AMC 10 · 2023 · #2
Grade 6 arithmeticPick an answer.
AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Turn each percent into a multiplier and compose: 20% off is × 0.8, 7.5% tax is × 1.075, so sticker → register is 0.86.
Each percent is a Grade 6 rate; chaining two of them is the same as multiplying the rates.
6.RP.A.3Analyze The UnitsThe register charge is 0.86x, so the budget condition is 0.86x ≤ 43 — plug in choices from largest down.
Once two percent moves collapse into one multiplier, plugging in answer choices is a single multiplication each.
6.RP.A.3Eliminate PossibilitiesTest the largest choice x = 50: 0.86 · 50 = 43, hitting the budget exactly — so it is affordable.
Multiplying 0.86 · 50 as 86 · 50 ÷ 100 = 4300 ÷ 100 = 43 is direct Grade 5 decimal arithmetic.
5.NBT.B.7Guess And CheckNo listed price exceeds 50, and any larger x would push 0.86x past 43 — so 50 is the max and the answer is (B).
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 10 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.