6.2 Discounts, Markups, and Sales Tax

A sale board reads final sale 30 percent, 50 percent, 60 percent, and 70 percent. — **Figure 6.3** Sale prices are often described as percent discounts.Sale prices are often described as percent discounts. (credit: "Close-up of a discount sign" by Ivan Radic/Flickr, CC BY 2.0)

Learning Objectives

After completing this section, you should be able to:

Calculate discounts.
Solve application problems involving discounts.
Calculate markups.
Solve application problems involving markups.
Compute sales tax.
Solve application problems involving sales tax.

Many people first encounter percentages during a retail transaction such as a percent discount (SALE! 25% off!!), or through sales tax ("Wait, I thought this was $1.99?"), a report that something has increased by some percentage of the previous value (NOW! 20% more!!). These are examples of percent decreases and percent increases. In this section, we discuss decrease, increase, and then the case of sales tax.

Calculating Discounts

Retailers frequently hold sales to help move merchandise. The sale price is almost always expressed as some amount off the original price. These are discounts, a reduction in the price of something. The price after the discount is sometimes referred to as the reduced price or the sale price.

When a reduction is a percent discount, it is an application of percent, which was introduced in Understanding Percent. The formula used was $part = percentage \times total$ . In a discount application, the discount plays the role of the part, the percent discount is the percentage, and the original price plays the role of the total.

When the original price and the percent discount are known, the discount and the sale price can be directly computed.

Example 1 Calculating Discount for a Percent Discount

Calculate the discount for the given price and discount percentage. Then calculate the sale price.

Original price = $75.80; percent discount is 25%
Original price = $168.90; percent discount is 30%

Substituting the values into the formula $discount = percent discount \times original price$ , we find that the discount is $discount = 0 .25 \times 75.80 = 18.95$ . The discount is $18.95.
The sale price of the item is then $sale price = original price - discount = 75.80 - 18.95 = 56.85$ , or $56.85.
Substituting the values into the formula $discount = percent discount \times original price$ , we find that the discount is $discount = 0 .30 \times 168.90 = 50.67$ . The discount is $50.67.
The sale price of the item is then $sale price = original price - discount = 168.90 - 50.67 = 118.23$ , or $118.23.

Sometimes the original price and the sale price of an item is known. From this, the percent discount can be computed using the formula $discount = percent discount \times original price$ , by solving for the percent discount.

Example 2 Calculating the Percent Discount from the Original and Sale Prices

Determine the percent discount based on the given original and sale prices.

Original price = $1,200.00; sale price = $900.00
Original price = $36.70; sale price = $29.52

Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $sale price = original price - discount$ . Substituting and calculating, we find the discount to be $900.00 = 1,200.00 - discount$ . Solving for the discount gives $300.00.
Step 2. Find the percent discount. Substituting the discount of $300.00 and the original price of $1,200.00, into the formula $discount = percent discount \times original price$ , we can find the percent discount.
$\begin{matrix} 300 .00 & = & percent discount \times 1,200 .00 \\ \frac{300 .00}{1,200.00} & = & percent discount \\ 0.25 & = & percent discount \end{matrix}$
Converting to percent form, the percent discount is 25%.
Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $sale price = original price - discount$ . Substituting and calculating, we find the discount to be $29.52 = 36.70 - discount$ . Solving for the discount gives $7.38.
Step 2. Find the percent discount. Substituting the discount of $7.38 and the original price of $36.70, into the formula $discount = percent discount \times original price$ , we can find the percent discount.
$\begin{matrix} 29 .52 & = & percent discount \times 36 .70 \\ \frac{29 .52}{36.70} & = & percent discount \\ 0.2 & = & percent discount \end{matrix}$
Converting to percent form, the percent discount is 20%.

Sometimes the sale price and the percent discount of an item are known. From this, the original price can be found. To avoid multiple steps, though, the formula that we will use is $sale price = original price \times (1 - percent discount)$ . The original price can be found by solving this equation for the original price.

Example 3 Calculating the Original Price from the Percent Discount and Sale Price

Determine the original price based on the percent discount and sale price.

Percent discount 10%; sale price = $450.00
Percent discount 75%, sale price = $90.00

Using the percent discount and the sale price, we can find the original price with the formula $sale price = original price \times (1 - percent discount)$ . Substituting and solving for the original price, we find

$\begin{matrix} sale price & = & original price \times (1 - percent discount) \\ 450 .00 & = & original price \times (1 - 0 .10) \\ 450 .00 & = & original price \times (0 .90) \\ 500 .00 & = & original price \end{matrix}$

The original price of the item was $500.00.

