AMC 10 · 2023 · #2
Grade 6 arithmeticPick an answer.
There are only five candidate prices and each one is easy to push through the "discount then tax" pipeline. Tool #3 (Eliminate) frames the work: plug each choice in and keep only the ones that produce ≤ 43. Tool #6 (Guess and Check) is the engine for each test, and the largest surviving candidate is the answer. Tool #8 (Units) keeps the percentages honest — in, × 0.8, × 1.075, $ out. Algebra (#13) also works but is heavier than just multiplying 5 numbers.
Combine the two factors
Discount and tax multiply.
Each percent is a Grade 6 rate; chaining two of them is the same as multiplying the rates.
Chaining two percent changes is the same as multiplying the two rates together.
▸ Why?
A percent is a count out of a hundred, so each change is one fixed multiplier.
▸ Why?
Applying one multiplier and then another scales the amount by their product.
Set up the inequality
The register total must be within budget.
Once two percent moves collapse into one multiplier, plugging in answer choices is a single multiplication each.
6.RP.A.3Eliminate PossibilitiesCheck the choices
Plug the choices in directly.
Multiplying 0.86 · 50 as 86 · 50 ÷ 100 = 4300 ÷ 100 = 43 is direct Grade 5 decimal arithmetic.
5.NBT.B.7Guess And CheckRead the answer
The largest affordable price is 50 dollars.
Among five candidates, the largest one that still satisfies 0.86x ≤ 43 is the answer — pure Grade 6 substitute-and-check.
6.EE.B.5Eliminate PossibilitiesThis AMC 12 problem only needs the Grade 6 idea that "20% off then 7.5% tax" is the same as multiplying by 0.86 — once the rate is collapsed, just plug in each choice.