Competition · AMC preparation · step 4 of 4
AMC 8 · 1999 · #4
Grade 5 rate-ratio
Pick an answer.
AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the 4-hour mark
Mark the vertical column at hour 4 — both riders' readings happen in that single column.
Reading a graph starts by fixing the x-value the question asks about. Everything else is a height in that single column.
5.G.A.2Draw A DiagramRead Alberto's distance
In that column Alberto's line meets the gridline for 60 miles.
The fourth gridline up sits at 4 × 15 = 60, which matches Alberto's line at hour 4.
5.G.A.2Draw A DiagramRead Bjorn's distance
In the same column Bjorn's line meets the gridline for 45 miles.
The third gridline up sits at 3 × 15 = 45, which matches Bjorn's line at hour 4.
5.G.A.2Draw A DiagramSubtract the two distances
"How many more" means subtract the smaller from the bigger: 60 - 45 = 15 miles, so the answer is (A).
Grade 4 "additive comparison" — to find how much more one quantity is than another, you subtract.
After 4 hours, the number of extra miles Alberto has biked over Bjorn equals Alberto's 60 miles minus Bjorn's 45 miles.
▸ Why?
Alberto's line at hour 4 sits on the fourth gridline up, and the axis rises in equal 15-mile steps, so his height is four of those steps: 4 × 15 = 60 miles.
▸ Why?
Bjorn's line at hour 4 sits on the third gridline up, three of the same equal 15-mile steps, so his height is 3 × 15 = 45 miles.
▸ Why?
"How many more" asks for the extra part, and Alberto's total is Bjorn's total plus that extra, so the extra is found by taking 45 away from 60.
▸ Why?
Alberto's 60 miles divides cleanly, with no gap or overlap, into Bjorn's 45 miles and the leftover extra, so those two parts add back to 60.
▸ Why?
Since 45 plus the extra makes 60, subtracting undoes that addition and pulls the extra out on its own.
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).
- Find the 4-hour mark
- Read Alberto's distance
- Read Bjorn's distance
- Subtract the two distances
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