AMC 8 · 2003 · #20

Grade 5 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

rateangle-sum-trianglefraction-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Medium solution 💡 3 insights

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Problem

At 4:20 PM, what is the measure of the acute angle between the hour hand and the minute hand of a standard 12-hour clock?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

A clock face is already a labeled circular diagram — Tool #1 (Draw a Diagram) lets us mark each hand's position on the dial and read the gap directly instead of juggling formulas. The trap is assuming the hour hand sits exactly on the 4 at 4{:}20; the diagram makes the small drift visible. Tool #9 (Solve an Easier Problem) supports it by splitting the question into two simpler sub-problems we already know how to do: first find where the minute hand is at 4{:}20, then find where the hour hand is, and subtract. Direct rates (6° per minute for the minute hand, 0.5° per minute for the hour hand) finish each sub-problem in one multiplication.

1STEP 1

Set the dial's scale: the 360° face splits into 12 equal sectors, so consecutive numbers sit 30° apart.

\frac{360°}{12}

= 30° between consecutive numbers

2STEP 2

The minute hand sweeps 6° per minute, so at 20 minutes it reaches 120° — exactly on the 4.

minute hand = 20 × 6° = 120°

3STEP 3

The hour hand creeps 0.5° per minute, so from 4{:}00 it drifts 10° past the 4 to 130°.

hour hand = 120° + 20 × 0.5° = 120° + 10° = 130°

4STEP 4

Both hands now have dial addresses, so the gap is one subtraction: 130° - 120° = 10°.

|130° - 120°| = 10° → (D)

Answer

The 10° answer matches the dial picture. At 4{:}20 the minute hand is exactly on the 4 and the hour hand sits a short way past the 4 toward the 5. The full gap from 4 to 5 is 30°, and 20 minutes is one-third of an hour, so the hour hand has covered one-third of that gap:

\frac{1}{3}

× 30° = 10°. That matches choice (D) and rules out (A) 0 (would require both hands on the same spot), (B) 5 and (C) 8 (too small for a one-third drift past the 4), and (E) 12 (too large).

💡Key takeaway

At 4{:}20 the minute hand is exactly on the 4, but the hour hand has already drifted one-third of the way toward the 5 — and one-third of the 30° gap between consecutive numbers is the 10° answer.

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