### Learning Outcomes

- Solve discount applications
- Solve mark-up applications

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.

### Discount

An amount of discount is a percent off the original price.

[latex]\begin{array}{ccc}\hfill \text{amount of discount}& =& \text{discount rate}\cdot \text{original price}\hfill \\ \hfill \text{sale price}& =& \text{original price}-\text{discount}\hfill \end{array}[/latex]

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]. |

### try it

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?

### try it

### Example

Jeff has a coupon at the Guitar Store for [latex]15\%[/latex] off any purchase of [latex]\$100[/latex] or more. He wants to buy a used guitar that has a price tag of [latex]\$220[/latex] on it. Jeff wonders how much money the coupon will take off of the [latex]\$220[/latex] original price*.*

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.

### try it

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.

### Mark-up

The mark-up is the amount added to the wholesale price.

[latex]\begin{array}{ccc}\hfill \text{amount of mark-up}& =& \text{mark-up rate}\cdot \text{wholesale price}\hfill \\ \hfill \text{list price}& =& \text{wholesale price}+\text{mark up}\hfill \end{array}[/latex]

The list price should always be more than the wholesale price.

### 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.