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Discounts, Markups, and Sales Tax

Calculate discounts.

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.

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 $\text{part}=\text{percentage}\times \text{total}$. 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.

Calculating Discount for a Percent Discount

Try it.

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%

Solution

Substituting the values into the formula $\text{discount}=percent discount\times original price$, we find that the discount is $\text{discount}=0.25\times 75.80=18.95$. The discount is $18.95.
The sale price of the item is then $sale price=original price-\text{discount}=75.80-18.95=56.85$, or $56.85.
Substituting the values into the formula $\text{discount}=percent discount\times original price$, we find that the discount is $\text{discount}=0.30\times 168.90=50.67$. The discount is $50.67.
The sale price of the item is then $sale price=original price-\text{discount}=168.90-50.67=118.23$, 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 $\text{discount}=percent discount\times original price$, by solving for the percent discount.

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.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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).

Determine Discount and New Price a Sale Rack Item

Try it.

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?

Solution

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 $\text{discount}=percent discount\times original price=0.30\times 80=24$.

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.

Determine the Percent Discount of a Bus Pass

Try it.

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?

Solution

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 $\text{\$}240-\text{\$}168=\text{\$}72$. Substituting these values into the formula $\text{discount}=percent discount\times original price$, we can find the percent discount.

\[\begin{array}{lll}\text{discount} & = & percent discount\times original price \\ 72 & = & percent discount\times 240 \\ \frac{72}{240} & = & \frac{percent discount\times 240}{240} \\ 0.3 & = & percent discount\end{array}\]

The student percent discount on the bus pass is 30%.

Finding the Original Price of a New Pair of Tires

Try it.

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?

Solution

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

\[\begin{array}{lll}sales price & = & original price\times (1-percent discount) \\ 189.00 & = & original price\times (1-0.10) \\ 189.00 & = & original price\times (0.90) \\ 210.00 & = & original price\end{array}\]

The original price of the two tires Kendra bought was $210.00.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

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

Try it.

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%

Solution

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

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 $retail price=\text{cost}\times (1+percent markup)$. The cost can be found by solving this equation for the cost.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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.

Determine Retail Price of a Power Bar

Try it.

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?

Solution

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 $\text{markup}=percent markup\times \text{cost}=0.60\times 1.30=0.78$ . The markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is $retail price=\text{cost}+\text{markup}=1.30+0.78=2.08$. The retail price is $2.08.

Determine the Percent Markup of a Phone

Try it.

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?

Solution

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 $\text{\$}840.00-\text{\$}480.00=\text{\$}360.00$. Substituting these values into the formula $\text{markup}=percent markup\times \text{cost}$, we can find the percent markup.

\[\begin{array}{lll}\text{markup} & = & percent markup\times \text{cost} \\ 360 & = & percent markup\times 480 \\ \frac{360}{480} & = & \frac{percent markup\times 480}{480} \\ 0.75 & = & percent markup\end{array}\]

The markup on the phone is 75%.

Finding the Cost of a T-Shirt

Try it.

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?

Solution

Using the percent markup and the retail price, $29.50, we can find the cost with the formula $retail price=original price\times (1+percent markup)$. Substituting and solving for cost we find

\[\begin{array}{lll}retail price & = & \text{cost}\times (1+percent markup) \\ 29.50 & = & \text{cost}\times (1+0.18) \\ 29.50 & = & \text{cost}\times (1.18) \\ 25.00 & = & \text{cost}\end{array}\]

The cost of the t-shirt was $25.00.

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.

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.

Condensed — the full section is in OpenStax Contemporary Mathematics.

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:

\[\text{total price}=\text{purchase price}+\text{ sales tax }\]

We can also use the formula:

\[\text{total price}=\text{purchase price}\times \text{(1}+\text{sales tax rate)}\]

This can be seen in the following examples.

Compute Sales Tax for Denver, Colorado

Try it.

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?

