AMC 10 · 2023 · #1
Grade 6 rate-ratioPick an answer.
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.
Draw the picture
They ride toward each other.
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 DiagramHow fast the gap closes
Facing each other, the speeds add.
When two things move toward each other, the gap shrinks at the sum of their speeds — a Grade 4 multi-step word-problem addition.
When two things move toward each other, the gap shrinks at the sum of their speeds.
▸ Why?
The gap changes at the difference of their motions, and moving in opposite directions makes that difference a sum.
▸ Why?
At a steady speed the distance covered is the speed multiplied by the time, so each rider's part is one product.
Find the meeting time
Divide the distance by that speed.
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 UnitsFind one rider's distance
Multiplying gives 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 12 problem only needs Grade 6 unit rates you already know — speeds add when two riders close in, then distance equals speed times time!