Applications of discount are very common in retail settings (see the image below). When you buy an item on sale, the original price of the item has been reduced by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price. We summarize the discount model in the box below.

Applications of discounts are common in everyday life. (credit: Charleston’s TheDigitel, Flickr)

The sale price should always be less than the original price. In some cases, the amount of discount is a fixed dollar amount. Then we just find the sale price by subtracting the amount of discount from the original price.

example

Jason bought a pair of sunglasses that were on sale for [latex]\text{\$10}[/latex] off. The original price of the sunglasses was [latex]\text{\$39}[/latex]. What was the sale price of the sunglasses?

Solution

Identify what you are asked to find.	What is the sale price?
Choose a variable to represent it.	Let [latex]s=[/latex] the sale price.
Write a sentence that gives the information to find it.	The sale price is the original price minus the discount.
Translate into an equation.
Simplify.	[latex]s=29[/latex]
Check if this answer is reasonable.
Yes. The sale price, [latex]\text{\$29}[/latex], is less than the original price, [latex]\text{\$39}[/latex].
Write a complete sentence that answers the question.	The sale price of the sunglasses was [latex]\text{\$29}[/latex].

In the first example, the amount of discount was a set amount, [latex]\text{\$10}[/latex]. In the next example, the discount is given as a percent of the original price.

exmaple

Elise bought a dress that was discounted [latex]\text{35%}[/latex] off of the original price of [latex]\text{\$140}[/latex]. What was ⓐ the amount of discount and ⓑ the sale price of the dress?

Show Solution

Solution
ⓐ Before beginning, you may find it helpful to organize the information in a list.

Original price = [latex]\text{\$140}[/latex]

Discount rate = [latex]35\text{%}[/latex]

Amount of discount = ?

Identify what you are asked to find.	What is the amount of discount?
Choose a variable to represent it.	Let [latex]d=[/latex] the amount of discount.
Write a sentence that gives the information to find it.	The discount is [latex]35\text{%}[/latex] of the original price.
Translate into an equation.
Simplify.	[latex]d=49[/latex]
Check if this answer is reasonable.
Yes. A [latex]\text{\$49}[/latex] discount is reasonable for a [latex]\text{\$140}[/latex] dress.
Write a complete sentence that answers the question.	The amount of discount was [latex]\text{\$49}[/latex].

ⓑ

Original price = [latex]\text{\$140}[/latex]

Amount of discount = [latex]\text{\$49}[/latex]

Sale price = ?

Identify what you are asked to find.	What is the sale price of the dress?
Choose a variable to represent it.	Let [latex]s=[/latex] the sale price.
Write a sentence that gives the information to find it.	The sale price is the original price minus the discount.
Translate into an equation.
Simplify.	[latex]s=91[/latex]
Check if this answer is reasonable.
Yes. The sale price, [latex]\text{\$91}[/latex], is less than the original price, [latex]\text{\$140}[/latex].
Write a complete sentence that answers the question.	The sale price of the dress was [latex]\text{\$91}[/latex].

You can estimate to see if the answer is reasonable. Since  [latex]15\%[/latex] is half way between  [latex]10\%[/latex] and [latex]20\%[/latex], find these numbers.

[latex]\begin{array}{c}10\%\,\,\text{of}\,\,220=0.1\cdot220=22\\20\%\,\,\text{of}\,\,220=0.2\cdot220=44\end{array}[/latex]

The answer, [latex]33[/latex], is between [latex]22[/latex] and [latex]44[/latex]. So [latex]$33[/latex] seems reasonable.

The example video that follows shows how to use the percent equation to find the amount of a discount from the price of a phone.

Calculate Discount Rate

There may be times when you buy something on sale and want to know the discount rate. The next example will show this case.

example

Jeannette bought a swimsuit at a sale price of [latex]\text{\$13.95}[/latex]. The original price of the swimsuit was [latex]\text{\$31}[/latex]. Find ⓐ the amount of discount and ⓑ the discount rate.

Show Solution

Solution
ⓐ Before beginning, you may find it helpful to organize the information in a list.

Original price = [latex]\text{\$31}[/latex]

Amount of discount = ?

Sale price = [latex]\text{\$13.95}[/latex]

Identify what you are asked to find.	What is the amount of discount?
Choose a variable to represent it.	Let [latex]d=[/latex] the amount of discount.
Write a sentence that gives the information to find it.	The discount is the original price minus the sale price.
Translate into an equation.
Simplify.	[latex]d=17.05[/latex]
Check if this answer is reasonable.
Yes. The [latex]\text{\$17.05}[/latex] discount is less than the original price.
Write a complete sentence that answers the question.	The amount of discount was [latex]$17.05[/latex].

ⓑ Before beginning, you may find it helpful to organize the information in a list.

Original price = [latex]$31[/latex]

Amount of discount = [latex]$17.05[/latex]

Discount rate = ?

Identify what you are asked to find.	What is the discount rate?
Choose a variable to represent it.	Let [latex]r=[/latex] the discount rate.
Write a sentence that gives the information to find it.	The discount is what percent of the original price?
Translate into an equation.
Divide.	[latex]\Large\frac{17.05}{31}\normalsize =\Large\frac{r(31)}{31}[/latex]
Simplify.	[latex]0.55=r[/latex]
Check if this answer is reasonable.
The rate of discount was a little more than [latex]50\text{%}[/latex] and the amount of discount is a little more than half of [latex]\text{\$31}[/latex].
Write a complete sentence that answers the question.	The rate of discount was [latex]55\text{%}[/latex].

In the following video we show another example of how to find the discount rate (also called the percent of change) given the original price and the marked-down price.

Solve Mark-up Applications

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the wholesale price. The retailer then adds a mark-up to the wholesale price to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the wholesale price. The percent is called the mark-up rate. To determine the amount of mark-up, multiply the mark-up rate by the wholesale price. We summarize the mark-up model in the box below.

example

Adam’s art gallery bought a photograph at the wholesale price of [latex]\text{\$250}[/latex]. Adam marked the price up [latex]\text{40%}[/latex]. Find ⓐ the amount of mark-up and ⓑ the list price of the photograph.

Show Solution

Solution

ⓐ
Identify what you are asked to find.	What is the amount of mark-up?
Choose a variable to represent it.	Let [latex]m=[/latex] the amount of each mark-up.
Write a sentence that gives the information to find it.	The mark-up is [latex]40\text{%}[/latex] of the wholesale price.
Translate into an equation.
Simplify.	[latex]m=100[/latex]
Check if this answer is reasonable.
Yes. The markup rate is less than [latex]50\text{%}[/latex] and [latex]\text{\$100}[/latex] is less than half of [latex]\text{\$250}[/latex].
Write a complete sentence that answers the question.	The mark-up on the photograph was [latex]\text{\$100}[/latex].

ⓑ
Identify what you are asked to find.	What is the list price?
Choose a variable to represent it.	Let [latex]p=[/latex] the list price.
Write a sentence that gives the information to find it.	The list price is the wholesale price plus the mark-up.
Translate into an equation.
Simplify.	[latex]p=350[/latex]
Check if this answer is reasonable.
Yes. The list price, [latex]\text{\$350}[/latex], is more than the wholesale price, [latex]\text{\$250}[/latex].
Write a complete sentence that answers the question.	The list price of the photograph was [latex]\text{\$350}[/latex].

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