Let's break down the problem step by step.

1. Determine the usual speed of the traveler.

We know the trip is 24 miles and normally takes 32 minutes. First, we need to convert 32 minutes into hours since speed is usually measured in miles per hour (mph): $32 \, \text{minutes} = \frac{32}{60} \, \text{hours} = \frac{8}{15} \, \text{hours}$

Now, we can calculate the usual speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{24 \, \text{miles}}{\frac{8}{15} \, \text{hours}} = 24 \times \frac{15}{8} = 45 \, \text{mph}$

2. Calculate the reduced speed due to the damaged tire.

The problem states that the traveler had to decrease her average speed by 18 miles per hour. Therefore, her new speed is: $45 \, \text{mph} - 18 \, \text{mph} = 27 \, \text{mph}$

3. Calculate the time required for the return trip.

Now, we can use the formula for time: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$ For the return trip: $\text{Time} = \frac{24 \, \text{miles}}{27 \, \text{mph}} = \frac{24}{27} \, \text{hours} = \frac{8}{9} \, \text{hours}$

To convert this into minutes: $\frac{8}{9} \, \text{hours} \times 60 \, \text{minutes/hour} = \frac{480}{9} \, \text{minutes} \approx 53.33 \, \text{minutes}$

Final Answer:

The traveler should budget approximately 53.33 minutes (or about 53 minutes and 20 seconds) for the return trip.

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Do you want more details on any part? Here are 5 related questions to expand on:

1. What if the reduced speed was 10 miles per hour instead of 18?
2. How would the solution change if the distance to the airport was 30 miles?
3. How can we calculate the time saved if the road conditions improve and allow higher speeds?
4. What is the percentage decrease in speed due to the damaged tire?
5. How does the relationship between speed, distance, and time work in general?

Tip: Always ensure time units are consistent (hours, minutes) when calculating speeds or travel times.