Solution

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 $purchase price\times tax rate=\$499\times 0.0881=\text{\$}43.96$ in sales tax for the TV.

The total price that Keven will pay is the purchase price plus the sales tax, or $\text{\$}499.00+\text{\$}43.96=\text{\$}542.96$.

Compute Sales Tax for Austin, Texas

Try it.

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?

Solution

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.

\[\begin{array}{lll}total price & = & purchase price\times (1+tax rate) \\ \text{\$}467.64 & = & purchase price\times (1+0.0825) \\ \text{\$}467.64 & = & purchase price\times (1.0825) \\ \text{\$}432 & = & purchase price\end{array}\]

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 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.

Formulas

$\text{discount}=\text{percent discount}\times \text{original price}$

$\text{sale price}=\text{original price}-\text{discount}$

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

$\text{markup}=\text{percent markup}\times \text{cost}$

$\text{retail price}=\text{cost}+\text{markup}$

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

$\text{sales tax}=\text{purchase price}\times \text{tax rate}$

Condensed — the full section is in OpenStax Contemporary Mathematics.

Practice (17)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Calculate the discount for the given price and discount percentage. Then calculate the sale price.
1. Original price = $75.80; percent discount is 25%
2. Original price = $168.90; percent discount is 30%
答えを明らかにしろ
1. Substituting the values into the formula $\text{discount}=percent discount\times original price$, we find that the discount is $\text{discount}=0.25\times 75.80=18.95$. The discount is $18.95.
  The sale price of the item is then $sale price=original price-\text{discount}=75.80-18.95=56.85$, or $56.85.
2. Substituting the values into the formula $\text{discount}=percent discount\times original price$, we find that the discount is $\text{discount}=0.30\times 168.90=50.67$. The discount is $50.67.
  The sale price of the item is then $sale price=original price-\text{discount}=168.90-50.67=118.23$, or $118.23.
Determine the percent discount based on the given original and sale prices.
1. Original price = $1,200.00; sale price = $900.00
2. Original price = $36.70; sale price = $29.52
答えを明らかにしろ
1. Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $sale price=original price-\text{discount}$. Substituting and calculating, we find the discount to be $900.00=1,200.00-\text{discount}$. 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 $\text{discount}=percent discount\times original price$, we can find the percent discount.
  \[\begin{array}{lll}300.00 & = & percent discount\times 1,200.00 \\ \frac{300.00}{1,200.00} & = & percent discount \\ 0.25 & = & percent discount\end{array}\]
  Converting to percent form, the percent discount is 25%.
2. Step 1. Find the discount. Using the original price and the sale price, we can find the discount with the formula $sale price=original price-\text{discount}$. Substituting and calculating, we find the discount to be $29.52=36.70-\text{discount}$. 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 $\text{discount}=percent discount\times original price$, we can find the percent discount.
  \[\begin{array}{lll}29.52 & = & percent discount\times 36.70 \\ \frac{29.52}{36.70} & = & percent discount \\ 0.2 & = & percent discount\end{array}\]
  Converting to percent form, the percent discount is 20%.
Determine the original price based on the percent discount and sale price.
1. Percent discount 10%; sale price = $450.00
2. Percent discount 75%, sale price = $90.00
答えを明らかにしろ
1. 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
  \[\begin{array}{lll}sale price & = & original price\times (1-percent discount) \\ 450.00 & = & original price\times (1-0.10) \\ 450.00 & = & original price\times (0.90) \\ 500.00 & = & original price\end{array}\]
  The original price of the item was $500.00.
1. Using the percent discount and the sale price, we can find the original price with the formula$sale price=original price\times (1-percent discount)$. Substituting and solving for the original price, we find
  \[\begin{array}{lll}sale price & = & original price\times (1-percent discount) \\ 90.00 & = & original price\times (1-0.75) \\ 90.00 & = & original price\times (0.25) \\ 360.00 & = & original price\end{array}\]
  The original price of the item was $360.00.
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?

