A can finish `3/5` of a task in 6 days and B can finish `2/3` of the same task in 12 days. A and B worked together for 5 days. C alone completed the remaining task in 8 days. If B and C working together will complete the same task in:

Options

A. 10 days
B. 18 days
C. 12 days
D. 15 days

Answer

To solve the problem step by step, let's break down the information provided and calculate the required values.

### Step 1: Determine the work rate of A
A can finish \( \frac{3}{5} \) of the task in 6 days.
To find out how much of the task A can complete in one day, we calculate:

\[ \text{Work rate of A} = \frac{\frac{3}{5}}{6} = \frac{3}{30} = \frac{1}{10} \]

This means A can complete \( \frac{1}{10} \) of the task in one day.

### Step 2: Determine the work rate of B
B can finish \( \frac{2}{3} \) of the task in 12 days.
To find out how much of the task B can complete in one day, we calculate:

\[ \text{Work rate of B} = \frac{\frac{2}{3}}{12} = \frac{2}{36} = \frac{1}{18} \]

This means B can complete \( \frac{1}{18} \) of the task in one day.

### Step 3: Calculate the combined work rate of A and B
Now, we add the work rates of A and B to find their combined work rate:

\[ \text{Combined work rate of A and B} = \frac{1}{10} + \frac{1}{18} \]

To add these fractions, we need a common denominator. The least common multiple of 10 and 18 is 90.

\[ \frac{1}{10} = \frac{9}{90}, \quad \frac{1}{18} = \frac{5}{90} \]

Now, adding these:

\[ \text{Combined work rate} = \frac{9}{90} + \frac{5}{90} = \frac{14}{90} = \frac{7}{45} \]

This means A and B together can complete \( \frac{7}{45} \) of the task in one day.

### Step 4: Calculate the work done by A and B in 5 days
Now, we calculate how much work A and B can complete together in 5 days:

\[ \text{Work done in 5 days} = 5 \times \frac{7}{45} = \frac{35}{45} = \frac{7}{9} \]

### Step 5: Calculate the remaining work
The total work is 1 (the whole task), so the remaining work after A and B have worked for 5 days is:

\[ \text{Remaining work} = 1 - \frac{7}{9} = \frac{2}{9} \]

### Step 6: Determine the work rate of C
C completes the remaining work in 8 days. Therefore, C's work rate is:

\[ \text{Work rate of C} = \frac{\frac{2}{9}}{8} = \frac{2}{72} = \frac{1}{36} \]

This means C can complete \( \frac{1}{36} \) of the task in one day.

### Step 7: Calculate the combined work rate of B and C
Now, we find the combined work rate of B and C:

\[ \text{Combined work rate of B and C} = \frac{1}{18} + \frac{1}{36} \]

Finding a common denominator (which is 36):

\[ \frac{1}{18} = \frac{2}{36} \]

Now, adding these:

\[ \text{Combined work rate} = \frac{2}{36} + \frac{1}{36} = \frac{3}{36} = \frac{1}{12} \]

### Step 8: Calculate the time taken by B and C to complete the task
If B and C together can complete \( \frac{1}{12} \) of the task in one day, then the time taken to complete the whole task (1 unit) is:

\[ \text{Time} = \frac{1}{\frac{1}{12}} = 12 \text{ days} \]

### Final Answer
B and C working together will complete the same task in **12 days**.