AMC 10 · 2023 · #1
Grade 6 rate-ratioPick an answer.
AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The problem describes positions on a road and motion along that road — Tool #1 (Draw a Diagram) is the natural lead. A simple labeled segment from A to B with arrows showing the two bikers makes the structure obvious: the gap of 45 miles closes at the combined speed 18 + 12 = 30 mph. Tool #8 (Analyze the Units) is the verification companion — tracking miles, mph, and hours through every multiplication makes sure the final number is in miles, which is what the problem asks for. Algebra (Tool #13) would also work but the diagram-plus-units path is faster and more visual for a Grade 6 rate concept.
Sketch the 45-mile road with A left and B right, then draw Alicia's and Beth's arrows closing the gap from both ends.
Drawing the road with two arrows turns a word problem into a closing-gap picture — the Grade 6 "unit rate" idea that miles-per-hour is how fast the gap shrinks.
6.RP.A.2Draw A DiagramThe two arrows close the gap together, so the speeds add — the combined closing speed is 30 mph.
When two things move toward each other, the gap shrinks at the sum of their speeds — a Grade 4 multi-step word-problem addition.
4.OA.A.3Draw A DiagramDivide the 45-mile gap by the 30-mph closing speed; miles over miles-per-hour leaves 1.5 hours until they meet.
Distance ÷ rate = time — the Grade 6 rate-reasoning that miles cancel and leave hours, exactly the unit the next step needs.
6.RP.A.3Analyze The UnitsThe meeting point is where Alicia rode in that time, so multiply her 18 mph by 1.5 hours to get her distance from A: 27 miles.
Rate × time = distance — the same Grade 6 unit-rate move in reverse; mph × hr cancels to mi, the requested unit.
6.RP.A.3Analyze The UnitsThis AMC 10 problem only needs Grade 6 unit rates you already know — speeds add when two riders close in, then distance equals speed times time!