6.2: Discounts, Markups, and Sales Tax
- 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 . 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.
The formula for a discount based on a percentage is , with the percent discount expressed as a decimal. The price of the item after the discount is .
These are often combined into the following formula
When the original price and the percent discount are known, the discount and the sale price can be directly computed.
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%
- Answer
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Substituting the values into the formula
, we find that the discount is
. The discount is $18.95.
The sale price of the item is then , or $56.85.
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Substituting the values into the formula
, we find that the discount is
. The discount is $50.67.
The sale price of the item is then , or $118.23.
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Substituting the values into the formula
, we find that the discount is
. The discount is $18.95.
Calculate the discount for the given original price and discount percentage. Then calculate the sale price.
Original price = $1,550.00; percent discount is 32%
Original price = $27.50; percent discount is 10%
Sometimes the original price and the sale price of an item is known. From this, the percent discount can be computed using the formula , by solving for the percent discount.
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
- Answer
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Step 1.
Find the discount. Using the original price and the sale price, we can find the discount with the formula
. Substituting and calculating, we find the discount to be
. 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 , we can find the percent discount.
Converting to percent form, the percent discount is 25%.
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Step 1.
Find the discount. Using the original price and the sale price, we can find the discount with the formula
. Substituting and calculating, we find the discount to be
. 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 , we can find the percent discount.
Converting to percent form, the percent discount is 20%.
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Step 1.
Find the discount. Using the original price and the sale price, we can find the discount with the formula
. Substituting and calculating, we find the discount to be
. Solving for the discount gives $300.00.
Determine the percent discount based on the given original and sale prices.
Original price = $250.00; sale price = $162.50
Original price = $19.50; sale price = $17.16
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 . The original price can be found by solving this equation for the original 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
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Using the percent discount and the sale price, we can find the original price with the formula
. Substituting and solving for the original price, we find
The original price of the item was $500.00.
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Using the percent discount and the sale price, we can find the original price with the formula
. Substituting and solving for the original price, we find
The original price of the item was $360.00.
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Using the percent discount and the sale price, we can find the original price with the formula
. Substituting and solving for the original price, we find
Determine the original price based on the percent discount and sale price.
Percent discount 15%; sale price = $11.05
Percent discount 9%; sale price = $200.20
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).
The sale rack at a clothing store is marked “All Items 30% off.” Ian finds a shirt that had an original price of $80.00. What is the discount on the shirt? What is the sale price of the shirt?
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We are asked to find the discount, and the sale price. We know the percent discount is 30%, or 0.30 in decimal form. The original price was $80.
Substituting into the percent discount formula, we find that the discount is .
The discount is $24 on that shirt. The sale price is the original price minus the discount, so the sale price is $80 – $24 = $56.
A bed originally priced at $550, but is on sale, with a 60% discount. What is the discount on the bed? What is the sale price of the bed?
An annual pass on the city bus is priced at $240. The student price, though, is $168. What is the percent discount for students for the bus pass?
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We know the original price of the item, $240. We also know the sale price of the item, $168. From this we know the discount is . Substituting these values into the formula , we can find the percent discount.
The student percent discount on the bus pass is 30%.
A pharmacy offers students at a nearby college a discount. Jerry purchases ibuprofen, which had an original price of $15.80. The cost to Jerry after the student discount was $13.43. What is the percent discount for students at the pharmacy?
Kendra’s car developed a flat, and the tire store told her that two tires had to be replaced. She got a 10% discount on the pair of tires, and the sale price came to $189.00. What was the original price of the tires?
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Using the percent discount and the sale price, we can find the original price with the formula . Substituting and solving for the original price, we find
The original price of the two tires Kendra bought was $210.00.
Marisol needed to buy a new microwave. She got a 26% discount. The sale price Marisol paid was $43.66. What was the original price of the microwave?
Computing Price Based on a Percent Off Coupon
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.
The formula for a markup based on a percentage is , with the percent markup expressed as a decimal. The price of the item after the markup is .
These are often combined into the following formula
It should be noted that the formulas used for a markup are very similar to those for a discount, with addition replacing the subtraction.
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%
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Substituting the values into the formula
, we find that the markup is
. The markup is $9.30.
The retail price of the item is then , or $62.00 + $9.30 = $71.30. -
Substituting the values into the formula
, we find that the markup is
. The markup is $337.50.
The retail price of the item is then , or $750.00 + $337.50 = $1,087.50.
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Substituting the values into the formula
, we find that the markup is
. The markup is $9.30.
Calculate the markup for the given cost and markup percentage. Then calculate the retail price.
Cost = $1,800.00; percent markup is 22%
Cost = $10.50; percent markup is 10%
Sometimes the cost and the retail price of an item are known. From this, the percent markup can be computed using the formula , by solving for the percent markup.
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
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Step 1:
Using the cost and the retail price, we can find the markup with the formula
. Substituting and calculating, we find the markup to be
. Solving for the markup gives $13.50.
Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula , we can find the percent markup.
Converting to percent form, the percent markup is 15%.
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Step 1:
Using the cost and the retail price, we can find the markup with the formula
. Substituting and calculating, we find the markup to be
. Solving for the markup gives $4.70.
Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula , we can find the percent markup.
Converting to percent form, the percent markup is 90.38%.
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Step 1:
Using the cost and the retail price, we can find the markup with the formula
. Substituting and calculating, we find the markup to be
. Solving for the markup gives $13.50.
Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places.
