Question
Soccer practice is 7 miles from Sam's home. So, first Sam ran to the bus stop at an average speed of 5 mph, and then he took a bus that went an average speed of 20 mph. For how long did Sam ride the bus if he ran 0.1 hours less than he rode? Write your answer in hours and then write it in minutes.
Answers
GPT-4o mini
Let \( t \) be the time in hours that Sam rode the bus. According to the problem, he ran for \( t - 0.1 \) hours.
First, we will calculate the distance Sam ran to the bus stop. The distance is given by the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
1. Sam's running speed is 5 mph, and he ran for \( t - 0.1 \) hours. Therefore, the distance he ran is:
\[
\text{Distance}_{\text{run}} = 5(t - 0.1)
\]
2. The bus travels at a speed of 20 mph for \( t \) hours, so the distance covered by the bus is:
\[
\text{Distance}_{\text{bus}} = 20t
\]
3. The total distance from Sam's home to the soccer practice is 7 miles. Thus, we can set up the equation:
\[
\text{Distance}_{\text{run}} + \text{Distance}_{\text{bus}} = 7
\]
Substituting in the distances calculated:
\[
5(t - 0.1) + 20t = 7
\]
4. Distributing the 5 in the first term gives:
\[
5t - 0.5 + 20t = 7
\]
5. Combining like terms results in:
\[
25t - 0.5 = 7
\]
6. Adding 0.5 to both sides provides:
\[
25t = 7.5
\]
7. Dividing both sides by 25, we find:
\[
t = \frac{7.5}{25} = 0.3 \text{ hours}
\]
Now, we can find the time Sam ran:
\[
t - 0.1 = 0.3 - 0.1 = 0.2 \text{ hours}
\]
Finally, let's convert the riding time \( t \) to minutes:
\[
0.3 \text{ hours} \times 60 \text{ minutes/hour} = 18 \text{ minutes}
\]
Thus, Sam rode the bus for
\[
\boxed{0.3} \text{ hours} \quad \text{or} \quad \boxed{18} \text{ minutes}.
\]
First, we will calculate the distance Sam ran to the bus stop. The distance is given by the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]
1. Sam's running speed is 5 mph, and he ran for \( t - 0.1 \) hours. Therefore, the distance he ran is:
\[
\text{Distance}_{\text{run}} = 5(t - 0.1)
\]
2. The bus travels at a speed of 20 mph for \( t \) hours, so the distance covered by the bus is:
\[
\text{Distance}_{\text{bus}} = 20t
\]
3. The total distance from Sam's home to the soccer practice is 7 miles. Thus, we can set up the equation:
\[
\text{Distance}_{\text{run}} + \text{Distance}_{\text{bus}} = 7
\]
Substituting in the distances calculated:
\[
5(t - 0.1) + 20t = 7
\]
4. Distributing the 5 in the first term gives:
\[
5t - 0.5 + 20t = 7
\]
5. Combining like terms results in:
\[
25t - 0.5 = 7
\]
6. Adding 0.5 to both sides provides:
\[
25t = 7.5
\]
7. Dividing both sides by 25, we find:
\[
t = \frac{7.5}{25} = 0.3 \text{ hours}
\]
Now, we can find the time Sam ran:
\[
t - 0.1 = 0.3 - 0.1 = 0.2 \text{ hours}
\]
Finally, let's convert the riding time \( t \) to minutes:
\[
0.3 \text{ hours} \times 60 \text{ minutes/hour} = 18 \text{ minutes}
\]
Thus, Sam rode the bus for
\[
\boxed{0.3} \text{ hours} \quad \text{or} \quad \boxed{18} \text{ minutes}.
\]