To solve this problem, we need to determine how long it would take for $A$ to complete the entire work based on the given information.

Step 1: Determine the fraction of work done by $A$.

• $A$ completes $\frac{1}{3} \%$ of the work in 17 days.
• $\frac{1}{3} \%$ can be written as $\frac{1}{3 \times 100} = \frac{1}{300}$.

This means $A$ does $\frac{1}{300}$ of the total work in 17 days.

Step 2: Determine the work rate of $A$.

• Work rate $R$ is given by work done per day.
• $R = \frac{\text{Work done}}{\text{Time taken}} = \frac{\frac{1}{300}}{17} = \frac{1}{300 \times 17}$.

Step 3: Calculate the time required to complete the whole work.

• To complete the whole work, $A$ would need $T$ days, where: $T = \frac{\text{Total Work}}{\text{Work rate}} = \frac{1}{\frac{1}{300 \times 17}} = 300 \times 17 = 5100 \text{ days}$

So, $A$ would take 5100 days to complete the whole work.

Would you like any further details or explanations?

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Here are 8 related questions for practice:

1. If $B$ does 25% of the work in 20 days, how long will $B$ take to complete the whole work?
2. What is the work rate of a person who completes 10% of a task in 5 days?
3. If $C$ can complete the entire work in 30 days, how much work does $C$ complete in 10 days?
4. Two people can complete a task together in 15 days. If one person works twice as fast as the other, how long would each take to complete the work alone?
5. If a task is 60% complete and 12 days have passed, how long will it take to finish the remaining work at the same rate?
6. How would the time change if $A$ worked 10% faster on the same task?
7. If the work rate doubles, how many days will it take to complete a task that originally took 20 days?
8. What fraction of the work is done after 25 days if a person completes the work in 100 days?

Tip: When dealing with work problems, it’s often useful to express rates as fractions of the whole task and to set up equations that relate these rates to time.