The problem involves calculating the fuel efficiency of two cars (blue and red) in terms of miles per gallon, based on their given fuel consumption and distance traveled.

Blue Car:

• The blue car travels $\frac{1}{2}$ miles on $\frac{1}{4}$ gallons of gasoline.

To find the unit rate (miles per gallon), we divide the distance by the amount of fuel consumed:

$\text{Miles per gallon for the blue car} = \frac{\frac{1}{2}}{\frac{1}{4}} = \frac{1}{2} \times \frac{4}{1} = 2 \text{ miles per gallon}.$

Red Car:

• The red car travels $20 \frac{4}{5}$ miles on $\frac{4}{5}$ gallons of gasoline. First, convert the mixed number $20 \frac{4}{5}$ to an improper fraction: $20 \frac{4}{5} = \frac{100}{5} + \frac{4}{5} = \frac{104}{5} \text{ miles}.$

Now, divide this distance by the fuel consumed:

$\text{Miles per gallon for the red car} = \frac{\frac{104}{5}}{\frac{4}{5}} = \frac{104}{5} \times \frac{5}{4} = \frac{104}{4} = 26 \text{ miles per gallon}.$

Conclusion:

• The blue car has a fuel efficiency of 2 miles per gallon.
• The red car has a fuel efficiency of 26 miles per gallon.

Thus, the red car can travel a greater distance on 1 gallon of gasoline.

Let me know if you want further details or have any questions!

Here are 5 related questions for deeper understanding:

1. How would the unit rate change if the blue car used $\frac{1}{3}$ gallons of gasoline instead of $\frac{1}{4}$?
2. Can you calculate the distance the red car would travel on 2 gallons of gasoline?
3. If the blue car had a fuel efficiency of 4 miles per gallon, how much fuel would it need to travel 8 miles?
4. How does the concept of a unit rate apply to real-world fuel economy ratings?
5. What is the mathematical method for converting mixed numbers to improper fractions?

Tip: When dividing fractions, remember that division is the same as multiplying by the reciprocal of the divisor.