AMC 10 · 2018 · #2
Grade 6 arithmeticPick an answer.
The question asks for a maximum under a weight cap, so Tool #14 (Extreme Principle) drives everything: push toward the boundary and prove nothing past it is reachable. Tool #8 (Analyze the Units) sets up the comparison, because the pack is limited in pounds, so dollars per pound is the honest measure of a rock. That immediately gives a ceiling, and Tool #3 (Eliminate Possibilities) turns the ceiling into two dead answer choices. The key move is Tool #15 (Organize Information in More Ways): rewriting all three prices with one rule, 3w-1, replaces a messy three-way choice with two clean goals. Tool #2 (Make a Systematic List) then checks all four cases so the clever argument is backed by a complete search.
Measure each rock per pound
Measure each rock by value per pound.
The pack is limited by pounds, not by rock count, so the fair way to rank rocks is dollars per pound.
The pack is limited by pounds, not by rock count, so the fair way to rank rocks is by value per pound.
▸ Why?
Each kind of rock carries a fixed value for a fixed weight, so its worth is a rate per pound.
▸ Why?
Once those rates are ranked, the best rate stays best however the pack is filled.
Set a ceiling on the total
The best rate gives a ceiling on the total.
If every pound is worth at most 2.8 dollars, eighteen pounds cannot break 50.4.
6.RP.A.3Eliminate PossibilitiesWrite all three prices with one rule
All three prices obey the same rule.
One rule covering all three prices turns a three-way choice into two simple goals: heavy load, few rocks.
6.EE.A.2Organize Information In More WaysPush weight up, rock count down
Fill the weight and use fewer rocks.
Each extra rock shaves a dollar off the flat three-dollars-per-pound rate, so fill the pack with as few rocks as possible.
6.EE.B.5Extreme PrincipleRule out a lighter pack
A lighter pack can never win.
Giving up a pound costs three dollars and almost never saves a rock, so a lighter pack is a worse pack.
6.EE.B.5Extreme PrincipleConfirm by listing every case
Checking every case gives 50 dollars.
Only four cases exist, so listing them all turns a clever argument into a guarantee.
4.OA.A.3Make A Systematic ListEvery rock here is worth 3 times its weight minus 1 dollar, so fill the pack right to the limit using as few rocks as you can.
- Measure each rock per pound
- Set a ceiling on the total
- Write all three prices with one rule
- Push weight up, rock count down
- Rule out a lighter pack
- Confirm by listing every case