The problem asks how long it will take for an object traveling at a speed of $\frac{1}{10}$ miles per hour (mph) to travel $\frac{9}{10}$ miles. You are tasked with estimating the answer first, then calculating the exact answer.

Step 1: Estimate the Answer

The object is traveling at $\frac{1}{10}$ mph, which means it covers $\frac{1}{10}$ miles in 1 hour. To travel $\frac{9}{10}$ miles, you can estimate that it will take about: $9 \times 1 = 9 \text{ hours}$ However, this is too large of an estimate because the question asks for the time in minutes, not hours. There are 60 minutes in an hour, so it will take approximately: $9 \times 60 = 540 \text{ minutes}$ This is also not accurate, so let's go for an exact calculation.

Step 2: Exact Calculation

The formula for time is: $\text{time} = \frac{\text{distance}}{\text{speed}}$ Using the given distance $\frac{9}{10}$ miles and speed $\frac{1}{10}$ mph, we substitute into the formula: $\text{time} = \frac{\frac{9}{10}}{\frac{1}{10}} = 9 \text{ hours}$ Convert this to minutes: $9 \times 60 = 540 \text{ minutes}$

Conclusion

The exact answer is that it will take 540 minutes to travel $\frac{9}{10}$ miles at a speed of $\frac{1}{10}$ mph.

Would you like any more details or clarifications?

Here are five related questions you could explore:

1. How would the time change if the speed increased to $\frac{2}{10}$ mph?
2. What would happen if the distance was doubled?
3. Can you derive the time for a general distance $d$ and speed $s$?
4. How would you calculate the time if the speed varied throughout the journey?
5. What if the object had to stop for 10 minutes after every mile?

Tip: Always check the units you're working with, especially when converting between hours and minutes!