Problem 14 · 2009 AMC 8
Medium
Ratios, Rates & Proportions
harmonic-mean
Austin and Temple are 50 miles apart along Interstate 35. Bonnie drove from Austin to her daughter's house in Temple, averaging 60 miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged 40 miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
Show answer
Answer: B — 48 mph.
Show hints
Hint 1 of 2
The trap answer is 50 (just averaging 60 and 40). But she spends MORE time at the slow 40 mph, so the average leans below 50. Average speed is always total distance ÷ total time — never the average of the speeds.
Still stuck? Show hint 2 →
Hint 2 of 2
Since the distance each way is the same, the actual 50 miles cancels out — the answer depends only on the two speeds 60 and 40.
Show solution
Approach: total distance ÷ total time
- Time there: 50/60 = 5/6 hr. Time back: 50/40 = 5/4 hr. Total time = 5/6 + 5/4 = 10/12 + 15/12 = 25/12 hr.
- Total distance = 2 × 50 = 100 miles. Average speed = 100 ÷ 25/12 = 100 × 12/25 = 48 mph.
- Sanity check: 48 is below the plain average of 50 — correct, because the slow leg eats more time.
- You'll see it again: equal-distance round trips give the harmonic mean of the speeds, 2×60×40/(60+40) = 4800/100 = 48 — always below the ordinary average, and the actual distance never matters.
Another way — pick a convenient distance:
- The distance cancels, so use the LCM of 60 and 40: pretend each leg is 120 miles.
- Out: 120/60 = 2 hr. Back: 120/40 = 3 hr. Total 240 miles in 5 hr = 48 mph.
Mark:
· log in to save