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.
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__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.
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__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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