Understanding Builder and Destroyer Work Rates

This problem involves calculating the time taken to complete a task (building a wall) based on the rates of work of different entities. In this scenario, we have a builder who constructs the wall and a destroyer who demolishes it.

• The builder can build a wall in 10 hours. This means the builder's rate of work is $\frac{1}{10}$ of the wall per hour.
• The destroyer can demolish the same wall in 20 hours. This means the destroyer's rate of work is $\frac{1}{20}$ of the wall per hour. Since demolition is the opposite of building, we consider this a negative rate when working together.

Calculating Combined Work Rate

Initially, both the builder and the destroyer work together for 15 hours. To find their combined effect, we subtract the destroyer's rate from the builder's rate:

Builder's Rate ($R_B$) = $\frac{1}{10}$ wall/hour

Destroyer's Rate ($R_D$) = $\frac{1}{20}$ wall/hour

Combined Rate ($R_{Combined}$) = $R_B - R_D$

Using LaTeX for the calculation:

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

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

$R_{Combined} = \frac{2}{20} - \frac{1}{20} = \frac{1}{20}$

So, when working together, they build $\frac{1}{20}$ of the wall every hour.

Work Done in the First 15 Hours

They work together for 15 hours. The amount of wall built during this time is:

Work Done = Combined Rate $\times$ Time

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

$ \text{Work Done} = \frac{15}{20} = \frac{3}{4} $

After 15 hours, $\frac{3}{4}$ of the wall is built.

Remaining Work and Builder's Solo Time

After 15 hours, the destroyer is withdrawn. The remaining portion of the wall that needs to be built is:

Remaining Work = Total Work - Work Done

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

$ \text{Remaining Work} = \frac{1}{4} $

Now, only the builder works to complete the remaining $\frac{1}{4}$ of the wall. The builder's rate is $\frac{1}{10}$ wall per hour.

Time taken by builder = $\frac{\text{Remaining Work}}{\text{Builder's Rate}}$

Time taken by builder = $\frac{1/4}{1/10}$ hours

$ \text{Time taken by builder} = \frac{1}{4} \times \frac{10}{1} = \frac{10}{4} = 2.5 \text{ hours} $

Total Time Calculation

The total time taken to build the wall is the sum of the time they worked together and the time the builder took to finish the remaining part.

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 is 17.5 hours.