

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