2023 AMC 10B 第 2 题

Question

Carlos 去体育用品店买跑鞋。跑鞋正在促销，每双价格降低

20\%

。Carlos 还知道，他必须按折后价支付

7.5\%

的销售税。他有

43

美元。他能买得起的最贵跑鞋的原价（折扣前价格）是多少？

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?

\$46

\$50

\$48

\$47

\$49

20\%

折扣把价格乘以

0.8

；

7.5\%

税把价格乘以

1.075

。

A

20\%

discount multiplies the price by

0.8;

a

7.5\%

tax multiplies by

1.075

令最终花费

0.8 \times 1.075 \times P

等于

43

，再解

P

。

Set the final cost

0.8 \times 1.075 \times P

equal to

43

and solve for

P

设原价为

P

。折扣和税后，Carlos 需要支付

0.8P \times 1.075 = 0.86P

。他能负担当且仅当

0.86P \le 43

，所以

P \le 50

。因此最贵的原价为

\$50

。所以正确答案是 B。

Let

P

be the original price. After the discount and tax, Carlos pays

0.8P \times 1.075 = 0.86P.

He can afford it when

0.86P \le 43,

which means

P \le 50.

So the priciest shoes he can swing start at

\$50.

Therefore, the answer is B.

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