A can finish `(2)/(5)` of a task in 12 days and B can finish `(2)/(3)` of the same task in 30 days. They work together for 10 days. C alone completes the remaining task in 16 days. B and C together will complete `(4)/(5)` of the same task in:

Options

A. 16 days
B. 12 days
C. 20 days
D. 15 days

Answer

To solve the problem step by step, we will first determine the work done by A, B, and C individually, then find out how much work they do together, and finally calculate how long it will take for B and C to complete \( \frac{4}{5} \) of the task.

### Step 1: Calculate A's one-day work
A can finish \( \frac{2}{5} \) of the task in 12 days.

\[ \text{A's one-day work} = \frac{2}{5} \div 12 = \frac{2}{5} \times \frac{1}{12} = \frac{2}{60} = \frac{1}{30} \]

### Step 2: Calculate B's one-day work
B can finish \( \frac{2}{3} \) of the task in 30 days.

\[ \text{B's one-day work} = \frac{2}{3} \div 30 = \frac{2}{3} \times \frac{1}{30} = \frac{2}{90} = \frac{1}{45} \]

### Step 3: Calculate the combined one-day work of A and B
Now, we add A's and B's one-day work:

\[ \text{A's one-day work} + \text{B's one-day work} = \frac{1}{30} + \frac{1}{45} \]

To add these fractions, we need a common denominator, which is 90:

\[ \frac{1}{30} = \frac{3}{90}, \quad \frac{1}{45} = \frac{2}{90} \]
\[ \text{Combined work of A and B} = \frac{3}{90} + \frac{2}{90} = \frac{5}{90} = \frac{1}{18} \]

### Step 4: Calculate the work done by A and B in 10 days
In 10 days, A and B together will complete:

\[ \text{Work done in 10 days} = 10 \times \frac{1}{18} = \frac{10}{18} = \frac{5}{9} \]

### Step 5: Calculate the remaining work
The total work is considered as 1 (the whole task). Therefore, the remaining work after A and B work for 10 days is:

\[ \text{Remaining work} = 1 - \frac{5}{9} = \frac{4}{9} \]

### Step 6: Calculate C's one-day work
C can complete the remaining \( \frac{4}{9} \) of the task in 16 days. Thus, C's one-day work is:

\[ \text{C's one-day work} = \frac{4}{9} \div 16 = \frac{4}{9} \times \frac{1}{16} = \frac{4}{144} = \frac{1}{36} \]

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

\[ \text{B's one-day work} + \text{C's one-day work} = \frac{1}{45} + \frac{1}{36} \]

The common denominator for 45 and 36 is 180:

\[ \frac{1}{45} = \frac{4}{180}, \quad \frac{1}{36} = \frac{5}{180} \]
\[ \text{Combined work of B and C} = \frac{4}{180} + \frac{5}{180} = \frac{9}{180} = \frac{1}{20} \]

### Step 8: Calculate the time taken by B and C to complete \( \frac{4}{5} \) of the task
To find out how long it will take for B and C to complete \( \frac{4}{5} \) of the task:

\[ \text{Time} = \text{Work} \div \text{Rate} = \frac{4}{5} \div \frac{1}{20} \]

This is equivalent to:

\[ \text{Time} = \frac{4}{5} \times 20 = 16 \text{ days} \]

### Final Answer
B and C together will complete \( \frac{4}{5} \) of the task in **16 days**.

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