Cost = $120.00; retail price = $190.00
Cost = $0.38; retail price = $1.14
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 . The cost can be found by solving this equation for the cost.
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
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Using the percent markup and the retail price, we can find the cost with the formula
. Substituting and solving for the cost, we find
The cost of the item was $8.85.
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Using the percent markup and the retail price, we can find the cost with the formula
. Substituting and solving for the original price, we find
The cost of the item was $11.80.
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Using the percent markup and the retail price, we can find the cost with the formula
. Substituting and solving for the cost, we find
Determine the cost based on the percent markup and retail price.
Percent markup 15%; retail price = $40.25
Percent markup 300%; retail price = $35.96
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.
Janice works at a convenience store near campus. It sells protein bars at a 60% markup. If a bar costs the store $1.30, how much is the retail price at the convenience store?
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We are asked to find the retail price. We know the percent markup is 60%. The cost of the bar was $1.30. Substituting into the percent markup formula, we find that the markup is . The markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is . The retail price is $2.08.
A furniture outlet spends $360.00 to buy a bed. The store marks up the bed by 250%. What is the retail price of the bed?
Javi began working at a phone outlet. In a recent shipment, he noticed that the cost of the phone to the store was $480.00. The phone sells for $840.00 in the store. What is the percent markup on the phone?
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We know the cost of the phone, $480. We also know the retail price of the phone, $840.00. From this we know the markup is . Substituting these values into the formula , we can find the percent markup.
The markup on the phone is 75%.
Maggie does some research into textbook costs. The Sociology of the Family text she finds sells for $234.36 but costs the store only $189.00. What is the percent markup on the sociology book?
Bob decided to order a t-shirt for his gaming friend online for $29.50. He knows the markup on such t-shirts is 18%. What was the t-shirt’s cost before the markup?
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Using the percent markup and the retail price, $29.50, we can find the cost with the formula . Substituting and solving for cost we find
The cost of the t-shirt was $25.00.
Tina has opened a retail shop and purchased a unique hat for resale. Tina uses a 50% markup and sells the hat for $57.00. How much did the hat cost Tina?
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.
To calculate the amount of sales tax paid on the purchase price in a locality with sales tax given in decimal form, calculate The total price is then
When the sales tax calculation results in a fraction of a penny, then normal rounding rules apply, round up for half a penny or more, but round down for less than half a penny.
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.
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
- Answer
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- The sales tax is found using . The purchase price is $428.99 and the tax rate is 8.25%. Substituting and calculating, the sales tax is . 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 .
- The sales tax on the item is found using . The purchase price is $34.88 and the tax rate is 8.25%. Substituting and calculating, the sales tax is . 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 .
The sales tax in Union County, Oregon, is 7%. Find the sales tax and total price of items based on the purchase price listed.
Purchase price = $1,499.00
Purchase price = $26.89
As before, the information available might be different than only the purchase price and the sales tax rate. In these cases, use either or 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.
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
- Answer
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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 , the sales tax rate can be found. Substituting and solving yields
Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%.
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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 , the sales tax rate can be found. Substituting and solving yields
Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%.
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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.
Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax percentages.
Purchase price = $83.90; total price = $88.30
Purchase price = $477.00; total price = $509.20
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
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When the sales tax rate and the total price are known, the formula
can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find
The purchase price, the price before tax, was $34.57.
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When the sales tax rate and the total price are known, the formula
can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find
The purchase price, the price before tax, was $93.21.
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When the sales tax rate and the total price are known, the formula
can be used to find the purchase price. Substituting the tax rate and total price into the formula and solving, we find
Find the purchase price for the indicated sales tax rate and total price.
Sales tax rate = 8.25%; total price = $157.81
Sales tax rate = 6.75%; total price = $522.01
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:
We can also use the formula:
.
This can be seen in the following examples.
The sales tax rate in Denver Colorado is 8.81%. Keven buys a TV in Denver, and the purchase price (before taxes) is $499.00. How much will Keven pay in sales tax and what will be the total amount he spends when he buys the TV?
- Answer
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The sales tax rate in Denver is 8.81%. To find the sales tax Keven will pay, find 8.81% of the purchase price. In decimal form, that sales tax rate is 0.0881. Using the formula and substituting 499.00 for purchase price, we find that Keven will pay in sales tax for the TV.
The total price that Keven will pay is the purchase price plus the sales tax, or .
Daryl decides to buy a new scooter in St. Louis, Missouri, where the sales tax is 9.68%. The scooter he chooses has a purchase price of $1,149. How much will Daryl pay in sales tax and what is the total price he spends on the scooter?
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?
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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 to find the purchase price.
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.
Elizabeth decides to buy new running shoes in her hometown of Springfield, Illinois, where the sales tax rate is 6.25%. If her total bill comes to $153, how much of the total price is sales tax?
Finding Sales Tax Percentage
West Virginia was the first state to impose a sales tax. This happened on May 3, 1921.
Look up your locality on this website that lists standard state-level sales tax rates and compare the sales tax structure in your state to two nearby states (for the lower 48) and for any two states (Alaska and Hawaii).
Check Your Understanding
- What is a discount?
- What is a markup?
- An item has a retail price of $45.00. What is the sale price after a 32% discount?
- A retailer buys an item for $311.00. What is the retail price if their markup is 60%?
- Does sales tax have the same formula as markup?
- If the sales tax is 6.8%, what is the total price for an item that has a purchase price of $39.95?