A can finish 80% of a task in 12 days and B can finish 20% of the same task in 2 days. They started the task together, but B left after 2 days and A continued to work. In how many days was the entire task completed?

Options

A. 17.5
B. 10
C. 15
D. 12

Answer

To solve the problem step by step, we will analyze the work done by A and B, calculate their individual work rates, and determine how long it takes to complete the entire task.

### Step 1: Determine the work rates of A and B
- A can finish 80% of the task in 12 days.
- Therefore, A can complete 100% of the task in:
\[ \text{Total days for A} = \frac{12 \text{ days}}{0.8} = 15 \text{ days} \]
- B can finish 20% of the task in 2 days.
- Therefore, B can complete 100% of the task in:
\[ \text{Total days for B} = \frac{2 \text{ days}}{0.2} = 10 \text{ days} \]

### Step 2: Calculate the work done by A and B in one day
- A's work rate:
\[ \text{Work rate of A} = \frac{1}{15} \text{ of the task per day} \]
- B's work rate:
\[ \text{Work rate of B} = \frac{1}{10} \text{ of the task per day} \]

### Step 3: Calculate the work done together in 2 days
- In 2 days, A and B together will complete:
\[ \text{Work done by A in 2 days} = 2 \times \frac{1}{15} = \frac{2}{15} \]
\[ \text{Work done by B in 2 days} = 2 \times \frac{1}{10} = \frac{2}{10} = \frac{1}{5} \]
- Total work done in 2 days:
\[ \text{Total work done} = \frac{2}{15} + \frac{1}{5} = \frac{2}{15} + \frac{3}{15} = \frac{5}{15} = \frac{1}{3} \]

### Step 4: Calculate the remaining work
- Total work is 1 (100% of the task), so the remaining work after 2 days is:
\[ \text{Remaining work} = 1 - \frac{1}{3} = \frac{2}{3} \]

### Step 5: Calculate how long A takes to finish the remaining work
- Since A can complete the entire task in 15 days, the time taken by A to complete \(\frac{2}{3}\) of the task is:
\[ \text{Time for A to complete remaining work} = \frac{2}{3} \times 15 = 10 \text{ days} \]

### Step 6: Calculate the total time taken for the entire task
- Total time taken = Time worked together (2 days) + Time taken by A to finish remaining work (10 days):
\[ \text{Total time} = 2 + 10 = 12 \text{ days} \]

Thus, the entire task was completed in **12 days**.