Competition · AMC preparation · step 4 of 4

AMC 8 · 1999 · #2

Grade 4 geometry-2d
Archetype Angle Chase in a Synthetic Figure └ Sub-archetype Fixed-Instrument Angle Composition
equal-spacingmental-arithmeticfraction-arithmetic identify-subproblems ↑ Prerequisites: fraction-arithmeticmulti-digit-arithmetic
📏 Short solution 💡 2 insights 📊 Diagram
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Problem
On a standard 12-hour clock, the hour hand points at the 10 and the minute hand points at the 12 when the time is 10/:00. What is the measure (in degrees) of the smaller of the two angles between the hands?

Pick an answer.

(A)
30
(B)
45
(C)
60
(D)
75
(E)
90

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

The clock face given in the problem is already a labeled circular diagram — Tool #1 (Draw a Diagram) lets us mark both hand positions on the dial and read the gap straight off, no formula needed. Tool #7 (Identify Subproblems) splits the work into two clean Grade 4 steps: first find the size of one sector between consecutive numbers, then count how many of those sectors sit between the 10 and the 12. Multiplying gives the answer, and the diagram confirms it is the smaller of the two possible angles.

1STEP 1

Find one sector's angle

The dial splits a full 360° turn into 12 equal wedges, so one number-to-number wedge is 360° ÷ 12 = 30°.

360°/12 = 30° per sector
2STEP 2

Count the sectors between the hands

At 10/:00 the minute hand is on the 12 and the hour hand on the 10; sweeping 10 → 11 → 12 clockwise crosses exactly 2 wedges.

sectors between 10 and 12 = 2
3STEP 3

Multiply to get the angle

Two 30° wedges give 2 × 30° = 60°; the other angle is 360° - 60° = 300°, so 60° is the smaller one — choice (C).

2 × 30° = 60° ( ＜ 300°) → (C)
Answer
60
Sanity check the answer against the diagram. The hands point at the 10 and the 12, only two numbers apart on the clock face — a small slice of the circle, nothing close to a half turn. A 90° angle would mean three sectors apart (like 12 and 3), and a 30° angle would mean just one sector apart (like 12 and 1). Two sectors must sit between those, giving 60°, which matches choice (C). The reflex angle on the other side is 360° - 60° = 300°, far larger, confirming 60° is the smaller angle the problem asks for.
💡Key takeaway

Each number-to-number gap on a clock is 30° (360° split into 12 equal pieces), and the 10 sits two gaps away from the 12 — so the smaller angle at 10 o'clock is 2 × 30° = 60°.

  • Find one sector's angle
  • Count the sectors between the hands
  • Multiply to get the angle

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