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.