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Percents: Percent Change

A retailer set the tag price of an item at $200. On a certain public holiday, the retailer set a 20% discount on the tag price, thinking that he will still make a profit equal to 25% of the price he had originally paid for the item. How much did the retailer originally pay?
__Alternative Method__: Set up the equation for the question: >$$\text{Cost} + 25\% \text{ of Cost} = \$160$$ >$$\text{Cost} + 0.25 \cdot \text{Cost} = \$160$$ >$$1.25 \cdot \text{Cost} = \$160$$ >$$\frac{5}{4} \cdot \text{Cost} = \$160$$ >$$\text{Cost} = \frac{4}{5} \cdot \$160$$ >$$\text{Cost} = \$128$$
Incorrect. [[Snippet]] __Reverse Plug In__: If cost is $144, the price paid by the customer should be >$$\text{Price} = \text{Cost} + 25\% \text{ of Cost}$$ >$$~~= \$144 + \frac{25}{100} \cdot \$144$$ >$$~~= \$144 + \frac{1}{4} \cdot \$144$$ >$$~~= \$144 + \$36$$ >$$~~= \$180$$ However, our previous calculation shows that the price should be $160. Hence, this answer choice is too big and should be eliminated.
Incorrect. [[Snippet]] __Reverse Plug In__: If cost is $96, the price paid by the customer should be >$$\text{Price} = \text{Cost} + 25\% \text{ of Cost}$$ >$$~~= \$96 + \frac{25}{100} \cdot \$96$$ >$$~~= \$96 + \frac{1}{4} \cdot \$96$$ >$$~~= \$96 + \$24$$ >$$~~= \$120$$ However, our previous calculation shows that the price should be $160. Hence, this answer choice is too small and should be eliminated.
Incorrect. Note that the question asked for the price to the retailer—not the price he ended up selling the item. [[Snippet]] __Reverse Plug In__: If cost is $160, the price paid by the customer should be >$$\text{Price} = \text{Cost} + 25\% \text{ of Cost}$$ >$$~~= \$160 + \frac{25}{100} \cdot \$160$$ >$$~~= \$160 + \frac{1}{4} \cdot \$160$$ >$$~~= \$160 + \$40$$ >$$~~= \$200$$ However, our previous calculation shows that the price should be $160 after a 20% discount. Therefore, if this answer choice were correct, the tag price on the item would be more than $200—clearly this answer choice is too big. Hence, eliminate this answer choice.
Incorrect. [[Snippet]] __Reverse Plug In__: If cost is $120, the price paid by the customer should be >$$\text{Price} = \text{Cost} + 25\% \text{ of Cost}$$ >$$~~= \$120 + \frac{25}{100} \cdot \$120$$ >$$~~= \$120 + \frac{1}{4} \cdot \$120$$ >$$~~= \$120 + \$30$$ >$$~~= \$150$$ However, our previous calculation shows that the price should be $160. Hence, this answer choice is too small and should be eliminated.
Correct. __Reverse Plug In__ the answer choices. If cost is $128, the price paid by the customer should be >$$\text{Price} = \text{Cost} + 25\% \text{ of Cost}$$ >$$~~= \$128 + \frac{25}{100} \cdot \$128$$ >$$~~= \$128 + \frac{1}{4} \cdot \$128$$ >$$~~= \$128 + \$32$$ >$$~~= \$160$$ This is the correct answer because the retailer can make a profit equal to 25% of the cost of a $128 item, despite giving a 20% discount, by setting a tag price of $200.
$96
$120
$128
$144
$160
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