2014年AMC10竞赛A第11题
$$2014-AMC10A-11$$  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 priceif 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 $$3$$?  打算购买电器的客户有三张优惠券，只能使用其中一张：  优惠券 $$1$$：如果标价至少为 ＄$$50$$，则比标价低 10\%。  优惠券$$2$$：如果标价至少为$$100$$，则标价可减$20。  优惠券 $$3$$：标价超过$$100$$的部分可享受 $$18$$\% 的折扣。  对于以下哪个列出的价格，优惠券 $$1$$ 比优惠券 $$2$$ 或优惠券 $$3$$ 的降价幅度更大？
课内体系->知识点->函数的应用->函数的实际应用->一次函数模型 美国AMC10/12->Knowledge Point->Algebra->Application->Equation Word Problems
Let the listed price be $$x$$. Since all the answer choices are above ＄$$100$$, we can assume $$x\textgreater{} 100$$. Thus the discounts after the coupons are used will be as follows:  Coupon $$1$$: $$x\times10$$\%$$=$$$$0.1 x$$.  Coupon $$2$$: $$20$$.  Coupon $$3$$: $$18$$\%$$\times (x-100)=0.18 x-18$$.  For coupon $$1$$ to give a greater price reduction than the other coupons, we must have $$0.1 x\textgreater20 \Rightarrow x\textgreater200$$ and $$0.1x\textgreater0.18 x-18 \Rightarrow0 .08 x\textless18 \Rightarrow x\textless225$$.  From the first inequality, the listed price must be greater than ＄$$200$$, so answer choices $$(\rm A)$$ and $$(\rm B)$$ are eliminated.  From the second inequality, the listed price must be less than ＄$$225$$, so answer choices $$(\rm D)$$ and $$(\rm E)$$ are eliminated.  The only answer choice that remains is $$(\rm C)$$＄$$219.95$$.  For coupon $$1$$ to be the most effective, we want $$10$$\% of the price to be greater than $$20$$. This clearly ~occurs if the value is over $$200$$. For coupon $$1$$ to be more effective than coupon $$3$$, we want to minimize the value over $$200$$, so $$(\rm C)$$＄$$219.95$$ is the smallest number over $$200$$. 