Surbhi can do a piece of work in 24 days. She completed 3/8 of the work and then left it. Amit can complete the remaining work in 10 days. Working together, they will complete 125% of the same work in:
12 days
Solving the Work and Time Problem: Surbhi and Amit
This problem involves calculating the time taken by individuals to complete a piece of work and then finding the time they take to complete a percentage of the work when working together. We will break down the problem step-by-step.
Understanding the Work Done Individually
- Surbhi's Work Capacity: Surbhi can do the entire work (100%) in 24 days. This means her work rate is $\frac{1}{24}$ of the work per day.
- Surbhi's Partial Work: Surbhi completed $\frac{3}{8}$ of the work.
- Remaining Work: The total work is considered as 1 unit. After Surbhi completed $\frac{3}{8}$, the remaining work is $1 - \frac{3}{8} = \frac{8}{8} - \frac{3}{8} = \frac{5}{8}$ of the work.
- Amit's Work Capacity: Amit completed this remaining $\frac{5}{8}$ of the work in 10 days.
Calculating Individual Work Rates
To find the time they take working together, we first need to know their individual daily work rates.
- Surbhi's Daily Work Rate: Surbhi does $\frac{1}{24}$ of the work in 1 day.
- Amit's Daily Work Rate: Amit does $\frac{5}{8}$ of the work in 10 days. To find his rate per day, we divide the work done by the number of days: $$ \text{Amit's Rate} = \frac{\text{Work Done}}{\text{Time Taken}} = \frac{\frac{5}{8}}{10} = \frac{5}{8 \times 10} = \frac{5}{80} = \frac{1}{16} $$ So, Amit does $\frac{1}{16}$ of the work in 1 day.
Calculating Combined Work Rate
When Surbhi and Amit work together, their daily work rates add up.
- Combined Daily Work Rate: $$ \text{Combined Rate} = \text{Surbhi's Rate} + \text{Amit's Rate} = \frac{1}{24} + \frac{1}{16} $$ To add these fractions, we find a common denominator. The least common multiple (LCM) of 24 and 16 is 48. $$ \text{Combined Rate} = \frac{1 \times 2}{24 \times 2} + \frac{1 \times 3}{16 \times 3} = \frac{2}{48} + \frac{3}{48} = \frac{2+3}{48} = \frac{5}{48} $$ Together, Surbhi and Amit complete $\frac{5}{48}$ of the work in 1 day.
Time to Complete the Work (100%) Together
The time taken to complete the entire work (1 unit) is the reciprocal of the combined daily work rate.
- Time for 100% Work: $$ \text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{5}{48}} = \frac{48}{5} \text{ days} $$ So, Surbhi and Amit together can complete the entire work in $\frac{48}{5}$ days, which is 9.6 days.
Time to Complete 125% of the Work Together
The question asks for the time taken to complete 125% of the same work. 125% of the work is equivalent to $1.25$ times the original work.
- Work Amount: 125% of the work is $1.25 \times 1 = \frac{125}{100} = \frac{5}{4}$ units of work.
- Time for 125% Work: Since they complete $\frac{5}{48}$ of the work per day, the time taken to complete $\frac{5}{4}$ of the work is: $$ \text{Time} = \frac{\text{Amount of Work}}{\text{Combined Rate}} = \frac{\frac{5}{4}}{\frac{5}{48}} = \frac{5}{4} \times \frac{48}{5} $$ We can cancel out the 5s: $$ \text{Time} = \frac{1}{4} \times \frac{48}{1} = \frac{48}{4} = 12 \text{ days} $$
Therefore, working together, Surbhi and Amit will complete 125% of the same work in 12 days.
Revision Table: Key Calculations
| Item | Calculation | Result |
|---|---|---|
| Surbhi's Daily Rate | $\frac{1}{\text{Total Days}}$ | $\frac{1}{24}$ work/day |
| Remaining Work | $1 - \frac{3}{8}$ | $\frac{5}{8}$ work |
| Amit's Daily Rate | $\frac{\text{Remaining Work}}{\text{Time Taken}}$ | $\frac{5/8}{10} = \frac{1}{16}$ work/day |
| Combined Daily Rate | Surbhi's Rate + Amit's Rate | $\frac{1}{24} + \frac{1}{16} = \frac{5}{48}$ work/day |
| Time for 100% Work | $\frac{1}{\text{Combined Rate}}$ | $\frac{48}{5}$ days |
| Time for 125% Work | Time for 100% Work $\times 1.25$ | $\frac{48}{5} \times \frac{5}{4} = 12$ days |
Additional Information: Work and Time Concepts
Work and time problems often rely on the concept of work rate. Understanding this is crucial.
- Work Rate: The amount of work done per unit of time. If a person can complete a total work in $T$ days, their daily work rate is $\frac{1}{T}$.
- Total Work: Often considered as 1 unit or equivalent to the LCM of the times taken by individuals if multiple people are involved.
- Work Done: Work Rate $\times$ Time Taken.
- Time Taken: $\frac{\text{Work Done}}{\text{Work Rate}}$.
- Combined Rate: When multiple people work together, their individual work rates add up. If person A has rate $R_A$ and person B has rate $R_B$, their combined rate is $R_A + R_B$.
- Percentage of Work: To calculate the time for a percentage of work (e.g., X%), you can calculate the time for 100% work and then multiply it by X/100. Alternatively, you can calculate the required amount of work (X/100 of total work) and divide it by the combined rate.
Important Questions from Time and Work
Rama and Hari can together finish a piece of work in 15 day. Rama works twice as fast as Hari, then Hari alone can finish work in :
Anil, Deepak and Dinesh together can complete a work in 35 days. Anil and Dinesh together can complete the same work in 60 days. In how many days Deepak alone can complete the same work?
Anu is four times as good as Binni in completing a task. Together they finish the same task in 7 hours. In how many hours will Anu alone complete the task?
P, Q and R can complete a work in 10 days, 20 days and 30 days, respectively, working alone. How soon can the work be completed if P is assisted by Q and R on alternate days?
A can complete a project in 10 days and B can complete the same project in 15 days. A and B start working on the project together and A quit 5 days before the project is completed. In how many days is the project completed?
