Miguel can drive 6
 times as fast as Oscar can ride his bicycle.  If it takes Oscar 3
 hours longer than Miguel to travel 54
 miles, how fast (in mph) can Oscar ride his bicycle?  Round your answer to two decimal places, if needed.

Let the speed of Oscar be \( v \) miles per hour (mph). Since Miguel can drive 6 times as fast, his speed is \( 6v \) mph.

The time it takes for Oscar to travel 54 miles is given by the formula:

\[
\text{Time for Oscar} = \frac{54}{v}
\]

Similarly, the time it takes for Miguel to travel 54 miles is:

\[
\text{Time for Miguel} = \frac{54}{6v}
\]

According to the problem, Oscar takes 3 hours longer than Miguel to travel the same distance, so:

\[
\frac{54}{v} = \frac{54}{6v} + 3
\]

### Step 1: Solve the equation

First, subtract \( \frac{54}{6v} \) from both sides:

\[
\frac{54}{v} - \frac{54}{6v} = 3
\]

Factor out \( \frac{54}{v} \):

\[
\frac{54}{v} \left(1 - \frac{1}{6}\right) = 3
\]

Simplify the expression inside the parentheses:

\[
\frac{54}{v} \times \frac{5}{6} = 3
\]

Multiply both sides by \( \frac{6}{5} \) to isolate \( \frac{54}{v} \):

\[
\frac{54}{v} = \frac{18}{5}
\]

Now, solve for \( v \) by multiplying both sides by \( v \) and dividing by \( \frac{18}{5} \):

\[
v = \frac{54 \times 5}{18}
\]

\[
v = 15 \, \text{mph}
\]

Thus, Oscar can ride his bicycle at 15 miles per hour.

### Do you want further details or have any questions?

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**5 Related Questions:**
1. How would the result change if Oscar could ride 4 times slower than Miguel?
2. What if Oscar's speed was reduced by 10%? How would that affect the time difference?
3. How would you model this scenario if the distance was unknown but the time difference was still 3 hours?
4. If the distance were doubled, how would the speeds affect the travel times?
5. Can we determine the speeds if the time difference between Oscar and Miguel was given as a percentage?

**Tip:** Always remember to carefully define variables when working with rates and times to avoid confusion.

Miguel can drive 6 times as fast as Oscar can ride his bicycle. If it takes Oscar 3 hours longer than Miguel to travel 54 miles, how fast (in mph) can Oscar ride his bicycle? Round your answer to two decimal places, if needed.

Find out how to calculate the speed of Oscar's bicycle given that Miguel drives 6 times faster and Oscar takes 3 hours longer to travel 54 miles. This problem explores key algebraic concepts and the formula for time based on distance and speed. Ideal for students in Grades 8-10.

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