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1. Real Numbers
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United States of AmericaVirginia
Grade 8

1.05 Discounts, fees, and markups

Lesson
Practice

Discounts, fees, and markups

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.

Discount

A percent decrease in the price of a product.

Markup

A percent increase in the price of a product.

Fee

A percent increase for the price of a service.

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:

Regular price

The price before any discount or markup is applied.

Sale price

The price after a discount or markup is applied.

Examples

Example 1

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?

Worked Solution
Create a strategy

To find the discount amount, multiply the original price by the percent discount. Remember that percent means 'divided by 100'.

Apply the idea
\displaystyle \text{Discount}\displaystyle =\displaystyle 36\cdot \dfrac{75}{100}Multiply the original price by the percent discount in fraction form
\displaystyle =\displaystyle \dfrac{2700}{100}Evaluate the multiplication
\displaystyle =\displaystyle 27Evaluate the division

The discount is \$27.

b

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

Worked Solution
Create a strategy

To find the discounted price, subtract the discount from the original price.

Apply the idea
\displaystyle \text{Discounted price}\displaystyle =\displaystyle \$36-\$27Subtract the discount from the original price
\displaystyle =\displaystyle \$9Evaluate

The discounted price is \$9.

Reflect and check

We have decreased the amount by 75\%, which means that Steph is paying the remaining 25\%, or 100\%-\text{percent decrease}.

We could have calculated the discounted price by:

\displaystyle \text{Discounted price}\displaystyle =\displaystyle 36\cdot \dfrac{25}{100}Rewrite 25\% as a fraction and multiply by the original price
\displaystyle =\displaystyle \dfrac{900}{100}Evaluate the multiplication
\displaystyle =\displaystyle 9Evaluate the division

We get the same answer. The discounted price is \$9.

Example 2

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

Worked Solution
Create a strategy

Remember that markup involves percent increase. Increasing by 20\% is the same as multiplying by 100\%+20\% or 120\%.

Apply the idea
\displaystyle \text{Markup price}\displaystyle =\displaystyle \$75\cdot \dfrac{120}{100} Convert 120\% to a fraction and multiply to the original price
\displaystyle =\displaystyle \dfrac{9000}{100} Evaluate the multiplication
\displaystyle =\displaystyle 90Evaluate the division

The markup price or new amount of the watch is \$90.

Reflect and check

We could have also found the new price by translating the statement into a proportion. To setup the proportion, we set the percent as a fraction equal to the new price over the original price.

\displaystyle \frac{\text{percent}}{100}\displaystyle =\displaystyle \frac{\text{part}}{\text{whole}}Proportion setup
\displaystyle \frac{ 120 }{100}\displaystyle =\displaystyle \frac{x}{75}Substitute 120 for the percent, x for the part, and 75 for the whole.
\displaystyle 100x\displaystyle =\displaystyle 9000Cross multiply.
\displaystyle \frac{100x}{100}\displaystyle =\displaystyle \frac{9000}{100}Evaluate the divison
\displaystyle x\displaystyle =\displaystyle 90The markup price is \$ 90

Example 3

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?

Worked Solution
Create a strategy

Find the percent rate as a ratio of the fee and the cost of the house.

Apply the idea
\displaystyle \dfrac{\text{percent}}{100}\displaystyle =\displaystyle \dfrac{\text{part}}{\text{whole}}Set up a proportion
\displaystyle \dfrac{x}{100}\displaystyle =\displaystyle \dfrac{1\,500}{4\,000}Substitute 1500 for the part, 4000 for the whole, and x for the percent
\displaystyle \dfrac{x}{100}\displaystyle =\displaystyle \dfrac{37.5}{100}Divide the numerator and denominator by 40
\displaystyle x\displaystyle =\displaystyle 37.5Multiply both sides by 100

The artist charges a percent fee of 37.5\%.

Reflect and check

Notice that the fee is an increase in the price of service. This means we could have used the percent increase formula to find the percent amount of the fee.

\displaystyle \text{Percent Increase}\displaystyle =\displaystyle \dfrac{\text{Increase}}{\text{Original Amount}} \cdot 100\%Percent increase formula
\displaystyle =\displaystyle \dfrac{1\,500}{4\,000} \cdot 100\%Subsitute known amounts
\displaystyle =\displaystyle 0.375\cdot 100\%Evaluate the division
\displaystyle =\displaystyle 37.5\%Evaluate the multiplication

Example 4

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.

Worked Solution
Create a strategy

A discount is a percent decrease so we can use the percent decrease formula: \text{Percent Decrease} = \dfrac{\text{Decrease}}{\text{Original Amount}} \cdot 100\%

Apply the idea

The discount amount is \$149.50 and the regular price is \$1792.94.

\displaystyle \text{Percent Discount}\displaystyle =\displaystyle \dfrac{149.50}{1792.94}\cdot 100\%Substitute the given values to the formula
\displaystyle =\displaystyle 8.34\%Evaluate
b

The TV is on sale for \$1428.74.

Worked Solution
Create a strategy

Calculate the discount amount and then use the percent decrease formula: \text{Percent Decrease} = \dfrac{\text{Decrease}}{\text{Original Amount}} \cdot 100\%

Apply the idea

The regular price is \$1792.94.

\displaystyle \text{Discount Amount}\displaystyle =\displaystyle 1792.94-1428.74Subtract 1428.74 from 1792.94
\displaystyle =\displaystyle \$364.20Evaluate
\displaystyle \text{Percent Discount}\displaystyle =\displaystyle \dfrac{364.20}{1792.94}\cdot 100\%Substitute the given values to the formula
\displaystyle =\displaystyle 20.31\%Evaluate
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.

Outcomes

8.CE.1

The student will estimate and apply proportional reasoning and computational procedures to solve contextual problems.

8.CE.1a

Estimate and solve contextual problems that require the computation of one discount or markup and the resulting sale price.

8.CE.1c

Estimate and solve contextual problems that require the computation of the percent increase or decrease.

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