答えを明らかにしろ

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 $\text{discount}=percent discount\times original price=0.30\times 80=24$.
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.
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?

答えを明らかにしろ

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 $\text{\$}240-\text{\$}168=\text{\$}72$. Substituting these values into the formula $\text{discount}=percent discount\times original price$, we can find the percent discount.
\[\begin{array}{lll}\text{discount} & = & percent discount\times original price \\ 72 & = & percent discount\times 240 \\ \frac{72}{240} & = & \frac{percent discount\times 240}{240} \\ 0.3 & = & percent discount\end{array}\]
The student percent discount on the bus pass is 30%.
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?

答えを明らかにしろ

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
\[\begin{array}{lll}sales price & = & original price\times (1-percent discount) \\ 189.00 & = & original price\times (1-0.10) \\ 189.00 & = & original price\times (0.90) \\ 210.00 & = & original price\end{array}\]
The original price of the two tires Kendra bought was $210.00.
Calculate the markup for the given cost and markup percentage. Then calculate the retail price.
1. Cost = $62.00; percent markup is 15%
2. Cost = $750.00; percent markup is 45%
答えを明らかにしろ
1. Substituting the values into the formula $\text{markup}=percent markup\times \text{cost}$, we find that the markup is $\text{markup}=0.15\times 62.00=9.30$. The markup is $9.30.
  The retail price of the item is then $retail price=\text{cost}+\text{markup}$, or $62.00 + $9.30 = $71.30.
2. Substituting the values into the formula $\text{markup}=percent markup\times \text{cost}$, we find that the markup is $\text{markup}=0.45\times 750.00=337.50$. The markup is $337.50.
  The retail price of the item is then $retail price=\text{cost}+\text{markup}$, or $750.00 + $337.50 = $1,087.50.
Determine the percent markup based on the given cost and retail price. Round percentages to two decimal places.
1. Cost = $90.00; retail price = $103.50
2. Cost = $5.20; retail price = $9.90
答えを明らかにしろ
1. Step 1: Using the cost and the retail price, we can find the markup with the formula $retail price=\text{cost}+\text{markup}$. Substituting and calculating, we find the markup to be $103.50=90.00+\text{markup}$. Solving for the markup gives $13.50.
  Step 2: After substituting the markup, $13.50, and the original price, $90.00, into the formula $\text{markup}=percent markup\times \text{cost}$, we can find the percent markup.
  \[\begin{array}{lll}13.50 & = & percent markup\times 90.00 \\ \frac{13.50}{90.00} & = & percent markup \\ 0.15 & = & percent markup\end{array}\]
  Converting to percent form, the percent markup is 15%.
1. Step 1: Using the cost and the retail price, we can find the markup with the formula $retail price=\text{cost}+\text{markup}$. Substituting and calculating, we find the markup to be $9.90=5.20+\text{markup}$. Solving for the markup gives $4.70.
  Step 2: After substituting the markup, $4.70, and the original price, $5.20, into the formula $\text{markup}=percent markup\times \text{cost}$, we can find the percent markup.
  \[\begin{array}{lll}4.70 & = & percent markup\times 5.20 \\ \frac{4.70}{5.20} & = & percent markup \\ 0.9038 & = & percent markup\end{array}\]
  Converting to percent form, the percent markup is 90.38%.
Determine the cost based on the percent markup and retail price.
1. Percent markup 20%; retail price = $10.62
2. Percent markup 125%; retail price = $26.55
答えを明らかにしろ
1. Using the percent markup and the retail price, we can find the cost with the formula $retail price=\text{cost}\times (1+percent markup)$. Substituting and solving for the cost, we find
  \[\begin{array}{lll}retail price & = & \text{cost}\times (1+percent markup) \\ 10.62 & = & \text{cost}\times (1+0.2) \\ 10.62 & = & \text{cost}\times (1.2) \\ 8.85 & = & \text{cost}\end{array}\]
  The cost of the item was $8.85.
1. Using the percent markup and the retail price, we can find the cost with the formula$retail price=\text{cost}\times (1+percent markup)$. Substituting and solving for the original price, we find
  \[\begin{array}{lll}retail price & = & \text{cost}\times (1+percent markup) \\ 26.55 & = & \text{cost}\times (1+2.25) \\ 26.55 & = & \text{cost}\times (3.25) \\ 11.80 & = & \text{cost}\end{array}\]
  The cost of the item was $11.80.
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?

