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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