Problem 16 · 2019 AMC 8
Medium
Ratios, Rates & Proportions
distance-speed-timesubstitution
Qiang drives 15 miles at an average speed of 30 miles per hour. How many additional miles will he have to drive at 55 miles per hour to average 50 miles per hour for the entire trip?
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Answer: D — 110 miles.
Show hints
Hint 1 of 2
Average speed is NOT the average of 30 and 55 — it's total distance ÷ total time. Time is the quantity you can actually add up across the two legs, so make time your handle.
Still stuck? Show hint 2 →
Hint 2 of 2
First leg takes 15÷30 = ½ hour. Let the extra distance be x (time x/55). Then set (total distance)÷(total time) = 50 and solve.
Show solution
Approach: total distance ÷ total time = 50
- The first leg takes 15 ÷ 30 = ½ hour. Driving x more miles at 55 adds x/55 hours, so total distance = 15 + x and total time = ½ + x/55.
- Average speed is the ratio: 15 + x½ + x/55 = 50, i.e. 15 + x = 25 + 10x/11.
- Multiply through by 11: 165 + 11x = 275 + 10x ⇒ x = 110.
- Why this transfers — and a sanity check: the answer 110 is far bigger than the 15-mile first leg because he must drive a long way at 55 to drag the average all the way up to 50 (just shy of 55). Never average speeds directly; only times and distances add.
Another way — fix the total time first:
- Driving 15 miles at 30 used ½ hour. Suppose the second leg also takes t hours; then total distance = 50(½ + t) and the second leg covers 55t miles.
- So 15 + 55t = 25 + 50t ⇒ 5t = 10 ⇒ t = 2 hours, and the extra distance is 55 × 2 = 110 miles.
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