Problem 24 · 2022 Math Kangaroo
Stretch
Ratios, Rates & Proportions
distance-speed-timeproportion
By bike it takes Marc 20 minutes to go from home to school and back; on foot the same round trip takes 60 minutes. He rides and walks at constant speeds. Yesterday Marc biked to Eva’s house (on the way to school), left the bike there, and walked the rest of the way to school. Coming home he first walked to Eva’s house, then biked the rest of the way. The whole journey took 52 minutes. What fraction of the journey did he cover by bike?
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Answer: B — 1⁄5
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Hint 1 of 2
One leg (home to school) takes 10 min by bike and 30 min on foot; the bike part is the same length each way.
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Hint 2 of 2
Let the bike part be a fraction \(f\) of one leg; he bikes it twice and walks the rest twice, totalling 52 min.
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Approach: mix bike and walking times over the route
- A round trip is two legs; by bike one leg takes 10 min, on foot one leg takes 30 min.
- Let the bike portion be a fraction \(f\) of one leg. He bikes that fraction on both legs and walks the rest on both legs, so the time is \(2(10f) + 2(30(1-f)) = 60 - 40f = 52\).
- Then \(40f = 8\), so \(f = \tfrac{1}{5}\), and the bike distance is \(\tfrac{1}{5}\) of the whole journey, so the answer is B.
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