2023 AMC 10B Problem 2

Start with a concrete price to see what the two changes do, then run it backwards from Carlos's $\$43$.

1. Try a $\$100$ pair. The $20\%$ discount takes off $\$20$, leaving $\$80$. The tax is $7.5\%$ of $\$80$, which is $\$6$. He would pay $\$86$.
2. So Carlos pays $86\%$ of the original price. In multiplication form, a $20\%$ discount means "times $0.8$" and a $7.5\%$ tax means "times $1.075$", and
$$0.8 \times 1.075 = 0.86.$$
3. Use the budget. Call the original price $p$. Carlos pays $0.86p$, and he has $\$43$, so he needs $0.86p \le 43$. The largest price allowed is
$$p = \frac{43}{0.86} = \frac{4300}{86} = 50.$$
4. Check. A $\$50$ pair is $\$40$ on sale. Tax is $7.5\%$ of $\$40$, which is $\$3$. Total: $\$43$, exactly his money.

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

Each percent change multiplies the price by a number: a discount of $r\%$ multiplies by $1 - \frac{r}{100}$, and a tax of $r\%$ multiplies by $1 + \frac{r}{100}$. Two changes in a row multiply those numbers together; you cannot just add or subtract the percents, because the tax is taken on the smaller, discounted price.

Work straight from the answer choices. The biggest choice is $\$50$: on sale it is $\$40$, and with tax it is $\$43$, which Carlos can just afford. Since that is the biggest choice and it fits, it is the answer.