Analyzing the Builder and Destroyer Rates

This problem involves calculating the time taken to complete a task (building a wall) where work is done by a builder and simultaneously undone by a destroyer.

• The builder's rate of work is the amount of wall they can build per hour. Since the builder builds one wall in 10 hours, their rate is: $ \frac{1}{10} \text{ wall/hour} $
• The destroyer's rate of work is the amount of wall they can demolish per hour. Since the destroyer demolishes one wall in 20 hours, their rate is: $ \frac{1}{20} \text{ wall/hour} $ This is considered negative work concerning the building process.

Calculating Work Done Together

For the first 15 hours, both the builder and the destroyer work together. To find the net rate of work during this period, we subtract the destroyer's rate from the builder'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} \text{ wall/hour} $

This means that while both were working, the effective progress towards building the wall was $ \frac{1}{20} $ of the wall per hour.

Now, let's calculate the amount of the wall built in the first 15 hours:

• Work Done = Combined Rate × Time
• Work Done = $ \frac{1}{20} \text{ wall/hour} \times 15 \text{ hours} $
• Work Done = $ \frac{15}{20} = \frac{3}{4} $ of the wall.

Calculating Remaining Work and Time

After 15 hours, the destroyer stops working. The remaining portion of the wall needs to be built solely by the builder.

• Remaining Work = Total Wall - Work Done
• Remaining Work = $ 1 - \frac{3}{4} = \frac{1}{4} $ of the wall.

The builder continues working alone to complete this remaining $ \frac{1}{4} $ of the wall. We use the builder's individual rate to find the time needed:

• Time = Work / Rate
• Time for Builder Alone = Remaining Work / Builder's Rate
• Time for Builder Alone = $ \frac{1/4}{1/10} $ hours
• Time for Builder Alone = $ \frac{1}{4} \times \frac{10}{1} = \frac{10}{4} = 2.5 $ hours.

Determining 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.

• Total Time = Time Worked Together + Time Builder Worked Alone
• Total Time = 15 hours + 2.5 hours
• Total Time = 17.5 hours.

Therefore, the total time taken to build the wall was 17.5 hours.