Problem 17 · 2018 AMC 8
Medium
Ratios, Rates & Proportions
ratiodistance-speed-time
Bella begins to walk from her house toward her friend Ella's house. At the same time, Ella begins to ride her bicycle toward Bella's house. They each maintain a constant speed, and Ella rides 5 times as fast as Bella walks. The distance between their houses is 2 miles, which is 10,560 feet, and Bella covers 2½ feet with each step. How many steps will Bella take by the time she meets Ella?
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Answer: A — 704 steps.
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Hint 1 of 2
They walk for the same amount of time until they meet, so the distances they cover split in the same ratio as their speeds — 1 to 5. Bella does the small share, Ella eats up the rest.
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Hint 2 of 2
The technique: for two movers closing a gap, "same time" means distance is shared in the speed ratio. Out of 1 + 5 = 6 equal parts, Bella covers 1 part — then convert her distance to steps.
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Approach: ratio of distances, then divide by step length
- In the equal time before they meet, Bella and Ella cover distances in their speed ratio 1 : 5. Together those parts make up the whole 10,560 feet, so Bella covers 1 of 6 parts.
- Bella's distance: 10,560 ÷ 6 = 1,760 feet.
- Each step is 2.5 ft, so steps = 1,760 ÷ 2.5 = 704. (The fact that 10,560 = 2 miles is just there to confirm the feet conversion.)
- You'll see it again: two objects meeting after starting at the same instant always divide the distance in their speed ratio — no need to find the meeting time at all.
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