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After 15 hours, the destroyer was withdrawn.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.
Important Questions from Time & Work (Notes)
- A fresh water tap fills a fish tank in 40 minutes. The same tank is filled by a salt water tap in 120 minutes. If both the taps are open, how many minutes will it take to fill the tank?
- A completes $\frac{7}{10}$ of a work in 15 days and then he completes the remaining work with the help of B in 5 days. In how many days can A and B together complete the entire work?
Aman can do 50% of the job in 16 days, and Bhanu can do 25% of the job in 24 days. In how many days can they do $\frac{1}{4}^{th}$ of the job working together ?
- X can finish a job in $141$ days. He worked for $57$ days alone and the remaining work was completed by Y, in $84$ days. How many days would both together take to complete the entire job?
Ravina, Sujata, and Saroj can complete a work of painting separately in 32, 48, and 64 hours, respectively. They started working together, but Saroj left after 5 hours. From the 6th hour, Ravina and Sujata decided to work on alternate hours starting with Ravina. In how much time will the entire work of painting be completed?
