AMC 10 · 2015 · #14

Grade 7 geometry-2d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series
arc-measureratio-proportionrotation-isometry physical-representation ↑ Prerequisites: arc-measureratio-proportion
📏 Medium solution 💡 3 insights 📊 Diagram
Problem
A small disk of radius 10 sits outside a clock of radius 20, touching it at the 12 o'clock mark, with a painted arrow pointing straight up. The disk rolls without slipping clockwise around the clock. Find the clock position the disk is touching at the first moment the arrow again points straight up.

Pick an answer.

(A)
2 o' clock
(B)
3 o' clock
(C)
4 o' clock
(D)
6 o' clock
(E)
8 o' clock

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

The whole question is about how much the disk has spun in space, so Tool #17 (Visualize Spatial Relationships) is primary: the arrow points up again exactly when the disk has made one full turn relative to the ground, and a rolling disk turns from TWO sources at once — it spins as it rolls along the surface AND it carries that spin around as it circles the clock (the coin-rotation effect). Tool #8 (Analyze the Units) makes this countable by comparing circumferences and the path the disk's center travels, turning "how much does it spin" into a clean ratio. Tool #1 (Draw a Diagram) anchors the final step: translate the fraction of the way around into an hour mark on the clock face.

1STEP 1

Arrow up means one full turn

The arrow points up again only after the disk completes a full 360° turn in space.

arrow up again ⇔ disk total rotation = 360°
2STEP 2

Count turns for a whole lap

The disk's center travels a circle 3× its own circumference, so it makes 3 full turns per lap.

(2π(20+10))/2π(10) = 60π/20π = 3 turns per lap
3STEP 3

One turn is a third of the lap

One turn is 1/3 of a lap, so the contact point sweeps 1/3× 360° = 120°.

(1 turn)/(3 turns) = 1/3 of the lap → 1/3× 360° = 120°
4STEP 4

Read off the clock position

A third of the 12-hour clock is 1/3× 12 = 4 hours clockwise from 12, landing at 4 o'clock — choice (C).

1/3× 12 hours = 4 hours from 12 → 4 o'clock → (C)
Answer
4 o' clock
The rolling-distance check alone would put the same point of the disk back in contact after 1/2 a lap (clock circumference is twice the disk's), i.e. at 6 o'clock — yet the arrow gets back up sooner, at 4 o'clock, because circling the clock adds an extra turn. That ordering (4 o'clock before 6 o'clock) is exactly what the coin-rotation effect predicts, so the answer is consistent rather than accidental. The 120° result also splits the clock into three equal 4-hour arcs, which matches the factor of 3 turns per lap. Choices (A) 2 o'clock and (B) 3 o'clock are too early for a full turn, and (D) 6 o'clock and (E) 8 o'clock would need more than one full turn.
💡Key takeaway

A rolling disk spins from two things at once, making 3 turns per lap — the arrow is back up after just the first, at 4 o'clock.

  • Arrow up means one full turn
  • Count turns for a whole lap
  • One turn is a third of the lap
  • Read off the clock position

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