Understanding the Project Work Problem
This problem involves calculating the total time needed to finish a project based on the individual work rates of three people: A, B, and C. The scenario describes a sequence where A and B start together, A leaves, and then C joins B to complete the remaining work.
Calculating Individual Work Rates
To solve this, we first find out how much of the project each person can complete in a single day. This is their 'work rate'.
- A completes the project in 10 days, so A's rate is $ \frac{1}{10} $ of the project per day.
- B completes the project in 15 days, so B's rate is $ \frac{1}{15} $ of the project per day.
- C completes the project in 20 days, so C's rate is $ \frac{1}{20} $ of the project per day.
Step-by-Step Solution: Project Phases
Phase 1: A and B Working Together
A and B begin the project and work side-by-side for the first 3 days.
- First, we find their combined work rate when they work together:
- $ \text{Combined Rate}_{A+B} = \text{Rate of A} + \text{Rate of B} = \frac{1}{10} + \frac{1}{15} $
- To add these, we find a common denominator, which is 30:
- $ \text{Combined Rate}_{A+B} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6} $ project per day.
- Next, we calculate the amount of work they complete in these 3 days:
- Work Done = $ \text{Combined Rate}_{A+B} \times \text{Time} $
- Work Done = $ \frac{1}{6} \times 3 = \frac{3}{6} = \frac{1}{2} $ of the project.
Phase 2: B and C Working Together
After 3 days, A leaves. The rest of the project work is completed by B and C working together.
- We first determine the amount of work remaining:
- Remaining Work = Total Work - Work Done in Phase 1
- Remaining Work = $ 1 - \frac{1}{2} = \frac{1}{2} $ of the project.
- Now, we find the combined work rate of B and C:
- $ \text{Combined Rate}_{B+C} = \text{Rate of B} + \text{Rate of C} = \frac{1}{15} + \frac{1}{20} $
- The common denominator for 15 and 20 is 60:
- $ \text{Combined Rate}_{B+C} = \frac{4}{60} + \frac{3}{60} = \frac{7}{60} $ project per day.
- Finally, we calculate the time B and C need to finish the remaining work:
- Time = $ \frac{\text{Remaining Work}}{\text{Combined Rate}_{B+C}} $
- Time = $ \frac{1/2}{7/60} = \frac{1}{2} \times \frac{60}{7} = \frac{60}{14} = \frac{30}{7} $ days.
Calculating Total Project Completion Time
The total time to complete the project is the sum of the durations of Phase 1 and Phase 2.
- Total Time = Time in Phase 1 + Time in Phase 2
- Total Time = 3 days + $ \frac{30}{7} $ days
- To add these, we express 3 days as a fraction with a denominator of 7: $ 3 = \frac{21}{7} $.
- Total Time = $ \frac{21}{7} + \frac{30}{7} = \frac{51}{7} $ days.
- Converting the improper fraction $ \frac{51}{7} $ into a mixed number gives us $ 7 \frac{2}{7} $ days.
Thus, the entire project work will be completed in $ 7 \frac{2}{7} $ days.
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?
