Problem 18 · 2020 Math Kangaroo
Stretch
Ratios, Rates & Proportions
distance-speed-timesubstitution
Roberto and Maria leave at the same time from the same point of a long circular track, he on foot and she by bike. Maria completes a lap 24 minutes before Roberto and waits for him while having an ice cream. When he reaches this point, Maria leaves on her bike in the opposite direction and Roberto continues walking without stopping in the same direction. They then meet 5 minutes later. Assuming the speeds are kept constant, how long does it take for Roberto to do a lap of the track?
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Answer: A — 30 min
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Hint 1 of 2
Let Roberto’s lap take T minutes; Maria’s lap takes T−24.
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Hint 2 of 2
When they move in opposite directions and meet in 5 minutes, together they cover one full lap.
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Approach: set up a lap-time equation from the opposite-direction meeting
- Roberto’s lap = T, Maria’s lap = T−24.
- Meeting in 5 min going opposite ways: 5(1/T + 1/(T−24)) = 1.
- This gives T² − 34T + 120 = 0, so T = 30 (rejecting T = 4).
- Roberto takes 30 min per lap.
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