AMC 8 · 1999 · #4

Grade 5 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

graph-readingratemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A line graph shows the distance (in miles) biked by Alberto and Bjorn over time (in hours). After 4 hours, how many more miles has Alberto biked than Bjorn?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

How to solve

Strategy Draw a Diagram

The graph is the problem. Tool #1 (Draw a Diagram) here means reading the given diagram carefully: find the vertical line at hour 4, see where Alberto's line and Bjorn's line cross it, and read each height off the miles axis. Once both numbers are read, the question "how many more" is a single subtraction. No algebra and no extra tool are needed — picking the wrong tool (like Algebra) would only add work.

1STEP 1

Mark the vertical column at hour 4 — both riders' readings happen in that single column.

Column at x = 4 hours

2STEP 2

In that column Alberto's line meets the gridline for 60 miles.

Alberto at x = 4: y = 60 miles

3STEP 3

In the same column Bjorn's line meets the gridline for 45 miles.

Bjorn at x = 4: y = 45 miles

4STEP 4

"How many more" means subtract the smaller from the bigger: 60 - 45 = 15 miles, so the answer is (A).

60 - 45 = 15 → (A)

Answer

Alberto's line is steeper than Bjorn's, so he must be ahead at any positive time — the answer should be a positive number, which 15 is. Also, the gap is exactly one gridline (15 miles) at hour 4, and looking at the picture, Alberto's line sits one gridline above Bjorn's at x = 4. That visual gap matches the computed answer, so 15 is consistent with both the numbers and the picture.

💡Key takeaway

When a problem hands you a graph, the graph is doing most of the work. Lock onto the x-value the question asks about (4 hours), read each line's height off the y-axis (60 and 45), and subtract. One gridline of difference here is exactly 15 miles — answer (A).

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