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After 15 hours, the destroyer is withdrawn.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.
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?
