A, B and C can do a work in 10, 12 and 15 days respectively. All three start the work together but A left 5 days before the completion of the work and after 2 days B left, how much time shall be taken to do whole work?

Answer

To solve the problem, we need to determine how long it will take A, B, and C to complete the work together, considering the conditions given in the question.

### Step-by-Step Solution:

1. **Calculate the Work Rates of A, B, and C**:
- A can complete the work in 10 days, so A's work rate is \( \frac{1}{10} \) of the work per day.
- B can complete the work in 12 days, so B's work rate is \( \frac{1}{12} \) of the work per day.
- C can complete the work in 15 days, so C's work rate is \( \frac{1}{15} \) of the work per day.

2. **Find the Combined Work Rate**:
- To find the combined work rate of A, B, and C working together, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]
- To add these fractions, we need a common denominator. The LCM of 10, 12, and 15 is 60.
- Converting each fraction:
\[ \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
- Now, adding them:
\[ \text{Combined work rate} = \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} = \frac{1}{4} \]
- This means together they can complete \( \frac{1}{4} \) of the work in one day.

3. **Determine the Total Work**:
- The total work can be considered as 1 unit of work.

4. **Calculate the Time Taken to Complete the Work**:
- Let \( T \) be the total time taken to complete the work.
- Since A leaves 5 days before completion and B leaves 2 days after A, we can break down the work:
- Let \( T - 5 \) be the time A worked (A works for \( T - 5 \) days).
- B works for \( T - 2 \) days (B works for 2 days longer than A).
- C works for the entire \( T \) days.

5. **Set Up the Equation**:
- The work done by A, B, and C can be expressed as:
\[ \text{Work done by A} = \frac{1}{10}(T - 5) \]
\[ \text{Work done by B} = \frac{1}{12}(T - 2) \]
\[ \text{Work done by C} = \frac{1}{15}(T) \]
- The total work done by A, B, and C together should equal 1:
\[ \frac{1}{10}(T - 5) + \frac{1}{12}(T - 2) + \frac{1}{15}(T) = 1 \]

6. **Solve the Equation**:
- Multiply through by 60 (the LCM of 10, 12, and 15) to eliminate the denominators:
\[ 6(T - 5) + 5(T - 2) + 4T = 60 \]
- Expanding this gives:
\[ 6T - 30 + 5T - 10 + 4T = 60 \]
- Combine like terms:
\[ 15T - 40 = 60 \]
- Add 40 to both sides:
\[ 15T = 100 \]
- Divide by 15:
\[ T = \frac{100}{15} = \frac{20}{3} \approx 6.67 \text{ days} \]

### Final Answer:
The total time taken to complete the work is approximately \( 6.67 \) days.