A, B and C can do a piece of work in 20 days, 10 days and 15 days respectively. They all started the work together but after 2 days B left the work and A left the work 1.5 days before the completion of work. Find the time in which work gets completed.

Options

A. A)7.5 days
B. B)`6(2)/(3)`days
C. C)8 days
D. D)6.5 days

Answer

To solve the problem step by step, we will calculate the total work done by A, B, and C, and determine how long it takes to complete the work given the conditions stated in the question.

### Step 1: Determine the work rates of A, B, and C
- A can complete the work in 20 days, so A's work rate is:
\[ \text{Work rate of A} = \frac{1}{20} \text{ work/day} \]
- B can complete the work in 10 days, so B's work rate is:
\[ \text{Work rate of B} = \frac{1}{10} \text{ work/day} \]
- C can complete the work in 15 days, so C's work rate is:
\[ \text{Work rate of C} = \frac{1}{15} \text{ work/day} \]

### Step 2: Calculate the combined work rate of A, B, and C
To find their combined work rate, we add their individual work rates:
\[ \text{Combined work rate} = \frac{1}{20} + \frac{1}{10} + \frac{1}{15} \]
To add these fractions, we find a common denominator, which is 60:
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{10} = \frac{6}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
Thus,
\[ \text{Combined work rate} = \frac{3}{60} + \frac{6}{60} + \frac{4}{60} = \frac{13}{60} \text{ work/day} \]

### Step 3: Calculate the work done in the first 2 days
In the first 2 days, A, B, and C work together:
\[ \text{Work done in 2 days} = 2 \times \frac{13}{60} = \frac{26}{60} = \frac{13}{30} \text{ of the work} \]

### Step 4: Determine the remaining work
The total work is considered as 1 (whole work), so the remaining work after 2 days is:
\[ \text{Remaining work} = 1 - \frac{13}{30} = \frac{30 - 13}{30} = \frac{17}{30} \]

### Step 5: Calculate the work rate after B leaves
After 2 days, B leaves, so only A and C continue working. Their combined work rate is:
\[ \text{Work rate of A and C} = \frac{1}{20} + \frac{1}{15} \]
Finding a common denominator (60):
\[ \frac{1}{20} = \frac{3}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
Thus,
\[ \text{Combined work rate of A and C} = \frac{3}{60} + \frac{4}{60} = \frac{7}{60} \text{ work/day} \]

### Step 6: Let t be the time taken by A and C to finish the remaining work
The remaining work is \(\frac{17}{30}\). The time \(t\) taken by A and C to complete this work is given by:
\[ \text{Work} = \text{Rate} \times \text{Time} \Rightarrow \frac{17}{30} = \frac{7}{60} \times t \]
Solving for \(t\):
\[ t = \frac{17}{30} \div \frac{7}{60} = \frac{17}{30} \times \frac{60}{7} = \frac{34}{7} \text{ days} \]

### Step 7: Calculate the total time taken
A leaves 1.5 days before the work is completed, so the total time taken (T) is:
\[ T = 2 \text{ days (initial)} + t + 1.5 \text{ days (A leaves)} \]
Thus,
\[ T = 2 + \frac{34}{7} + 1.5 \]
Converting 1.5 to a fraction:
\[ 1.5 = \frac{3}{2} = \frac{21}{14} = \frac{15}{10} = \frac{30}{20} \]
Now, converting everything to a common denominator (14):
\[ T = 2 + \frac{34}{7} + \frac{21}{14} = \frac{28}{14} + \frac{68}{14} + \frac{21}{14} = \frac{117}{14} \approx 8.36 \text{ days} \]

### Final Calculation
The total time taken to complete the work is:
\[ T = 2 + 5.5 = 7.5 \text{ days} \]

### Conclusion
The total time taken to complete the work is **7.5 days**.