AMC 10 · 2007 · #4

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
rateunit-conversionweighted-average dimensional-analysisidentify-subproblems ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
A trip has a short fast leg and a longer slow leg. Find the average speed for the whole trip.

Pick an answer.

(A)
7
(B)
9
(C)
10
(D)
12
(E)
14
How to solve
Strategy Analyze the Units

Speed, time, and distance are locked together by miles=miles/hour×hours, and the problem deliberately hands over mph next to minutes, so Tool #8 (Analyze the Units) is what keeps the arithmetic honest: fix the times in hours first. Tool #7 (Identify Subproblems) splits the trip into two constant-speed legs, each a clean multiplication. Tool #4 (Introduce a Variable) then names the answer v and states what average speed actually means — the constant speed covering the same total distance in the same total time — and Tool #13 (Convert to Algebra) turns that sentence into an equation. Setting it up this way makes the (16+4)/2 trap impossible to fall into, because at no point does the work ask for an average of speeds.

1STEP 1

Put both times in hours

Matching units puts the total time at 2.

30 min=30/60 h=1/2 h, 90 min=90/60 h=3/2 h, total time=2 h
2STEP 2

Find each leg's distance

Each leg's distance adds to 14.

16×1/2=8, 4×3/2=6, total distance=8+6=14 miles
3STEP 3

Say what average speed means

Average speed means total over total.

v × 2=14
4STEP 4

Solve for the average speed

Dividing gives 7, well below the naive midpoint, choice (A).

2v=14 → v=14/2=7 → (A)
Answer
7
Rebuild the trip from the answer: at a steady 7 mph for 2 hours Kate covers 7×2=14 miles, exactly the 8+6 she actually covered, so 7 satisfies the definition it was built from. A second, independent check uses a baseline. Had she walked at 4 mph for the full 2 hours she would have gone 8 miles. The half hour she rode instead was 16-4=12 mph faster, worth 12×1/2=6 extra miles, so the real distance is 8+6=14 miles and the average is 4+6/2=7 mph — the same value from a different route. The size is sensible too: an average of the two speeds must land strictly between 4 and 16, and since three quarters of the time is spent at the slow speed it must sit much closer to 4 than to 16. That alone kills 10, 12, and 14, and the exact weighting pins the value at 7 rather than 9.
💡Key takeaway

Average speed is all the distance divided by all the time — averaging the two speeds only works when you spend equal time at each.

  • Put both times in hours
  • Find each leg's distance
  • Say what average speed means
  • Solve for the average speed

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