Competition · AMC preparation · step 4 of 4
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 percents into multipliers
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.3Introduce A VariableWrite Jack's total
Jack rings up $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.2Introduce A VariableWrite Jill's total
Jill swaps the order: $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.2Introduce A VariableCompare the two products
Same 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.
The total Jack rings up equals the total Jill rings up.
▸ Why?
Each total is the product of the same three numbers — the $ 90 price, the tax factor 1.06, and the discount factor 0.80 — and the two clerks differ only in the order they multiply them.
▸ Why?
Regrouping the product lets us pull the shared 90 to the front and treat the tax and discount factors as one bundle, so each clerk's total is 90 times that bundle.
▸ Why?
Inside that bundle Jack forms 1.06 × 0.80 and Jill forms 0.80 × 1.06, and reversing the order of two multiplied numbers leaves the product unchanged.
Both clerks multiply 0.
- Turn percents into multipliers
- Write Jack's total
- Write Jill's total
- Compare the two products
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