# Prices Involving Discounts

## Determine the value of a number that has been reduced by a given percentage.

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Prices Involving Discounts

In 2002, a Cook County home had an assessed value of 100,000. By 2008, the assessed value of the home had been marked 160% by Cook County assessors. What was the assessed value of the home in 2008? In this concept, you will learn how to use percents in everyday life for situations involving discounts and markups. ### Finding Prices Involving Discounts A store owner buys an item at a wholesale cost and marks it up by adding a percent of the price to the price he paid for it. If you buy that item when it is on sale, you can find the amount of the discount if you know the rate of discount the store is offering. A markup is an increase in the price of an item. A discount is a decrease in the price of an item. Both of these changes can be expressed as an amount of money or as a percent of the original price of the item. A markup is often how stores or businesses make a profit. They buy merchandise at one price and sell it for another price. The difference in the price they bought it for and the price they sell it for is the profit margin. Stores discount items all the time. Often, they are still making a profit, but they discount an item by a specific percentage to try to sell more of that item. To figure out a discount or a markup, you need to know the percent of change The percent of change is the percent the price is changing by. If the markup is 10%, then that is the percent of change (an increase). If the discount is 10%, then that is the percent of change (a decrease). You can use the following formula to solve markup and discount problems: Percent of change×original amount=amount of the change\begin{align*}\text{Percent of change} \times \text{original amount} = \text{amount of the change}\end{align*} Let’s look at an example. A camera store buys a camera for149 and marks up the price by 35%. What price does the camera sell for?

First, find the amount of the markup.

Amount of change==percent of change×original amount35%×149\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 35\% \times \149 \end{array}\end{align*} Next, convert the percent to a decimal and multiply. ==0.35×14952.15\begin{align*}\begin{array}{rcl} &=& 0.35 \times \149\\ &=& \52.15 \end{array}\end{align*} The markup is52.15.

Now find the selling price by adding the markup to the wholesale price.

$149+$52.15=201.15\begin{align*}149 + 52.15 = 201.15\end{align*} The answer is the selling price is201.15.

Let’s look at another example.

A camera that normally sells for $189 is on sale at a 20% discount. What is the sale price? First, find the amount of the discount. Amount of change==percent of change×original amount20%×$189\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 20\% \times 189 \end{array}\end{align*}

Next, convert the percent to a decimal and multiply.

==0.20×$189$37.80\begin{align*}\begin{array}{rcl} &=& 0.20 \times 189\\ &=& 37.80 \end{array}\end{align*}

The discount is $37.80. Now find the sale price by subtracting the discount from the original price.$189$37.80=$151.20\begin{align*}189 - 37.80 = 151.20\end{align*}

The answer is the sale price is $151.20. ### Examples #### Example 1 Earlier, you were given a problem about the Cook County home assessment. The assessors valued the home at$100,000 in 2002. By 2008, Cook County assessors had marked up the assessed value of the home by 160%. What was the assessed value of the home in 2008?

First, find the amount of the markup.

Amount of change==percent of change×original amount160%×100,000\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 160\% \times 100,000 \end{array}\end{align*} Next, convert the percent to a decimal and multiply. ==1.60×100,000160,000\begin{align*}\begin{array}{rcl} &=& 1.60 \times 100,000\\ &=& 160,000 \end{array}\end{align*} The markup is160,000.

Now find the new assessed value by adding the markup to the original assessed value.

$100,000+$160,000=260,000\begin{align*}100,000 + 160,000 = 260,000\end{align*} The answer is the assessed value of the home in 2008 was260,000.

#### Example 2

A pair of shoes with a wholesale price of $40 was marked up 50%. During a sale, the shoes were discounted 33%. What was the sale price? First, find the amount of markup. Amount of change====percent of change×original amount50%×$400.50×$40$20\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 50\% \times 40\\ &=& 0.50 \times 40\\ &=& 20 \end{array}\end{align*}

The markup is $20. Next, find the regular selling price by adding the markup to the wholesale price. The selling price is$40+$20=$60\begin{align*}40 + 20 = 60\end{align*}.

Now find the amount of the discount.

Amount of change====percent of change×original amount33%×$600.33×$6019.80\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 33\% \times 60\\ &=& 0.33 \times 60\\ &=& 19.80 \end{array}\end{align*} The discount is19.80.

Finally, find the sale price by subtracting the discount from the selling price.

$60$19.80=40.20\begin{align*}60 - 19.80 = 40.20\end{align*} The sale price is40.20.

#### Example 3

A camera store buys a camera for $159.00 and marks it up 10%. What is the new price of the camera? First, find the amount of the markup. Amount of change==percent of change×original amount10%×$159\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 10\% \times 159\\ \end{array}\end{align*}

Next, convert the percent to a decimal and multiply.

==0.10×$159$15.90\begin{align*}\begin{array}{rcl} &=& 0.10 \times 159\\ &=& 15.90 \end{array}\end{align*}

The markup is $15.90. Now find the selling price by adding the markup to the wholesale price.$159+$15.90=$174.90\begin{align*}159 + 15.90 = 174.90\end{align*}

The answer is the new price is $174.90. #### Example 4 A chair that costs$199.00 is on sale for 15% off. What is the sale price of the chair?

First, find the amount of the discount.

Amount of change==percent of change×original amount15%×199\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 15\% \times 199 \end{array}\end{align*} Next, convert the percent to a decimal and multiply. ==0.15×19929.85\begin{align*}\begin{array}{rcl} &=& 0.15 \times 199\\ &=& 29.85 \end{array}\end{align*} The discount is29.85.

Now find the sale price by subtracting the discount from the original price.

$199$29.85=169.15\begin{align*}199 - 29.85 = 169.15\end{align*} The answer is the sale price is169.15.

#### Example 5

A shirt that costs $15.00 is on sale for 30% off. What is the sale price of the shirt? First, find the amount of the discount. Amount of change==percent of change×original amount30%×$15\begin{align*}\begin{array}{rcl} \text{Amount of change} &=& \text{percent of change} \times \text{original amount}\\ &=& 30\% \times 15 \end{array}\end{align*}

Next, convert the percent to a decimal and multiply.

==0.30×$15$4.50\begin{align*}\begin{array}{rcl} &=& 0.30 \times 15\\ &=& 4.50 \end{array}\end{align*}

The discount is $4.50. Now find the sale price by subtracting the discount from the original price.$15$4.50=$10.50\begin{align*}15 - 4.50 = 10.50\end{align*}

The answer is the sale price is $10.50. ### Review Here is a list of wholesale prices. Figure out each sale price if the markup is 20%. 1.$19.00
2. $12.00 3.$18.00
4. $25.00 5.$13.50
6. $9.95 7.$45.00
8. $90.00 9.$85.00
10. $17.00 Each item is discounted 15%. Figure out each new sale price; the original prices are listed below. 1.$55.00
2. $35.50 3.$18.00
4. $8.75 5.$25.00

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### Vocabulary Language: English

TermDefinition
Percent of Change The percent of change is the percent that a value has increased or decreased by.