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:
\[ \boxed{40} \text{ kilometers per hour} \]
1 answer