Competition · AMC preparation · step 4 of 4

AMC 8 · 2009 · #3

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rategraph-readingpattern-recognitionunit-conversion pattern-recognitiondimensional-analysis ↑ Prerequisites: multi-digit-arithmeticunit-conversion

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A graph shows Suzanna's biking distance over time: at 5, 10, 15, 20 minutes she has gone 1, 2, 3, 4 miles respectively. If she keeps that same constant speed for a full half hour (30 minutes), how many miles will she have ridden in total?

Pick an answer.

(A)

(B)

5.5

(C)

(D)

6.5

(E)

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

The graph hands us a clean numerical pattern: every extra 5 minutes adds exactly 1 mile. Tool #5 (Look for a Pattern) lets us extend that rule from the graph's last point at 20 min straight up to 30 min without any algebra. Tool #8 (Analyze the Units) is the safety check — we are mixing "minutes" with "half an hour", so we make sure both are in the same unit before extending the pattern.

1STEP 1

Convert hours to minutes

Half an hour is 30 minutes — the same unit the graph uses on its x-axis.

1/2 hour × (60 min)/(1 hour) = 30 min

2STEP 2

Find the pattern in the points

The four plotted points give a clean rule: every +5 minutes adds exactly +1 mile.

5 → 1, 10 → 2, 15 → 3, 20 → 4 (every +5 min gives +1 mi)

3STEP 3

Extend the pattern to 30

Extend past the graph: 20 min → 4 mi, then two more +5-min steps reach 30 min → 6 miles.

20 → 4, 25 → 5, 30 → 6

4STEP 4

Check with the rate

Rate check: $\frac{1}{5}$ mile per minute × 30 min = 6 miles, confirming choice (C).

1/5 mi/min × 30 min = 6 mi → (C)

Answer

At 20 minutes she has 4 miles, so her speed averages

\frac{4}{20}

= 0.2 mile per minute, which is 0.2 × 60 = 12 mph — a realistic easy bike pace. In 30 minutes at 12 mph she covers

\frac{1}{2}

× 12 = 6 miles. That matches the answer (C) 6 and rules out the close decoys 5.5, 6.5.

💡Key takeaway

If you can read "every 5 minutes she gains 1 mile" off a graph, you can already finish this AMC 8 problem — that's Grade 4 pattern-spotting plus a quick Grade 6 rate check.

Convert hours to minutes
Find the pattern in the points
Extend the pattern to 30
Check with the rate

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Convert hours to minutes

Find the pattern in the points

Extend the pattern to 30

Check with the rate

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