Using the percent discount and the sale price, we can find the original price with the formula $sale price = original price \times (1 - percent discount)$ . Substituting and solving for the original price, we find

$\begin{matrix} sale price & = & original price \times (1 - percent discount) \\ 90 .00 & = & original price \times (1 - 0 .75) \\ 90 .00 & = & original price \times (0 .25) \\ 360 .00 & = & original price \end{matrix}$

The original price of the item was $360.00.

Solve Application Problems Involving Discounts

In application problems, identify what is given and what is to be found, using the terms that have been learned, such as discount, original price, percent discount, and sale price. Once you have identified those, use the appropriate formula (or formulas) to find the solution(s).

Work It Out

There are cases where retailers allow multiple discounts to be applied. However, it is rare that the discount percentages are added together. For example, if you have a 15% coupon and qualify for a 20% price reduction, the retailer typically does not add those two percentages together to determine the new price. The retailer instead applies one discount, then applies the second discount to the price obtained after the first discount was deducted.

Research the original prices of two different laptops offered by one retail outlet. Assume you will receive a student discount of 12% and your outlet of choice is having a 15% off sale on all laptops.

For each laptop:

List the original price and calculate the price after applying the student discount (12%) only.
Then find the price after applying the sale discount (15% off) to the price found in Step 1.
Determine the total saved on the laptop and what percent discount the total savings represents.
Now, apply the discounts in reverse order (first the sale discount, then the student discount).
Note anything interesting about your findings.

Calculate Markups

When retailers purchase goods to sell, they pay a certain price, called the cost. The retailer then charges more than that amount for the goods. This increase is called the markup. This selling price, or retail price, is what the retailer charges the consumer in order to pay their own costs and make a profit. Markup, then is very similar to discount, except we add the markup, while we subtract the discount.

It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing the subtraction.

Example 7 Determining the Retail Price Based on the Cost and the Percent Markup

Calculate the markup for the given cost and markup percentage. Then calculate the retail price.

Cost = $62.00; percent markup is 15%
Cost = $750.00; percent markup is 45%

Substituting the values into the formula $markup = percent markup \times cost$ , we find that the markup is $markup = 0.15 \times 62.00 = 9.30$ . The markup is $9.30.
The retail price of the item is then $retail price = cost + markup$ , or $62.00 + $9.30 = $71.30.
Substituting the values into the formula $markup = percent markup \times cost$ , we find that the markup is $markup = 0.45 \times 750.00 = 337.50$ . The markup is $337.50.
The retail price of the item is then $retail price = cost + markup$ , or $750.00 + $337.50 = $1,087.50.

Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the formula $markup = percent markup \times cost$ , by solving for the percent markup.

Example 8 Calculating the Percent Markup from the Cost and Retail Price

Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places.

Cost = $90.00; retail price = $103.50
Cost = $5.20; retail price = $9.90

Step 1: Using the cost and the retail price, we can find the markup with the formula $retail price = cost + markup$ . Substituting and calculating, we find the markup to be $103.50 = 90.00 + markup$ . Solving for the markup gives $13.50.
Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula $markup = percent markup \times cost$ , we can find the percent markup.

$\begin{matrix} 13 .50 & = & percent markup \times 90.00 \\ \frac{13 .50}{90.00} & = & percent markup \\ 0.15 & = & percent markup \end{matrix}$

Converting to percent form, the percent markup is 15%.

Step 1: Using the cost and the retail price, we can find the markup with the formula $retail price = cost + markup$ . Substituting and calculating, we find the markup to be $9.90 = 5.20 + markup$ . Solving for the markup gives $4.70.
Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula $markup = percent markup \times cost$ , we can find the percent markup.

$\begin{matrix} 4 .70 & = & percent markup \times 5.20 \\ \frac{4.70}{5.20} & = & percent markup \\ 0.9038 & = & percent markup \end{matrix}$

Converting to percent form, the percent markup is 90.38%.

Sometimes the retail price and the percent markup of an item are known. From this, the cost can be found. To avoid multiple steps, though, the formula that we will use is $retail price = cost \times (1 + percent markup)$ . The cost can be found by solving this equation for the cost.

Example 9 Calculating the Cost from the Percent Markup and Retail Price

Determine the cost based on the percent markup and retail price.

