Let's analyze the problem step-by-step.
Problem Restatement
- A completes 8?\frac{8}{?}?8 of a work in 20 days. (Assuming the problem means "A completes 810\frac{8}{10}108 or 45\frac{4}{5}54 of the work in 20 days" or "A completes 8 units of work in 20 days" - but since the problem is unclear, let's assume A completes 810=45\frac{8}{10}=\frac{4}{5}108=54 of the work in 20 days.)
- Then B joins A, and together they finish the remaining work in 3 days.
- We need to find how long B alone will take to complete the entire work.
Step 1: Clarify the problem
Since the problem states "a completes 8 of a work in 20 days," it likely means:
- A completes 810\frac{8}{10}108 (or 80%) of the work in 20 days.
- Remaining work = 1−810=210=151-\frac{8}{10}=\frac{2}{10}=\frac{1}{5}1−108=102=51.
Step 2: Calculate A's work rate
- A completes 810\frac{8}{10}108 of the work in 20 days.
- So, A's work rate = 8/1020=810×20=8200=250=125\frac{8/10}{20}=\frac{8}{10\times 20}=\frac{8}{200}=\frac{2}{50}=\frac{1}{25}208/10=10×208=2008=502=251 work per day.
Step 3: Calculate combined work rate of A and B
- Remaining work = 15\frac{1}{5}51.
- A and B together finish this in 3 days.
- So, combined work rate of A and B = 1/53=115\frac{1/5}{3}=\frac{1}{15}31/5=151 work per day.
Step 4: Calculate B's work rate
- Combined rate = A's rate + B's rate
- 115=125+B’s rate\frac{1}{15}=\frac{1}{25}+\text{B's rate}151=251+B’s rate
- B's rate = 115−125=5−375=275\frac{1}{15}-\frac{1}{25}=\frac{5-3}{75}=\frac{2}{75}151−251=755−3=752 work per day.
Step 5: Calculate how long B alone takes to finish the work
- B's rate = 275\frac{2}{75}752 work per day
- Time taken by B alone = 1B’s rate=12/75=752=37.5\frac{1}{\text{B's rate}}=\frac{1}{2/75}=\frac{75}{2}=37.5B’s rate1=2/751=275=37.5 days.
Final Answer:
B alone will take 37.5 days to complete the entire work. If you want me to clarify any step or if the problem meant something else, please let me know!
