AMC10 2010 B

AMC10 2010 B · Q11

AMC10 2010 B · Q11. It mainly tests Linear equations, Percent.

A shopper plans to purchase an item that has a listed price greater than \$100 and can use any one of three coupons. Coupon A gives 15\% off the listed price, Coupon B gives \$30 off the listed price, and Coupon C gives 25\% off the amount by which the listed price exceeds \$100. Let $x$ and $y$ be the smallest and largest prices, respectively, for which Coupon A saves at least as many dollars as Coupon B or C. What is $y-x$?

一位购物者计划购买一件标价大于 $100$ 的商品，可以使用以下三种优惠券中的任意一种。优惠券 A 打八五折（15% off），优惠券 B 减 $30$，优惠券 C 对标价超过 $100$ 的部分打七五折（25% off）。设 $x$ 和 $y$ 分别为优惠券 A 节省的金额至少与 B 或 C 一样多的最小和最大价格。求 $y-x$？

(A) 50 50

(B) 60 60

(D) 80 80

(E) 100 100

Answer

Correct choice: (A)

正确答案：（A）

Solution

Let $p$ dollars be the purchase price of the stem. The savings provided by Coupon A, B, and C respectively are $0.15p$, 30, and $0.25(p-100)$. Coupon A saves at least as much as Coupon B if $0.15p\ge30$, so $p\ge200$. Coupon A saves at least as much as Coupon C if $0.15p\ge0.25(p-100)$, so $p\le250$. Therefore $x=200$, $y=250$, and $y-x=50$.

设购买价格为 $p$ 美元。优惠券 A、B、C 分别节省 $0.15p$、$30$ 和 $0.25(p-100)$ 美元。优惠券 A 至少与 B 一样多当且仅当 $0.15p\ge30$，即 $p\ge200$。优惠券 A 至少与 C 一样多当且仅当 $0.15p\ge0.25(p-100)$，即 $p\le250$。因此 $x=200$，$y=250$，$y-x=50$。

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