Percent markup 20%; retail price = $10.62
Percent markup 125%; retail price = $26.55

Using the percent markup and the retail price, we can find the cost with the formula $retail price = cost \times (1 + percent markup)$ . Substituting and solving for the cost, we find

$\begin{matrix} retail price & = & cost \times (1 + percent markup) \\ 10 .62 & = & cost \times (1 + 0.2) \\ 10 .62 & = & cost \times (1 .2) \\ 8 .85 & = & cost \end{matrix}$

The cost of the item was $8.85.

Using the percent markup and the retail price, we can find the cost with the formula $retail price = cost \times (1 + percent markup)$ . Substituting and solving for the original price, we find

$\begin{matrix} retail price & = & cost \times (1 + percent markup) \\ 26 .55 & = & cost \times (1+2 .25) \\ 26 .55 & = & cost \times (3 .25) \\ 11 .80 & = & cost \end{matrix}$

The cost of the item was $11.80.

Solve Application Problems Involving Markups

As before when working with application problems, be sure to look for what is given and identify what you are to find. Once you have evaluated the problem, use the appropriate formula to find the solution(s). These application problems address markups.

Compute Sales Tax

Sales tax is applied to the sale or lease of some goods and services in the United States but is not determined by the federal government. It is most often set, collected, and spent by individual states, counties, parishes, and municipalities. None of these sales tax revenues go to the federal government.

For example, North Carolina has a state sales tax of 4.75% while New Mexico has a state sales tax of 5%. Additionally, many counties in North Carolina charge an additional 2% sales tax, bringing the total sales tax for most (72 of the 100) counties in North Carolinians to 6.75%. However, in Durham, the county sales tax is 2.25% plus an additional 0.5% tax used to fund public transportation, bringing Durham County’s sales tax to 7%. To find the sales tax in a particular place, then, add other locality sales taxes to the base state sales tax rate.

How much we pay in sales tax depends on where we are, and what we are buying.

To determine the amount of sales tax on taxable purchase, we need to find the product of the purchase price, or marked price, and the sales tax rate for that locality.

You should notice that this the same as markup, except using a different term. Sales tax plays the role of markup, the purchase price plays the role of cost, and the tax rate plays the role of percent markup. This means all the strategies developed for markups apply to this situation, with the changes indicated.

Example 13 Sales Tax in Kankakee Illinois

The sales tax in Kankakee, Illinois, is 8.25%. Find the sales tax and total price of items based on the purchase price listed.

Purchase price = $428.99
Purchase price = $34.88

The sales tax is found using $sales tax = purchase price \times tax rate$ . The purchase price is $428.99 and the tax rate is 8.25%. Substituting and calculating, the sales tax is $sales tax = $ 428.99 \times 0.0825 = $ 35.391675$ . The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 1, we round down and the sales tax is $35.39. The total price is the sales tax plus the purchase price, so is $$ 428.99 + $ 35.88 = $ 464.87$ .
The sales tax on the item is found using $sales tax = purchase price \times tax rate$ . The purchase price is $34.88 and the tax rate is 8.25%. Substituting and calculating, the sales tax is $sales tax = $ 34.88 \times 0.0825 = $ 2.8776$ . The sales tax needs to be rounded off. Since the third decimal place (fraction of a penny) is 7, we round up and the sales tax is $2.88. The total price of the item is the sales tax plus the purchase price, so is $$ 34.88 + $ 2.88 = $ 37.76$ .

As before, the information available might be different than only the purchase price and the sales tax rate. In these cases, use either $sales tax = purchase price \times tax rate$ or $Total price = purchase price \times (1 + tax rate)$ and solve for the indicated tax, price, or rate. These problems mirror those for percent markup.

Be aware, almost all sales tax rates are structured as full percentages, or half percent, or one-quarter percent, or three-quarter percent. This means the decimal value of the sales tax rate, written as a percent, will be either 0, as in 5.0%, 5 as in 7.5%, 25 as in 3.25%, or 75 as in 4.75%. When rounding for the sales tax percentage, be sure to use this guideline.

Example 14 Calculating the Sales Tax from the Purchase Price and the Total Price

Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax percentages.

