AMC 10 · 2012 · #7
Grade 5 patternrate-ratioPick an answer.
The string repeats with period 5, so Tool #5 (Look for a Pattern) turns "which light is the 21st red?" into a division question instead of a listing question. Tool #1 (Draw a Diagram) fixes the one thing that sinks most attempts: distance is counted in gaps between lights, not in lights. Tool #9 (Solve an Easier Related Problem) supplies the cross-check — work out the tiny case of moving forward by just two reds, then repeat it. Tool #8 (Analyze the Units) keeps the inches-to-feet conversion honest at the end.
Distance counts gaps, not lights
Distance counts the gaps, not the lights.
Lights are fenceposts and the 6 inches are the fence panels between them, so it is the panels you count.
Lights are fenceposts and the spacings are the panels between them, so it is the panels you count.
▸ Why?
Each gap is one space between two neighbouring lights, so there is always one fewer gap than light.
▸ Why?
The lights are set at the same fixed spacing, so the distance is that step counted once per gap.
Locate the two red lights
Each light is placed by its block.
Two reds per block of five means the red count and the block count run in lockstep, so you can jump straight to block 11 instead of listing lights.
4.OA.C.5Look For A PatternCross-check the span of 45
A second count confirms the span.
Every two reds you pass costs exactly one full block of five lights, so nine repeats of that small move covers the whole trip.
4.OA.C.5Solve An Easier Related ProblemTurn 45 gaps into inches
Those gaps come to a fixed length.
Equal gaps mean one multiplication replaces adding 6 inches forty-five times.
4.NBT.B.5Analyze The UnitsConvert to feet and answer
Converting gives 22.5, choice (E).
Feet are bigger units than inches, so the number has to shrink — dividing by 12 is the only move that makes sense.
5.MD.A.1Analyze The UnitsNumber the lights first, then count the gaps between the two you care about — with a block that repeats every 5 lights and 2 reds inside it, jumping ahead 18 reds always means jumping ahead exactly 45 lights.
- Distance counts gaps, not lights
- Locate the two red lights
- Cross-check the span of 45
- Turn 45 gaps into inches
- Convert to feet and answer