A can do a piece of work in 80 days and B in 100 days. They work together for first 20 days before B goes away. In how many days will A manage to finish the remaining work?

Answer

To solve the problem step by step, we will first determine the work done by A and B, then calculate the remaining work after they work together for 20 days, and finally find out how long it will take A to finish the remaining work alone.

### Step 1: Determine the work done by A and B

A can complete the work in 80 days, and B can complete it in 100 days.

The efficiency (work done per day) of A is:
\[ \text{Efficiency of A} = \frac{1}{80} \text{ (work units per day)} \]

The efficiency of B is:
\[ \text{Efficiency of B} = \frac{1}{100} \text{ (work units per day)} \]

### Step 2: Find the combined efficiency of A and B

To find the combined efficiency of A and B when they work together, we add their efficiencies:
\[ \text{Combined Efficiency} = \frac{1}{80} + \frac{1}{100} \]

To add these fractions, we need a common denominator. The least common multiple of 80 and 100 is 400. Therefore:
\[ \frac{1}{80} = \frac{5}{400} \quad \text{and} \quad \frac{1}{100} = \frac{4}{400} \]
\[ \text{Combined Efficiency} = \frac{5}{400} + \frac{4}{400} = \frac{9}{400} \]

### Step 3: Calculate the work done in the first 20 days

Now, we calculate the total work done by A and B together in the first 20 days:
\[ \text{Work done in 20 days} = \text{Combined Efficiency} \times \text{Time} = \frac{9}{400} \times 20 \]
\[ \text{Work done in 20 days} = \frac{180}{400} = \frac{9}{20} \text{ of the total work} \]

### Step 4: Determine the total work

Since A can complete the total work in 80 days, the total work can be represented in terms of work units:
\[ \text{Total Work} = 80 \times \text{Efficiency of A} = 80 \times \frac{1}{80} = 1 \text{ (total work units)} \]

### Step 5: Calculate the remaining work

The remaining work after A and B have worked together for 20 days is:
\[ \text{Remaining Work} = 1 - \frac{9}{20} = \frac{20}{20} - \frac{9}{20} = \frac{11}{20} \]

### Step 6: Calculate how long it will take A to finish the remaining work

Now, we need to find out how long it will take A to finish the remaining \(\frac{11}{20}\) of the work alone. A's efficiency is \(\frac{1}{80}\), so we can set up the equation:
\[ \text{Time} = \frac{\text{Remaining Work}}{\text{Efficiency of A}} = \frac{\frac{11}{20}}{\frac{1}{80}} = \frac{11}{20} \times 80 = 44 \text{ days} \]

### Final Answer

A will take **44 days** to finish the remaining work alone.

---