Question

A cyclist starts a 30-mile race. She races at the same constant speed for 40 minutes down a highway and for 12 miles down a country road, and then she crosses the finish line. What is the speed of the cyclist in miles per hour?

Answer 1

To determine the speed of the cyclist, we first need to analyze the total distance covered and the time spent for each segment of the race.

1. **Down the highway:** The cyclist travels for 40 minutes. To convert this to hours, we use the conversion \( 40 \text{ minutes} = \frac{40}{60} \text{ hours} = \frac{2}{3} \text{ hours} \).

2. **Speed during the highway segment:** Let the speed of the cyclist be \( v \) miles per hour. The distance covered on the highway can be calculated using the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
Therefore, the distance covered on the highway is:
\[
\text{Distance}_{\text{highway}} = v \times \frac{2}{3} \text{ hours}
\]

3. **Down the country road:** The cyclist then travels 12 miles.

4. **Total distance of the race:** The total distance of the race is 30 miles. Therefore, we can set up the equation based on the total distance:
\[
\text{Distance}_{\text{highway}} + \text{Distance}_{\text{country road}} = 30 \text{ miles}
\]
Substituting in the known distances:
\[
v \times \frac{2}{3} + 12 = 30
\]

5. **Solving for \( v \):**
First, isolate the term involving \( v \):
\[
v \times \frac{2}{3} = 30 - 12
\]
\[
v \times \frac{2}{3} = 18
\]
Next, multiply both sides by \( \frac{3}{2} \) to solve for \( v \):
\[
v = 18 \times \frac{3}{2}
\]
\[
v = 27
\]

Thus, the speed of the cyclist is \( \boxed{27} \) miles per hour.