A boat traveled 1/4 of the distance between two cities in 2/3 of an hour. At this rate how many hours will it take the boat to travel the entire distance between the two cities

To determine how long it will take the boat to travel the entire distance between the two cities, we can follow these steps:

1. **Determine the rate of travel:**
   - The boat traveled \( \frac{1}{4} \) of the total distance in \( \frac{2}{3} \) of an hour.

   - The rate of travel \( R \) can be calculated as:
   \[
   R = \frac{\text{Distance traveled}}{\text{Time taken}} = \frac{\frac{1}{4} \text{ of the total distance}}{\frac{2}{3} \text{ hour}} = \frac{\frac{1}{4}}{\frac{2}{3}} = \frac{1}{4} \times \frac{3}{2} = \frac{3}{8} \text{ of the total distance per hour}
   \]

2. **Determine the total time needed to travel the entire distance:**
   - Let \( T \) be the total time required to travel the entire distance.
   - The rate \( R \) is \( \frac{3}{8} \) of the distance per hour, so to cover the entire distance \( (1 \text{ full distance}) \), we have:
   \[
   1 = R \times T = \frac{3}{8} \times T
   \]

   - Solving for \( T \):
   \[
   T = \frac{1}{\frac{3}{8}} = \frac{8}{3} \text{ hours}
   \]

So, it will take the boat **\(\frac{8}{3}\) hours** (or approximately 2.67 hours) to travel the entire distance between the two cities.

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### Related Questions:
1. How would the time change if the boat's speed was increased by 20%?
2. What is the boat's speed in terms of distance per minute?
3. How long would it take the boat to travel half the distance?
4. What if the boat took 1 hour to travel \( \frac{1}{4} \) of the distance? How would that change the total time?
5. If the boat traveled \( \frac{1}{3} \) of the distance in 1 hour, how much time would the full trip take?
6. How would the total time change if the boat took \( \frac{4}{3} \) hours to travel \( \frac{1}{4} \) of the distance?
7. Can you convert the total travel time into minutes?
8. What would be the time taken if the boat traveled \( \frac{1}{6} \) of the distance in 1 hour?

### Tip:
When dealing with ratios or proportions, always simplify fractions and convert complex ratios to make calculations easier.

Learn how to calculate the time required for a boat to travel the entire distance between two cities using ratio and proportion. This problem involves determining the boat's speed and applying it to find the total travel time. Suitable for Grades 7-9.

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