Solving Work Rate Problems with Detailed Steps and Analysis

Work rate problems are a common type of mathematical problem that can be found in various contexts, such as time management, project planning, and even general day-to-day scenarios. The problem discussed here involves two individuals, A and B, who work together on a piece of work. This article will provide a step-by-step solution to determine how long it takes to complete the work under specific conditions.

Problem Description

Two individuals, A and B, are capable of completing a piece of work in 10 days and 20 days, respectively. They begin working together but A leaves 2 days before the completion of the work. The task is to determine how many days it will take to complete the work under these conditions.

Analysis and Solution

Let's analyze this step by step:

Simplified Solution Method

In 20 days, both A and B would have completed 3 works (since A in 10 days completes 1 work, B in 20 days completes 1 work, and together they complete 3 works in 20 days).

Hence, 1 work is completed in 20/3 days.

Now, let's calculate how much work is done in the last 2 days.

Work completed in 2 days 2 × (3/20) 3/10

The remaining work to be completed is:

Balance work 1 - 3/10 7/10

This 7/10 of the work is completed by B alone. Since B completes 1 work in 20 days, the time taken by B to complete 7/10 of the work is:

(7/10) × 20 14 days

Therefore, the total time taken to complete the work is:

14 2 16 days

Detailed Mathematical Approach

Let's denote the work rates of A and B as follows:

A's one day work 1/10 B's one day work 1/20 Combined one day work (1/10 1/20) 3/20

B works alone for 2 days before A leaves, so B's work done in 2 days is:

B's 2 days work 2 × (1/20) 1/10

The remaining work is:

Remaining work 1 - 1/10 9/10

This remaining 9/10 of the work is completed by both working together. Since they complete 3/20 of the work in one day, the time taken to complete 9/10 of the work is:

(9/10) / (3/20) (9/10) × (20/3) 6 days

Hence, the total time taken to complete the work is:

6 2 8 days

Taking It a Step Further

Consider another scenario where A, B, and C need to work together to complete a piece of work. Let their individual work rates be 1/10, 1/12, and 1/15 respectively. If A leaves 4 days before the completion of the work, we need to find out the total number of days the work will be completed.

First, calculate the combined work rate of A, B, and C:

Combined work rate 1/10 1/12 1/15

Find a common denominator (60) to simplify the combined work rate:

Combined work rate (6/60 5/60 4/60) 15/60 1/4

Let t be the total number of days they work together before A leaves. Since A leaves 4 days before the completion, B and C work together for t and A works with B and C for t-4 days.

Work done by B and C in t days:

(1/12 1/15)t (5/60 4/60)t 9t/60 3t/20

Work done by A, B, and C together for t-4 days:

(1/10 1/12 1/15)(t-4) (6/60 5/60 4/60)(t-4) 15(t-4)/60 (t-4)/4

The total work done is 1, so:

3t/20 (t-4)/4 1

Multiply through by 20 to eliminate the denominator:

3t 5(t-4) 20

Simplify:

3t 5t - 20 20

8t 40

t 5

Hence, the total number of days the work is completed is:

5 4 9 days

Conclusion

This problem solving method is quite versatile and can be applied to various similar work rate problems. Understanding the combined work rate and the individual work rates of individuals helps in solving complex scenarios effectively.