Problem 17 · 2005 AMC 8
Easy
Ratios, Rates & Proportions
slope-as-speed
The results of a cross-country team's training run are graphed below. Which student has the greatest average speed?
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Answer: E — Evelyn.
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Hint 1 of 2
Speed = distance ÷ time. On a distance-vs-time graph, that ratio for each runner is the steepness of the line from the origin O out to their dot.
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Hint 2 of 2
You don't need numbers off the axes. Just eyeball which dot's line from O tilts up most steeply — high distance for little time.
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Approach: fastest = steepest line from the origin
- Average speed is distance ÷ time, which is exactly the slope of the segment joining O to a runner's dot. Steeper line = more distance per unit time = faster.
- Evelyn's dot sits high (large distance) and far left (small time), so her line from O is the steepest ⇒ Evelyn.
- Watch the trap: Carla is the highest dot, but she took the most time, so she isn't fastest — 'farthest' is not 'fastest.' Only the slope from the origin tells you speed.
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