Understanding the Wall Building and Demolishing Problem

This question is a classic example of a work and time problem, specifically involving two entities with opposing actions: a builder constructing a wall and a destroyer demolishing it. We need to figure out the total time taken to build the wall under changing conditions.

Defining the Rates of Work

First, let's determine the rate at which the builder builds and the destroyer demolishes the wall. The rate is typically defined as the amount of work done per unit of time.

• The builder can build a wall in 10 hours. So, the builder's rate is $\frac{1}{\text{10}}$ of the wall per hour.
• The destroyer can demolish the same wall in 14 hours. Since demolishing is the opposite of building, the destroyer's rate is $-\frac{1}{\text{14}}$ of the wall per hour (negative sign indicates work being undone).

Calculating Combined Work Rate

Initially, both the builder and the destroyer work together. To find their combined rate, we add their individual rates:

Combined rate = Builder's rate + Destroyer's rate

Combined rate = $\frac{1}{\text{10}} + \left(-\frac{1}{\text{14}}\right) = \frac{1}{\text{10}} - \frac{1}{\text{14}}$

To subtract these fractions, we find a common denominator, which is the least common multiple (LCM) of 10 and 14. The LCM of 10 and 14 is 70.

Combined rate = $\frac{1 \times 7}{\text{10} \times 7} - \frac{1 \times 5}{\text{14} \times 5} = \frac{7}{\text{70}} - \frac{5}{\text{70}} = \frac{7 - 5}{\text{70}} = \frac{2}{\text{70}} = \frac{1}{\text{35}}$

So, when working together, the net rate of building is $\frac{1}{\text{35}}$ of the wall per hour.

Work Done in the First 7 Hours

Both worked together for the first 7 hours. We can calculate the amount of the wall built during this time using the combined rate:

Work done in 7 hours = Combined rate $\times$ Time

Work done in 7 hours = $\frac{1}{\text{35}}$ wall/hour $\times$ 7 hours = $\frac{7}{\text{35}} = \frac{1}{\text{5}}$ of the wall.

So, after 7 hours, $\frac{1}{5}$ of the wall was built.

Calculating Remaining Work

The total work required is to build 1 complete wall. After 7 hours, $\frac{1}{5}$ of the wall is built. The remaining work is:

Remaining work = Total wall - Work done

Remaining work = $1 - \frac{1}{5} = \frac{5}{\text{5}} - \frac{1}{\text{5}} = \frac{4}{\text{5}}$ of the wall.

Time Taken to Finish Remaining Work

After 7 hours, the destroyer was taken out. Only the builder continues to work on the remaining $\frac{4}{5}$ of the wall. The builder's rate is $\frac{1}{\text{10}}$ of the wall per hour.

Time taken by builder for remaining work = Remaining work / Builder's rate

Time taken = $\frac{4}{\text{5}} \div \frac{1}{\text{10}} = \frac{4}{\text{5}} \times \frac{\text{10}}{1} = \frac{4 \times \text{10}}{5 \times 1} = \frac{\text{40}}{5} = 8$ hours.

The builder took an additional 8 hours to complete the wall.

Total Time to Build the Wall

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 working together + Time builder worked alone

Total time = 7 hours + 8 hours = 15 hours.

Thus, the total time taken to build the wall was 15 hours.

Revision Table: Wall Building Problem

Here's a quick review of the key information used in solving this wall building problem:

Additional Information: Understanding Work and Time Concepts

Work and time problems are common in quantitative aptitude. The core concept is the relationship between work done, the rate of work, and the time taken.

• Work, Rate, and Time: These are related by the formula: Work = Rate $\times$ Time. This formula can be rearranged to find Rate = Work / Time or Time = Work / Rate.
• Rate: The rate is the amount of work done per unit of time. It is often expressed as a fraction (e.g., $\frac{1}{\text{10}}$ of the job per hour). A higher rate means the work is done faster.
• Combined Rates: When multiple individuals or entities work together, their rates are usually added if they are working towards the same goal. If one entity is undoing the work of another (like the destroyer here), their rate is subtracted. If Person A does $\frac{1}{a}$ of the work per unit time and Person B does $\frac{1}{b}$ of the work per unit time, their combined rate is $\frac{1}{a} + \frac{1}{b}$ when working together constructively. If one is destructive, the combined rate becomes $\frac{1}{a} - \frac{1}{b}$.
• Total Work: In many problems, the total work is considered as '1 unit' (like building 1 wall). The fraction of work done is then represented as a fraction of 1.

Solving work and time problems often involves: identifying individual rates, calculating combined rates (if applicable), determining the work done in a specific time period, finding remaining work, and finally calculating the time needed for the remaining work at the relevant rate.