AMC 10 · 2012 · #5
Grade 6 algebraPick an answer.
All three comparisons multiply, and they chain: each one rewrites the next fruit in terms of the one before it. Tool #4 (Introduce a Variable) names the count at the start of that chain, the blueberries, and Tool #13 (Convert to Algebra) turns each English sentence into an equation, so the whole salad becomes one expression in a single letter and the total 280 becomes one equation. The direction of each comparison is the only place this problem can break, so each translation gets checked against a tiny example. Finally, solving equations only shows what the counts must be, so the salad is built and re-checked against all four facts, and Tool #3 (Eliminate Possibilities) clears the choices that are other fruits' counts.
Translate each comparison exactly
Each comparison becomes one exact equation.
"Four times as many X as Y" always makes X the bigger pile, no matter which word comes first.
4.OA.A.1Convert To AlgebraMeasure every fruit in blueberries
Every count is then written in one letter.
When every comparison points back to the same fruit, that one fruit's count controls the whole salad.
When every comparison points back to the same fruit, that one fruit's count controls the whole salad.
▸ Why?
Each comparison is a fixed multiplier, so every count is that one count scaled by a known factor.
▸ Why?
Chaining the comparisons passes the description along, so each fruit ends up written in the same terms.
Let the total pin down b
The total pins that letter at 8.
Once the whole salad is one multiple of b, the given total is a single division away from b.
6.EE.B.7Introduce A VariableRead off the cherry count
Reading off the wanted count gives 64.
The chain built cherries out of blueberries, so knowing the blueberries hands you the cherries at once.
6.EE.A.2Introduce A VariableBuild the salad, clear the traps
Rebuilding the whole set confirms it, choice (D).
An answer you can actually build out of whole fruit is proved, not merely implied.
4.OA.B.4Eliminate PossibilitiesWhen every clue compares one fruit to another, rewrite the whole salad in terms of a single fruit — here it comes to 35 blueberries' worth — and let the total finish the job: 280 ÷ 35 = 8 blueberries, so 64 cherries.
- Translate each comparison exactly
- Measure every fruit in blueberries
- Let the total pin down b
- Read off the cherry count
- Build the salad, clear the traps