If A can complete a project in 3 days, B can complete the same project in 4 days while C can complete it in 5 days. All of them stated the project together but after 3 days C left, after 4 days B left and after 5 days A left, then how many projects they have finished in 5 days?

Options

A. 2
B. 1
C. 3
D. 4

Answer

To solve the problem step by step, we first need to determine the work done by each person (A, B, and C) in a day and then calculate how much work they complete together over the specified time.

### Step 1: Determine the work rate of A, B, and C
- A can complete the project in 3 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1 \text{ project}}{3 \text{ days}} = \frac{1}{3} \text{ projects per day} \]
- B can complete the project in 4 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1 \text{ project}}{4 \text{ days}} = \frac{1}{4} \text{ projects per day} \]
- C can complete the project in 5 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1 \text{ project}}{5 \text{ days}} = \frac{1}{5} \text{ projects per day} \]

### Step 2: Find the combined work rate of A, B, and C
To find the total work rate when all three work together, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{3} + \frac{1}{4} + \frac{1}{5} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 3, 4, and 5 is 60. Thus, we convert each fraction:
\[ \frac{1}{3} = \frac{20}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60} \]
Now we can add them:
\[ \text{Combined work rate} = \frac{20}{60} + \frac{15}{60} + \frac{12}{60} = \frac{47}{60} \text{ projects per day} \]

### Step 3: Calculate the work done in the first 3 days
In the first 3 days, all three work together:
\[ \text{Work done in 3 days} = 3 \times \frac{47}{60} = \frac{141}{60} \text{ projects} \]

### Step 4: Calculate the work done by B and A after C leaves
After 3 days, C leaves. Now A and B continue working together for the next day (Day 4):
\[ \text{Work done by A and B in 1 day} = \frac{1}{3} + \frac{1}{4} = \frac{20}{60} + \frac{15}{60} = \frac{35}{60} \text{ projects} \]
So, after Day 4, the total work done is:
\[ \text{Total work after 4 days} = \frac{141}{60} + \frac{35}{60} = \frac{176}{60} \text{ projects} \]

### Step 5: Calculate the work done by A alone on Day 5
On Day 5, A works alone:
\[ \text{Work done by A in 1 day} = \frac{1}{3} = \frac{20}{60} \text{ projects} \]
Now, adding this to the total work done:
\[ \text{Total work after 5 days} = \frac{176}{60} + \frac{20}{60} = \frac{196}{60} \text{ projects} \]

### Step 6: Calculate the number of complete projects finished
To find out how many complete projects they finished, we divide the total work done by the work required for one project:
\[ \text{Number of projects completed} = \frac{196}{60} \approx 3.27 \]
Since we can only count complete projects, they finished 3 complete projects.

### Final Answer
Thus, the total number of projects they have finished in 5 days is **3**.

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