3 men A, B and C can complete the work in 10, 12 and 15 days. A, B, C starts work together, A and B left the work 2 days before the comple tion of the work, then the whole work will be finished in how many days?

Options

A. `3(7)/(15)`
B. `5(7)/(15)`
C. 5
D. 7

Answer

To solve the problem, we need to determine how many days it takes for A, B, and C to complete the work together, given that A and B leave 2 days before the work is finished.

### Step-by-Step Solution:

1. **Determine 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. **Calculate the Combined Work Rate of A, B, and C:**
\[ \text{Combined Rate} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 10, 12, and 15 is 60.
- Convert each rate:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)
- Now, add them:
\[ \text{Combined Rate} = \frac{6 + 5 + 4}{60} = \frac{15}{60} = \frac{1}{4} \]
This means A, B, and C together can complete \( \frac{1}{4} \) of the work in one day.

3. **Let the Total Work be 1 Unit:**
- If the total work is 1 unit, then the time taken to complete the work by A, B, and C together is:
\[ \text{Time} = \frac{1 \text{ unit}}{\frac{1}{4} \text{ unit/day}} = 4 \text{ days} \]

4. **Determine the Work Done Before A and B Leave:**
- A and B leave 2 days before the work is completed. Therefore, they work together for \( x - 2 \) days, where \( x \) is the total number of days to complete the work.
- In the first \( x - 2 \) days, A, B, and C work together:
\[ \text{Work done in } (x - 2) \text{ days} = (x - 2) \times \frac{1}{4} \]

5. **Calculate the Remaining Work Done by C:**
- In the last 2 days, only C works:
\[ \text{Work done by C in 2 days} = 2 \times \frac{1}{15} = \frac{2}{15} \]

6. **Set Up the Equation for Total Work:**
- The total work done is equal to 1 unit:
\[ (x - 2) \times \frac{1}{4} + \frac{2}{15} = 1 \]

7. **Solve for x:**
- Multiply the entire equation by 60 (the common denominator) to eliminate fractions:
\[ 15(x - 2) + 8 = 60 \]
\[ 15x - 30 + 8 = 60 \]
\[ 15x - 22 = 60 \]
\[ 15x = 82 \]
\[ x = \frac{82}{15} = 5 \frac{7}{15} \]

### Final Answer:
The whole work will be finished in \( 5 \frac{7}{15} \) days.