AMC12 2014 A

AMC12 2014 A · Q8

AMC12 2014 A · Q8. It mainly tests Linear inequalities, Percent.

A customer who intends to purchase an appliance has three coupons, only one of which may be used: Coupon 1: $10\%$ off the listed price if the listed price is at least $\$50$ Coupon 2: $\$ 20$ off the listed price if the listed price is at least $\$100$ Coupon 3: $18\%$ off the amount by which the listed price exceeds $\$100$ For which of the following listed prices will coupon $1$ offer a greater price reduction than either coupon $2$ or coupon $3$?

一位顾客打算购买一件电器，有三个优惠券，只能使用其中一个：优惠券1：标价至少$\$50$时，标价9折（10\% off）。优惠券2：标价至少$\$100$时，减$\$ 20$。优惠券3：标价超过$\$100$的部分的18\%折扣。对于下列哪个标价，优惠券1提供的折扣大于优惠券2或3？

(A) \$179.95 \$179.95

(B) \$199.95 \$199.95

(D) \$239.95 \$239.95

(E) \$259.95 \$259.95

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let the listed price be $x$. Since all the answer choices are above $\$100$, we can assume $x > 100$. Thus the discounts after the coupons are used will be as follows: Coupon 1: $x\times10\%=.1x$ Coupon 2: $20$ Coupon 3: $18\%\times(x-100)=.18x-18$ For coupon $1$ to give a greater price reduction than the other coupons, we must have $.1x>20\implies x>200$ and $.1x>.18x-18\implies.08x<18\implies x<225$. The only choice that satisfies such conditions is $\boxed{\textbf{(C)}\ \$219.95}$

设标价为$x$。所有选项均超过$\$100$，故$x > 100$。各优惠券折扣为：优惠券1：$x\times10\%=.1x$ 优惠券2：$20$ 优惠券3：$18\%\times(x-100)=.18x-18$ 优惠券1折扣最大需$.1x>20\implies x>200$且$.1x>.18x-18\implies.08x<18\implies x<225$。唯一满足的是$\boxed{\textbf{(C)}\ \$219.95}$

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