Competition · AMC preparation · step 4 of 4

AMC 10 · 2023B · #2

Grade 6 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In
percentageratedecimal-arithmeticlinear-equations-one-var dimensional-analysisguess-and-checkidentify-subproblems ↑ Prerequisites: percentagedecimal-arithmetic
📏 Short solution 💡 2 insights
Problem
Shoes are listed at $x, marked down 20%, then taxed 7.5% at the register. Carlos has exactly $43. Find the largest original price x he can afford.

Pick an answer.

(A)
$\$46$
(B)
$\$50$
(C)
$\$48$
(D)
$\$47$
(E)
$\$49$

AMC 10 2023 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Eliminate Possibilities

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.

1STEP 1

Turn the percents into multipliers

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.

0.8 × 1.075 = 0.86
2STEP 2

Write the budget condition

The register charge is 0.86x, so the budget condition is 0.86x ≤ 43 — plug in choices from largest down.

register = 0.86 · x ; 0.86 · x ≤ 43
3STEP 3

Test the largest choice

Test the largest choice x = 50: 0.86 · 50 = 43, hitting the budget exactly — so it is affordable.

0.86 · 50 = 43.00 ≤ 43 ✓
4STEP 4

Rule out anything larger

No listed price exceeds 50, and any larger x would push 0.86x past 43 — so 50 is the max and the answer is (B).

x = 50 → (B)
Answer
$50
Magnitude check: a 20% cut on $50 gives $40, and a 7.5% tax on $40 is $3, total $43 — matches Carlos's wallet exactly. Smaller choices ($46, $47, $48, $49) would also fit but aren't the most expensive, and the problem asks for the maximum, so (B) is correct.
💡Key takeaway

This 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.

  • Turn the percents into multipliers
  • Write the budget condition
  • Test the largest choice
  • Rule out anything larger

A parent dashboard for the family lives at sensimlab.com.

More like this

Same problem type. Tags show what each one shares with this problem.