AMC 10 · 2017 · #4

Grade 7 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateunit-conversionfraction-arithmetic convert-to-algebradimensional-analysis ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Samia rides her bike at 17 kilometers per hour for the first half of the trip, then walks at 5 kilometers per hour for the second half. The whole trip takes 44 minutes. Find how far she walked, in kilometers rounded to the nearest tenth.

Pick an answer.

(A)
2.0
(B)
2.2
(C)
2.8
(D)
3.4
(E)
4.4
How to solve
Strategy Introduce a Variable

Tool #4 (Introduce a Variable): the bike half and the walk half cover the same unknown distance, so name that one distance d and both parts are described at once. Tool #8 (Analyze the Units): the speeds are kilometers per hour, so each time must be d ÷ speed in hours, which means the 44 minutes has to become a fraction of an hour first. Tool #7 (Identify Subproblems): split the 44 minutes into the bike time plus the walk time, write each as d over its speed, add them, and solve the single equation for d.

1STEP 1

Name the equal half-distance

The key is that both legs are the same distance.

d = bike distance = walk distance
2STEP 2

Write each part's time

Each leg's time is that distance over its speed.

t_bike = d/17, t_walk = d/5
3STEP 3

Turn 44 minutes into hours

The given 44 minutes must become hours.

44 min = 44/60 = 11/15 hour
4STEP 4

Add the times and solve

Adding the times and solving gives the distance.

22d/85 = 11/15 → d = 11/15·85/22 = 935/330
5STEP 5

Round to the nearest tenth

Rounding gives 2.8.

d = 935/330 = 2.83 ≈ 2.8 → (C)
Answer
2.8
Plug d = 2.83 back in: bike time 2.8333/17 ≈ 0.1667 hour and walk time 2.8333/5 ≈ 0.5667 hour, totaling about 0.7333 hour, which is 0.7333 × 60 ≈ 44 minutes. It also makes sense that the walk eats most of the time: the two halves are the same length, but walking is more than three times slower than biking, so the walk should take far longer — and 0.57 hour versus 0.17 hour matches that.
💡Key takeaway

Call the equal half-distance d; biking takes d/17 hour and walking takes d/5 hour, and together they fill 44/60 hour, which gives d ≈ 2.8 kilometers — choice (C).

  • Name the equal half-distance
  • Write each part's time
  • Turn 44 minutes into hours
  • Add the times and solve
  • Round to the nearest tenth

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