AMC10 2014 A

AMC10 2014 A · Q11

AMC10 2014 A · Q11. It mainly tests Linear equations, Interest / growth (simple).

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台电器，有三个优惠券，只能使用其中一个：优惠券1：标价至少\$50时，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的部分打8折（18% off）对于下列哪个标价，优惠券1提供的折扣金额大于优惠券2或优惠券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

Answer (C): Let $P>100$ be the listed price. Then the price reductions in dollars are as follows: Coupon 1: $\frac{P}{10}$ Coupon 2: $20$ Coupon 3: $\frac{18}{100}(P-100)$ Coupon 1 gives a greater price reduction than coupon 2 when $\frac{P}{10}>20$, that is, $P>200$. Coupon 1 gives a greater price reduction than coupon 3 when $\frac{P}{10}>\frac{18}{100}(P-100)$, that is, $P<225$. The only choice that satisfies these inequalities is \$219.95.

答案（C）：设标价为 $P>100$。则折扣金额（美元）如下：优惠券1：$\frac{P}{10}$ 优惠券2：$20$ 优惠券3：$\frac{18}{100}(P-100)$ 当 $\frac{P}{10}>20$（即 $P>200$）时，优惠券1的降价幅度大于优惠券2。当 $\frac{P}{10}>\frac{18}{100}(P-100)$（即 $P<225$）时，优惠券1的降价幅度大于优惠券3。唯一满足这些不等式的选项是 \\\$219.95。

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