Let's analyze the problem step-by-step.

Problem Restatement

• A and B together can finish a work in 30 days.
• They work together for 20 days.
• Then B leaves.
• A works alone for another 20 days to finish the remaining work.
• We need to find how many days A alone can finish the entire work.

Step 1: Define variables

• Let the total work be 1 unit.
• Let A's rate of work be aaa (work per day).
• Let B's rate of work be bbb (work per day).

Since A and B together can finish the work in 30 days:

a+b=130a+b=\frac{1}{30}a+b=301​

Step 2: Work done in the first 20 days together

In 20 days, A and B together complete:

20×(a+b)=20×130=2030=2320\times (a+b)=20\times \frac{1}{30}=\frac{20}{30}=\frac{2}{3}20×(a+b)=20×301​=3020​=32​

So, 23\frac{2}{3}32​ of the work is done.

Step 3: Remaining work after 20 days

Remaining work = 1−23=131-\frac{2}{3}=\frac{1}{3}1−32​=31​

Step 4: A finishes the remaining work alone in 20 days

So A's work rate is:

a=work donetime=1320=160a=\frac{\text{work done}}{\text{time}}=\frac{\frac{1}{3}}{20}=\frac{1}{60}a=timework done​=2031​​=601​

Step 5: Calculate how many days A alone takes to finish the entire work

Since A's rate is 160\frac{1}{60}601​ work per day, time taken by A alone to finish the whole work is:

Time=1a=60 days\text{Time}=\frac{1}{a}=60\text{ days}Time=a1​=60 days

Answer:

A alone can finish the work in 60 days.