AMC 10 · 2017 · #7

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

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

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

Let d be each equal half — the biked part and the walked part — so she bikes d and walks d, and d is the answer.

d = bike distance = walk distance

2STEP 2

Write each part's time

Time = distance ÷ speed, so biking d at 17 km/h takes $\frac{d}{17}$ hour and walking d at 5 km/h takes $\frac{d}{5}$ hour.

t_bike =

\frac{d}{17}

, t_walk =

\frac{d}{5}

3STEP 3

Turn 44 minutes into hours

Speeds are per hour, so the time must be too: 44 minutes = $\frac{44}{60}$ = $\frac{11}{15}$ hour.

44 min =

\frac{44}{60}

\frac{11}{15}

hour

4STEP 4

Add the times and solve

Bike time + walk time = trip: $\frac{d}{17}$ + $\frac{d}{5}$ = $\frac{11}{15}$ , i.e. $\frac{22d}{85}$ = $\frac{11}{15}$ , so d = $\frac{935}{330}$ .

\frac{22d}{85}

\frac{11}{15}

→ d =

\frac{11}{15}

\frac{85}{22}

\frac{935}{330}

5STEP 5

Round to the nearest tenth

Divide: $\frac{935}{330}$ = 2.8333… km. The next digit is 3, so it rounds down to 2.8 — choice (C).

d =

\frac{935}{330}

= 2.83 ≈ 2.8 → (C)

Answer

2.8

Plug d = 2.83 back in: bike time

\frac{2.8333}{17}

≈ 0.1667 hour and walk time

\frac{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 $\frac{d}{17}$ hour and walking takes $\frac{d}{5}$ hour, and together they fill $\frac{44}{60}$ 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

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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