AMC 10 · 2003 · #4

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
rateunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Anna walks 1 km uphill from home to school in 30 minutes, then walks the same 1 km back downhill in 10 minutes. Find her average speed, in km/hr, for the whole round trip.

Pick an answer.

(A)
3
(B)
3.125
(C)
3.5
(D)
4
(E)
4.5

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

The word "speed" with a target unit of km/hr is a signal for Tool #8 (Analyze the Units): average speed is defined as (total distance in km)/(total time in hr), so the units themselves tell us exactly what to collect — a total distance and a total time — and warn us to convert minutes to hours before dividing. Tool #7 (Identify Subproblems) then splits the job cleanly: first add up the distance, separately add up the time, and only at the end combine them. The units framing also guards against the classic trap of averaging the two speeds (2 and 6) to get 4, which mixes rates instead of dividing one total by another.

1STEP 1

Read the target unit

The answer is in km/hr, so it is total distance over total time — one division at the end, not an average of two speeds.

average speed = (total distance)/(total time)
2STEP 2

Add distance and time

Total distance is 1 + 1 = 2 km; total time is 30 + 10 = 40 min, which is 4060=23\frac{40}{60}=\frac{2}{3} hr.

total distance = 1 + 1 = 2 km, total time = 40 min = 40/60 = 2/3 hr
3STEP 3

Divide totals for the speed

Dividing by 23\frac{2}{3} means multiplying by the reciprocal 32\frac{3}{2}, so the average speed is 3 km/hr, choice (A).

(2 km)/(2/3 hr) = 2 × 3/2 = 3 km/hr → (A)
Answer
3
The two actual speeds are 2 km/hr uphill (1 km in half an hour) and 6 km/hr downhill (1 km in a sixth of an hour), so the round-trip average must land between 2 and 6 — and 3 does. It sits closer to the slow speed than to 4, which is correct because she spends much more time crawling uphill than coasting downhill. This also exposes the trap answer (D) 4: that is the plain average (2+6)/2, which wrongly ignores that the slow leg eats up most of the time.
💡Key takeaway

Average speed is all your distance divided by all your time — never just the average of the two speeds, because the slow part eats up more of the clock.

  • Read the target unit
  • Add distance and time
  • Divide totals for the speed

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