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2023 AMC 10B Problem 2

2023 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 2 min · position P1-10

Carlos went to a sports store to buy running shoes. Running shoes were on sale, with prices reduced by $20\%$ on every pair of shoes. Carlos also knew that he had to pay a $7.5\%$ sales tax on the discounted price. He had $43$ dollars. What is the original (before discount) price of the most expensive shoes he could afford to buy?

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

Answer

Key insight

A 20% discount then 7.5% tax multiplies the price by 0.8 times 1.075 = 0.86, so the budget allows 43/0.86 = 50 dollars.

Solution

Let $p$ be the original price. After the $20\%$ discount the shoes cost $0.8p$ ; adding $7.5\%$ tax gives
$1.075 \cdot 0.8p = 0.86p.$
Carlos needs $0.86p \le 43$ , so
$p \le \frac{43}{0.86} = 50.$
The most expensive pair he can afford is $\$50$ : indeed $\$50$ drops to $\$40$ on sale, and $\$40$ plus $7.5\%$ tax is $\$43$ exactly.

The answer is $\boxed{\textbf{(B)}\ \$50}$ .

Why this works

Successive percent changes are successive multiplications by $(1 \pm r)$ , and multiplication is commutative, so the order of discount and tax does not matter; only the product $0.8 \times 1.075$ does. Once the price is a single multiple of $p$ , the budget becomes a one-line inequality.

Alternative approach

Test the choices from the top: $\$50 \to \$40 \to \$43$ fits exactly. Any larger price already exceeds the budget, so $50$ is the maximum.

The trap

Combining the two percents additively (a net 12.5% cut) and dividing 43 by 0.875, which points to 49.

Common mistakes

Combining the two percents additively (a net 12.5% cut) and dividing 43 by 0.875, which points to 49.
Applying the tax to the original price instead of the discounted price, or subtracting the tax rather than adding it.

Techniques

Set up the equation/formula and compute; no special trick needed