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.
