AMC 8 · 2005 · #11
Grade 6 rate-ratioPick an answer.
AMC 8 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The two clerks apply the same two multipliers (1.06 for tax and 0.80 for discount) to the same starting price; only the order is swapped. Tool #11 (Find an Invariant) spots the unchanging quantity: when you multiply a string of numbers, reordering the factors never changes the product. That property — commutativity of multiplication — is the invariant, so Jack's total and Jill's total must be equal before we ever compute a dollar amount. Tool #4 (Introduce a Variable) lets us write both totals as products of 90, 1.06, and 0.80 in two orders so the invariance is visible at a glance. No arithmetic is required to pick the answer; the structure decides it.
Turn each percent change into one multiplier: adding 6% tax means × 1.06, taking 20% off means × 0.80.
A percent change is just one multiplication — turning each step into a clean factor is what makes the order question tractable.
6.RP.A.3Use Matrix LogicJack rings up textdollar 90, then × 1.06 (tax), then × 0.80 (discount): 90 × 1.06 × 0.80.
Each operation in order becomes one factor in a single product expression.
6.EE.A.2Use Matrix LogicJill swaps the order: textdollar 90, then × 0.80 (discount), then × 1.06 (tax): 90 × 0.80 × 1.06.
Same starting price, same two factors — only the order of the last two has changed.
6.EE.A.2Use Matrix LogicSame three numbers, so by commutativity the two products are equal and the difference = 0 → (C).
Once both totals are written as the same product of 90, 1.06, and 0.80, the commutative property makes them identical without computing the dollar amount.
6.EE.A.3Work BackwardsBoth clerks multiply textdollar 90 by the same two factors — 1.06 for tax and 0.80 for discount. The commutative property of multiplication says order does not matter, so the totals are equal and the difference is textdollar 0.