Everyone has seen stores advertising, "25\% off", or "pay less when you pay cash." These are both examples of discounts. A **discount** is a reduction in price, or a** percent decrease**. Businesses often use discount sales to encourage people to buy from them, so it's important to be able to calculate discounts to make sure you're getting a great deal.

A **markup** is a **percent increase** that indicates the price of the product has increased from its original price, and the percent of the markup indicates by how much. For example, a retailer might purchase an item and then resell it at a markup of 20\% in order to make a profit.

Similar to a markup, a **fee** can also be represented by a **percent increase**.

Since markups, discounts, and fees are simply percent increases and decreases, we can calculate them using the same methods. Recall that we can apply percent increases and decreases in one step:

For a percent increase, add the percent to 100\% and multiply the result by the original price.

For a percent decrease, subtract the percent from 100\% and multiply the result by the original price.

The calculations involve for discounts, markups, and fees are the same except that markup and fees involve percent increase while markdown and discount involve percent decrease.

Some terms related to discounts, markups and fees are the following:

Steph is going to buy a hat that is marked at 75\% off. The original price is \$36.

a

What is the value of the discount in dollars?

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b

What is the price that Steph will pay for the hat?

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A watch that normally costs \$75 is marked up by 20\%. What is the new price of the watch?

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An artist was hired to paint a portrait which will cost \$4\,000. The contractor also includes a service fee of \$1\,500 for the overall cost of the contract.

What percent of the cost of the portrait is the artist's fee?

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A TV normally sells for \$1792.94, but is currently on sale.

In each of the following scenarios, calculate the percent discount correct to two decimal places.

a

The TV is discounted by \$149.50.

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b

The TV is on sale for \$1428.74.

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Idea summary

Discounts are examples of percent decreases. We subtract the amount of discount from the original price to find the sale price. We can also multiply the original price by {100\%-\text{percent decrease}} to get the sale price.

Fees and markups are examples of percent increases. We add the fee or the markup to the original price to find the full price. We can also multiply the original cost by {100\% + \text{percent increase}} to find the full price.