Problem 28 · 2024 Math Kangaroo
Stretch
Ratios, Rates & Proportions
distance-speed-timetotal-then-divide
Olga walked in the park. For half of the time she walked at 2 km/h. For half of the distance she walked at 3 km/h. For the rest of the time she walked at 4 km/h. For what fraction of the time did she walk at 4 km/h?
Show answer
Answer: A — \(\frac{1}{14}\)
Show hints
Hint 1 of 2
Name the total time T and total distance S, then write the distance covered in each phase.
Still stuck? Show hint 2 →
Hint 2 of 2
The three phases must account for all of S, which gives one equation linking S and T.
Show solution
Approach: write each phase, force the distances to add to S, then read off the time share
- In the first half of the time (time \(\tfrac{T}{2}\)) at 2 km/h she covers \(2\cdot\tfrac{T}{2}=T\); the 3 km/h stretch covers half the distance \(\tfrac{S}{2}\); the leftover time \(\tfrac{T}{2}-\tfrac{S}{6}\) at 4 km/h covers \(4(\tfrac{T}{2}-\tfrac{S}{6})\).
- Adding all three distances to \(S\) gives \(S=T+\tfrac{S}{2}+(2T-\tfrac{2S}{3})\), which solves to \(S=\tfrac{18T}{7}\).
- Then the 4 km/h time is \(\tfrac{T}{2}-\tfrac{S}{6}=\tfrac{T}{2}-\tfrac{3T}{7}=\tfrac{T}{14}\), so its share of the total time is \(\frac{1}{14}\), answer A.
Mark:
· log in to save