Average Speed on a Two-Part Trip
Average speed on a two-part trip is a Grade 7 applied math skill from Yoshiwara Intermediate Algebra teaching that overall average speed is not simply the arithmetic mean of the two speeds. Students use the formula: average speed = total distance / total time, requiring separate calculation of each segment.
Key Concepts
Property The average speed for a trip is the total distance divided by the total time. For a two part trip with distances $d 1$ and $d 2$ traveled at rates $r 1$ and $r 2$, the times for each part are $t 1 = \frac{d 1}{r 1}$ and $t 2 = \frac{d 2}{r 2}$.
The average speed for the entire trip is given by the formula: $$\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{d 1 + d 2}{\frac{d 1}{r 1} + \frac{d 2}{r 2}}$$.
Examples A car travels 150 miles at 50 mph and returns the same distance at 30 mph. Total distance is 300 miles. Total time is $\frac{150}{50} + \frac{150}{30} = 3 + 5 = 8$ hours. Average speed is $\frac{300}{8} = 37.5$ mph.
Common Questions
How do you calculate average speed on a two-part trip?
Find the total distance of both parts, find the total time for both parts, then divide: average speed = total distance / total time.
Why is the average speed not just the average of the two speeds?
You spend different amounts of time at each speed, so the time-weighted average is needed. Simply averaging the speeds ignores how long each speed was maintained.
If you drive 60 mph for 1 hour and 30 mph for 2 hours, what is the average speed?
Total distance = 60 + 60 = 120 miles. Total time = 3 hours. Average speed = 120/3 = 40 mph.
Where does the harmonic mean come in for average speed?
When traveling equal distances at two speeds, the average speed equals the harmonic mean: 2/(1/v1 + 1/v2).