Competition · AMC preparation · step 4 of 4

AMC 8 · 2006 · #4

Grade 4 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

fraction-arithmeticmodular-arithmeticspatial-visualization identify-subproblemsmodular-arithmetic ↑ Prerequisites: fraction-arithmeticmodular-arithmetic

📏 Short solution 💡 3 insights 📊 Diagram

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Problem

A spinner starts pointing west. It is rotated clockwise 2

\frac{1}{4}

revolutions, then counterclockwise 3

\frac{3}{4}

revolutions. Which direction does it point at the end?

Pick an answer.

(A)

north

(B)

east

(C)

south

(D)

west

(E)

northwest

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

A compass rose is the natural picture for this problem — tool #1 (Draw a Diagram) lets us walk around the four cardinal points and watch quarter-turns happen. Tool #7 (Identify Subproblems) splits the work into two cleaner pieces: first combine the two rotations into a single net revolution, then apply only its fractional part to the starting direction. Whole revolutions can be thrown away because they bring the spinner back to where it was.

1STEP 1

Combine the two turns

Treat CCW as positive and CW as negative and subtract: the net is 1 $\frac{1}{2}$ revolutions CCW.

Net = +3 3/4 - 2 1/4 = (3-2) + (3/4-1/4) = 1 1/2 revolutions CCW

2STEP 2

Drop the whole revolutions

One full turn returns to the start, so 1 $\frac{1}{2}$ revolutions is just $\frac{1}{2}$ revolution CCW.

1 1/2 rev ≡ 1/2 rev CCW

3STEP 3

Turn the compass from West

From West, a half turn CCW walks West → South → East, which is choice (B).

West → {1/4 CCW} South → {1/4 CCW} East → (B)

Answer

east

Check by adding the fractional parts only. The clockwise 2

\frac{1}{4}

is the same as

\frac{1}{4}

CW (drop the 2 full spins): West → North. The counterclockwise 3

\frac{3}{4}

is the same as

\frac{3}{4}

CCW (drop the 3): from North, CCW goes North → West → South → East. Final direction is East, matching answer (B). The two methods agree, which is the cross-check we want.

💡Key takeaway

Cancel the full spins first, then walk the leftover fraction around the compass. Half a revolution counterclockwise from West lands on East — a Grade 4 mixed-number subtraction plus a compass walk.

Combine the two turns
Drop the whole revolutions
Turn the compass from West

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Combine the two turns

Drop the whole revolutions

Turn the compass from West

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