Purchase price = $329.50; total price = $354.21
Purchase Price = $13.77; total price = $14.39

Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $24.71.
Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula $sales tax = purchase price \times tax rate$ , the sales tax rate can be found. Substituting and solving yields

$\begin{matrix} Sales Tax & = & purchase price \times tax rate \\ $ 24.71 & = & $ 329.50 \times tax rate \\ \frac{$ 24.71}{$ 329.50} & = & tax rate \\ 0 .07499 & = & tax rate \end{matrix}$

Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%.

Step 1. Find the sales tax paid. First, the amount of sales tax must be found. Subtracting the purchase price from the total price, the amount of sales tax is $0.62.
Step 2. Find the sales tax rate. Using the purchase price, the sales tax, and the formula $sales tax = purchase price \times tax rate$ , the sales tax rate can be found. Substituting and solving yields

$\begin{matrix} sales tax & = & purchase price \times tax rate \\ $ 0.62 & = & $ 13.77 \times tax rate \\ \frac{$ 0.62}{$ 13.77} & = & tax rate \\ 0 .04503 & = & tax rate \end{matrix}$

Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%.

Example 15 Calculating the Purchase Price from the Sales Tax and Total Price

Find the purchase price for the indicated sales tax rate and total price.

Sales tax rate = 5.75%; total price = $36.56
Sales tax rate = 4.25%; total price = $97.17

When the sales tax rate and the total price are known, the formula $total price = purchase price \times (1 + tax rate)$ can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find

$\begin{matrix} Total price & = & purchase price \times (1 + tax rate) \\ $ 36.56 & = & purchase price \times (1 + 0 .0575) \\ \frac{$ 36.56}{1.0575} & = & purchase price \\ $ 34.57 & = & purchase price \end{matrix}$

The purchase price, the price before tax, was $34.57.

When the sales tax rate and the total price are known, the formula $total price = purchase price \times (1 + tax rate)$ can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find

$\begin{matrix} Total price & = & purchase price \times (1 + tax rate) \\ $ 97.17 & = & purchase price \times (1 + 0 .0425) \\ \frac{$ 97.17}{1.0425} & = & purchase price \\ $ 93.21 & = & purchase price \end{matrix}$

The purchase price, the price before tax, was $93.21.

Solve Application Problems Involving Sales Tax

Solving problems involving sales tax follows the same ideas and steps as solving problems for markups. But here we will use the following formula:

$total price = purchase price + sales tax$

We can also use the formula:

$total price = purchase price \times (1 + sales tax rate)$

This can be seen in the following examples.

Example 17 Compute Sales Tax for Austin, Texas

Jillian visits Austin, Texas, and purchases a new set of weights for her home. She spends, including sales tax, $467.64. The sales tax rate in Austin Texas is 8.25%. How much of the total price is sales tax?

The sales tax paid for this purchase is the difference in the total price and the purchase price. We know the total price is $467.64. We also know the sales tax rate, which is 8.25%. In decimal form, this is 0.0825. Using these values and the formula $total price = purchase price \times (1 + tax rate)$ to find the purchase price.

$\begin{matrix} total price & = & purchase price \times (1 + tax rate) \\ $ 467.64 & = & purchase price \times (1 + 0.0825) \\ $ 467.64 & = & purchase price \times (1.0825) \\ $ 432 & = & purchase price \end{matrix}$

Knowing both the total price and the now the purchase price, we can find the difference, which is the sales tax.

The total price was $467.64. The purchase price was $432. The difference of the total price and the purchase price, or the sales tax, is then $467.64 − $432.00, which is $35.64. Jillian pays $35.64 in sales tax.

Key Terms

Discount
Cost
Markup
Retail price

Key Concepts

Discounts are markdowns from an original price.
Mark-ups are increases to the price paid by a retailer to cover their costs.
be able to calculate the markup based on a percentage of the cost
Sales taxes vary from state to state and often county to county.
Retail prices, sales prices and percent discounts can be calculated if the other two values are known.
Original costs, retail prices, and percent markup can be calculated if the other two values are known.
In calculations, sales tax acts like a markup.

Videos

Formulas

$discount = percent discount \times original price$

$sale price = original price - discount$

$sale price = original price - percent discount \times original price = original price \times (1 - percent discount)$

$markup = percent markup \times cost$

$retail price = cost + markup$

$retail price = cost + percent markup \times cost = cost \times (1 + percent markup)$

$sales tax = purchase price \times tax rate$

$Total price = purchase price + purchase price \times tax rate = purchase price \times (1 + tax rate)$

Adapted from Contemporary Mathematics by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.