Solving the Builder and Destroyer Work Rate Problem

This problem involves calculating the time taken to complete a task (building a wall) when different entities work at different rates, and one entity stops working partway through. We need to understand the concept of work rates and how to combine them.

Understanding Work Rates

In problems like this, we often think about how much of the task can be completed in one unit of time (in this case, one hour).

• Builder's Rate: The builder can build a wall in 10 hours. This means the builder completes $ \frac{1}{10} $ of the wall every hour.
• Destroyer's Rate: The destroyer can demolish (undo the work) a wall in 20 hours. This means the destroyer undoes $ \frac{1}{20} $ of the wall every hour.

Step-by-Step Solution for Time Calculation

Let's break down the problem into steps:

Step 1: Calculate the Combined Work Rate

When the builder and the destroyer work together, the builder is building, and the destroyer is demolishing. So, their rates work against each other. The net rate at which the wall is being built is the builder's rate minus the destroyer's rate.

Combined Rate = Builder's Rate - Destroyer's Rate

Combined Rate = $ \frac{1}{10} - \frac{1}{20} $

To subtract these fractions, we find a common denominator, which is 20:

Combined Rate = $ \frac{2}{20} - \frac{1}{20} = \frac{1}{20} $ of the wall per hour.

This means that when they work together, $ \frac{1}{20} $ of the wall is built each hour.

Step 2: Calculate Work Done in the First 15 Hours

Both the builder and the destroyer worked together for the first 15 hours. We can calculate how much of the wall was built during this time.

Work Done = Combined Rate $ \times $ Time Worked Together

Work Done = $ \frac{1}{20} \text{ wall/hour} \times 15 \text{ hours} $

Work Done = $ \frac{15}{20} = \frac{3}{4} $ of the wall.

So, after 15 hours, $ \frac{3}{4} $ of the wall was built.

Step 3: Calculate the Remaining Work

The total work required is to build 1 complete wall. We need to find out how much work is left to be done.

Remaining Work = Total Work - Work Done

Remaining Work = $ 1 - \frac{3}{4} $

Remaining Work = $ \frac{1}{4} $ of the wall.

Step 4: Calculate Time Taken by the Builder Alone for Remaining Work

After 15 hours, the destroyer was withdrawn. The builder continued alone to finish the remaining $ \frac{1}{4} $ of the wall. We use the builder's individual rate for this calculation.

Time = Work / Rate

Time for Remaining Work = Remaining Work / Builder's Rate

Time for Remaining Work = $ \frac{1/4}{1/10} $ hours

Time for Remaining Work = $ \frac{1}{4} \times \frac{10}{1} = \frac{10}{4} = 2.5 $ hours.

Step 5: Calculate the Total Time Taken

The total time taken to build the wall is the sum of the time they worked together and the time the builder worked alone to finish the job.

Total Time = Time Worked Together + Time for Remaining Work

Total Time = $ 15 \text{ hours} + 2.5 \text{ hours} $

Total Time = $ 17.5 $ hours.

Final Answer

The total time taken to build the wall was 17.5 hours.