# 2.2: Discount Problems

- Page ID
- 45782

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Learning outcome

- Use mathematical notation to solve discount problems

A store called Feminine Furnishings is having a sale on bedding. Their queen sheet sets have a list price of \($126\). During the first two weeks of February, the store has a—Snuggle Up In Love Sale— where customers save \(14\%\) on any white, pink, or red bedding set. Carly selects a bedding set with pink stripes and pays \($108.36\).

Applications of discount are very common in retail settings. When you buy an item on sale, the **original **or **list 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 subtracted from the original price.

**amount of discount = discount rate \(\cdot\) original price**

**sale price = original price \(-\) discount**

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. For example, as you walk into a store you may see a sign that says “Spend \($100\) or more, and receive a \($20\) discount!”

example

Jason bought a pair of sunglasses from a rack that was labeled \(\text{\$10}\) off all Retrop brand glasses. The original price of the sunglasses was \(\text{\$39}\). What was the sale price of the sunglasses?

[reveal-answer q=”967734″]Show Answer[/reveal-answer]

[hidden-answer a=”967734″]

Solution

Identify what you are asked to find. | What is the sale price? |

Choose a variable to represent it. | Let \(s=\) 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. | \(s=29\) |

Check if this answer is reasonable. | |

Yes. The sale price, \(\text{\$29}\), is less than the original price, \(\text{\$39}\). | |

Write a complete sentence that answers the question. | The sale price of the sunglasses was \(\text{\$29}\). |

[/hidden-answer]

try it

[ohm_question]146772[/ohm_question]

In the first example, the amount of discount was a set or static amount. In the next example, the discount is given as a percent of the original price.

Example

Elise bought a dress that was discounted \(\text{35%}\) off of the original price of \(\text{\$140}\). What was the amount of discount and the sale price of the dress?

[reveal-answer q=”967374″]Show Answer[/reveal-answer]

[hidden-answer a=”967374″]

Solution

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

Original price = \(\text{\$140}\)

Discount rate = \(35\text{%}\)

Amount of discount = ?

Identify what you are asked to find. | What is the amount of discount? |

Choose a variable to represent it. | Let \(d=\) the amount of discount. |

Write a sentence that gives the information to find it. | The discount is \(35\text{%}\) of the original price. |

Translate into an equation. | |

Simplify. | \(d=49\) |

Check if this answer is reasonable. | |

Yes. A \(\text{\$49}\) discount is reasonable for a \(\text{\$140}\) dress. | |

Write a complete sentence that answers the question. | The amount of discount was \(\text{\$49}\). |

ⓑ

Original price = \(\text{\$140}\)

Amount of discount = \(\text{\$49}\)

Sale price = ?

Identify what you are asked to find. | What is the sale price of the dress? |

Choose a variable to represent it. | Let \(s=\) 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. | \(s=91\) |

Check if this answer is reasonable. | |

Yes. The sale price, \(\text{\$91}\), is less than the original price, \(\text{\$140}\). | |

Write a complete sentence that answers the question. | The sale price of the dress was \(\text{\$91}\). |

[/hidden-answer]

try it

[ohm_question]146775[/ohm_question]

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 \(\text{\$13.95}\). The original price of the swimsuit was \(\text{\$31}\). Find the ⓐ amount of discount and ⓑ discount rate.

[reveal-answer q=”123339″]Show Answer[/reveal-answer]

[hidden-answer a=”123339″]

Solution

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

Original price = \(\text{\$31}\)

Amount of discount = ?

Sale price = \(\text{\$13.95}\)

Identify what you are asked to find. | What is the amount of discount? |

Choose a variable to represent it. | Let \(d=\) 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. | \(d=17.05\) |

Check if this answer is reasonable. | |

Yes. The \(\text{\$17.05}\) discount is less than the original price. | |

Write a complete sentence that answers the question. | The amount of discount was \($17.05\). |

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

Original price = \($31\)

Amount of discount = \($17.05\)

Discount rate = ?

Identify what you are asked to find. | What is the discount rate? |

Choose a variable to represent it. | Let \(r=\) 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. | \(\Large\frac{17.05}{31}\normalsize =\Large\frac{r(31)}{31}\) |

Simplify. | \(0.55=r\) |

Check if this answer is reasonable. | |

The rate of discount was a little more than \(50\text{%}\) and the amount of discount is a little more than half of \(\text{\$31}\). | |

Write a complete sentence that answers the question. | The rate of discount was \(55\text{%}\). |

[/hidden-answer]

try it

[ohm_question]156971[/ohm_question]

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.