AMC12 2023 B

AMC12 2023 B · Q2

AMC12 2023 B · Q2. It mainly tests Percent, Interest / growth (simple).

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?

卡洛斯去体育用品店买跑鞋。跑鞋打折，每双鞋的价格降低 $20\%$ 。卡洛斯还知道他需要支付折扣价的 $7.5\%$ 销售税。他有 $\$$43 。他能买得起的最贵的鞋的原价（打折前）是多少？

(A) \$46 \$46

(B) \$50 \$50

(D) \$47 \$47

(E) \$49 \$49

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let the price originally be x. Then, after a 20 percent discount, the price is now x−15x=45x. From the discounted price 45x, we now take 7.5100 of 45x and add it to 45x, giving us 45x+(7.5100)(45x)=45x+30500x=45x+350x=4350x. Now we write the inequality 43/50x≤43 and multiply by 5043 on both sides to get x≤50. We want the greatest original price, which would be $50 or option choice B.

设原价为 $x$ 。打折 $20\%$ 后，价格变为 $x - 0.2x = 0.8x$ 。在折扣价 $0.8x$ 的基础上，加 $7.5\%$ 的销售税，即 $0.8x + 0.075 \times 0.8x = 0.8x + 0.06x = 0.86x$ 。不等式 $0.86x \leq 43$ ，解得 $x \leq 43 / 0.86 = 50$ 。最大的原价是 $\$$50，即选项 B 。

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