Question
A professional cyclist is training for the Tour de France. Yesterday, he biked 140 kilometers in 3 hours and 25 minutes. Today, he biked 150 kilometers in 3 hours and 50 minutes. What was the cyclist's average speed over his two rides?
Write your answer as a whole number, a decimal, or a simplified fraction or mixed number. Do not round.
kilometers per hour
Answers
Answered by
GPT-4o mini
To find the cyclist's average speed over the two rides, we first need to calculate the total distance and the total time taken for both rides.
**Step 1: Calculate the total distance.**
The distance for the first ride is 140 kilometers, and the distance for the second ride is 150 kilometers.
\[
\text{Total Distance} = 140 \text{ km} + 150 \text{ km} = 290 \text{ km}
\]
**Step 2: Calculate the total time.**
For the first ride, the time taken is 3 hours and 25 minutes. We can convert this into hours.
\[
3 \text{ hours} + \frac{25 \text{ minutes}}{60} = 3 + \frac{25}{60} = 3 + \frac{5}{12} = \frac{36}{12} + \frac{5}{12} = \frac{41}{12} \text{ hours}
\]
For the second ride, the time taken is 3 hours and 50 minutes. We can convert this into hours as well.
\[
3 \text{ hours} + \frac{50 \text{ minutes}}{60} = 3 + \frac{50}{60} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \text{ hours}
\]
**Step 3: Add the total time.**
Now we need to add both times together:
\[
\text{Total Time} = \frac{41}{12} \text{ hours} + \frac{23}{6} \text{ hours}
\]
To add these fractions, we first need a common denominator. The least common multiple of 12 and 6 is 12.
Convert \(\frac{23}{6}\) to have a denominator of 12:
\[
\frac{23}{6} = \frac{23 \times 2}{6 \times 2} = \frac{46}{12}
\]
Now, we can add:
\[
\text{Total Time} = \frac{41}{12} + \frac{46}{12} = \frac{41 + 46}{12} = \frac{87}{12} \text{ hours}
\]
**Step 4: Calculate the average speed.**
To find the average speed, we divide the total distance by the total time:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{290 \text{ km}}{\frac{87}{12} \text{ hours}}
\]
This division can be simplified by multiplying by the reciprocal:
\[
\text{Average Speed} = 290 \text{ km} \times \frac{12}{87}
\]
Calculating this gives:
\[
\text{Average Speed} = \frac{290 \times 12}{87} = \frac{3480}{87}
\]
To simplify \(\frac{3480}{87}\):
Calculating the division yields:
\[
3480 \div 87 = 40
\]
Thus, the average speed of the cyclist over the two rides is:
\[
\boxed{40} \text{ kilometers per hour}
\]
**Step 1: Calculate the total distance.**
The distance for the first ride is 140 kilometers, and the distance for the second ride is 150 kilometers.
\[
\text{Total Distance} = 140 \text{ km} + 150 \text{ km} = 290 \text{ km}
\]
**Step 2: Calculate the total time.**
For the first ride, the time taken is 3 hours and 25 minutes. We can convert this into hours.
\[
3 \text{ hours} + \frac{25 \text{ minutes}}{60} = 3 + \frac{25}{60} = 3 + \frac{5}{12} = \frac{36}{12} + \frac{5}{12} = \frac{41}{12} \text{ hours}
\]
For the second ride, the time taken is 3 hours and 50 minutes. We can convert this into hours as well.
\[
3 \text{ hours} + \frac{50 \text{ minutes}}{60} = 3 + \frac{50}{60} = 3 + \frac{5}{6} = \frac{18}{6} + \frac{5}{6} = \frac{23}{6} \text{ hours}
\]
**Step 3: Add the total time.**
Now we need to add both times together:
\[
\text{Total Time} = \frac{41}{12} \text{ hours} + \frac{23}{6} \text{ hours}
\]
To add these fractions, we first need a common denominator. The least common multiple of 12 and 6 is 12.
Convert \(\frac{23}{6}\) to have a denominator of 12:
\[
\frac{23}{6} = \frac{23 \times 2}{6 \times 2} = \frac{46}{12}
\]
Now, we can add:
\[
\text{Total Time} = \frac{41}{12} + \frac{46}{12} = \frac{41 + 46}{12} = \frac{87}{12} \text{ hours}
\]
**Step 4: Calculate the average speed.**
To find the average speed, we divide the total distance by the total time:
\[
\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{290 \text{ km}}{\frac{87}{12} \text{ hours}}
\]
This division can be simplified by multiplying by the reciprocal:
\[
\text{Average Speed} = 290 \text{ km} \times \frac{12}{87}
\]
Calculating this gives:
\[
\text{Average Speed} = \frac{290 \times 12}{87} = \frac{3480}{87}
\]
To simplify \(\frac{3480}{87}\):
Calculating the division yields:
\[
3480 \div 87 = 40
\]
Thus, the average speed of the cyclist over the two rides is:
\[
\boxed{40} \text{ kilometers per hour}
\]