Question

一位打算购买电器的顾客有三张优惠券，但只能使用其中一张：

优惠券 $1$ ：标价至少为 $\$50$ 时，减价 $10\%$

优惠券 $2$ ：标价至少为 $\$100$ 时，减 $\$ 20$

优惠券 $3$ ：对标价超过 $\$100$ 的部分减价 $18\%$

对下列哪个标价，优惠券 $1$ 的减价幅度会同时大于优惠券 $2$ 和优惠券 $3$ ？

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 $3?$

$\$ 179.95$

$\$ 199.95$

$\$ 219.95$

$\$ 239.95$

$\$ 259.95$

比较实际减掉的美元数，而不是最终价格。

Compare the actual dollar discount, not the final price

解出优惠券 $1$ 同时优于优惠券 $2$ 和优惠券 $3$ 的价格范围。

Solve the inequalities where coupon $1$ beats coupons $2$ and $3$

设标价为 $x$ 。

优惠券 $1$ 使价格变为 $0.9x$ ，优惠券 $2$ 使价格变为 $x - 20$ ，优惠券 $3$ 使价格变为 $x - 0.18(x - 100) = 0.82x + 18\text{。}$

需要同时满足 $0.9x \lt x - 20$ 和 $0.9x \lt 0.82x + 18\text{。}$ 解这两个不等式，得到 $200 \lt x \lt 225\text{。}$

选项中只有 $\$ 219.95$ 符合。

所以正确答案是 C。

Let us analyze what these coupons do to an arbitrary price, $x.$

Coupon $1$ changes this price to $0.9x.$ Coupon $2$ changes the price to $x - 20.$ Coupon $3$ changes the price to $x - 0.18(x - 100) = 0.82x + 18.$

We want $0.9x \lt x - 20$ and $0.9x \lt 0.82x + 18.$ Solving both gives us $200 \lt x \lt 225.$

The only answer choice that works is $\$ 219.95.$

Thus, C is the correct answer.