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

The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about 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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Horizontal axis: hours, labeled $0, 1, 2, 3, 4, 5$; Vertical axis: miles, with gridlines at $0, 15, 30, 45, 60, 75$ (each gridline = $15$ miles); Alberto's line and Bjorn's line both start at $(0, 0)$ and rise to the right; Answer choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$

Unknowns: Alberto's miles minus Bjorn's miles at hour $4$

Understand

Restated: 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?

Plan

Primary tool: #1 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.

Execute — Answer: A

#1 Draw a Diagram 5.G.A.2 Step 1

Mark the column at hour $4$.
On the horizontal axis, locate the tick labeled $4$ and follow a vertical line upward.
This is the column where both readings happen.

$$\text{Column at } x = 4 \text{ hours}$$

💡 Reading a graph starts by fixing the $x$-value the question asks about. Everything else is a height in that single column.

#1 Draw a Diagram 5.G.A.2 Step 2

Read Alberto's height.
Alberto's line in that column meets the gridline labeled $60$, so Alberto has biked $60$ miles after $4$ hours.

$$\text{Alberto at } x = 4: \; y = 60 \text{ miles}$$

💡 The fourth gridline up sits at $4 \times 15 = 60$, which matches Alberto's line at hour $4$.

#1 Draw a Diagram 5.G.A.2 Step 3

Read Bjorn's height.
Bjorn's line in the same column meets the gridline labeled $45$, so Bjorn has biked $45$ miles after $4$ hours.

$$\text{Bjorn at } x = 4: \; y = 45 \text{ miles}$$

💡 The third gridline up sits at $3 \times 15 = 45$, which matches Bjorn's line at hour $4$.

#1 Draw a Diagram 4.OA.A.2 Step 4

Subtract to find how many more miles Alberto has biked.
The question "how many more" is a comparison: bigger minus smaller.

$$60 - 45 = 15 \;\Rightarrow\; \textbf{(A)}$$

💡 Grade 4 "additive comparison" — to find how much more one quantity is than another, you subtract.

[1] #1 5.G.A.2 Mark the column at hour $4$. On the horizontal axis, locate the tick labeled $4$

[2] #1 5.G.A.2 Read Alberto's height. Alberto's line in that column meets the gridline labeled

[3] #1 5.G.A.2 Read Bjorn's height. Bjorn's line in the same column meets the gridline labeled

[4] #1 4.OA.A.2 Subtract to find how many more miles Alberto has biked. The question "how many m

Review

Reasonableness: 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.

Alternative: Tool #5 (Find a Pattern): compute each rider's speed from the graph. Alberto reaches $60$ miles in $4$ hours, so his pace is $60 \div 4 = 15$ miles per hour. Bjorn reaches $45$ miles in $4$ hours, so his pace is $45 \div 4 = 11.25$ miles per hour. The hourly gap is $15 - 11.25 = 3.75$ miles, and after $4$ hours the total gap is $3.75 \times 4 = 15$ miles — same answer (A).

CCSS standards used (min grade 5)

5.G.A.2 Represent real-world and mathematical problems by graphing points in the first quadrant of the coordinate plane (Reading each rider's distance at hour $4$ as the $y$-coordinate of a point on a coordinate plane, using the gridline spacing of $15$ miles.)
4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, distinguishing multiplicative from additive comparison (Recognizing "how many more" as an additive comparison and computing $60 - 45 = 15$.)

⭐ 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).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 5)

More like this