答えを明らかにしろ

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 $\text{markup}=percent markup\times \text{cost}=0.60\times 1.30=0.78$ . The markup is $0.78 on that protein bar. The retail price is the cost plus the markup, so the retail price is $retail price=\text{cost}+\text{markup}=1.30+0.78=2.08$. The retail price is $2.08.
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?

答えを明らかにしろ

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 $\text{\$}840.00-\text{\$}480.00=\text{\$}360.00$. Substituting these values into the formula $\text{markup}=percent markup\times \text{cost}$, we can find the percent markup.
\[\begin{array}{lll}\text{markup} & = & percent markup\times \text{cost} \\ 360 & = & percent markup\times 480 \\ \frac{360}{480} & = & \frac{percent markup\times 480}{480} \\ 0.75 & = & percent markup\end{array}\]
The markup on the phone is 75%.
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?

答えを明らかにしろ

Using the percent markup and the retail price, $29.50, we can find the cost with the formula $retail price=original price\times (1+percent markup)$. Substituting and solving for cost we find
\[\begin{array}{lll}retail price & = & \text{cost}\times (1+percent markup) \\ 29.50 & = & \text{cost}\times (1+0.18) \\ 29.50 & = & \text{cost}\times (1.18) \\ 25.00 & = & \text{cost}\end{array}\]
The cost of the t-shirt was $25.00.
The sales tax in Kankakee, Illinois, is 8.25%. Find the sales tax and total price of items based on the purchase price listed.
1. Purchase price = $428.99
2. Purchase price = $34.88
答えを明らかにしろ
1. 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 $sales tax=\text{\$}428.99\times 0.0825=\text{\$}35.391675$. 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 $\text{\$}428.99+\text{\$}35.88=\text{\$}464.87$.
2. 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 $sales tax=\text{\$}34.88\times 0.0825=\text{\$}2.8776$. 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 $\text{\$}34.88+\text{\$}2.88=\text{\$}37.76$.
Find the sales tax rate for the indicated purchase price and total price. Round using the guideline for sales tax percentages.
1. Purchase price = $329.50; total price = $354.21
2. Purchase Price = $13.77; total price = $14.39
答えを明らかにしろ
1. 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
  \[\begin{array}{lll}Sales Tax & = & purchase price\times tax rate \\ \text{\$}24.71 & = & \text{\$}329.50\times tax rate \\ \frac{\text{\$}24.71}{\text{\$}329.50} & = & tax rate \\ 0.07499 & = & tax rate\end{array}\]
  Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 7.5%.
1. 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
  \[\begin{array}{lll}sales tax & = & purchase price\times tax rate \\ \text{\$}0.62 & = & \text{\$}13.77\times tax rate \\ \frac{\text{\$}0.62}{\text{\$}13.77} & = & tax rate \\ 0.04503 & = & tax rate\end{array}\]
  Keeping in mind the guideline for rounding sales tax rate, the sales tax rate is 4.5%.
Find the purchase price for the indicated sales tax rate and total price.
1. Sales tax rate = 5.75%; total price = $36.56
2. Sales tax rate = 4.25%; total price = $97.17
答えを明らかにしろ
1. 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
  \[\begin{array}{lll}Total price & = & purchase price\times (1+tax rate) \\ \text{\$}36.56 & = & purchase price\times (1+0.0575) \\ \frac{\text{\$}36.56}{1.0575} & = & purchase price \\ \text{\$}34.57 & = & purchase price\end{array}\]
  The purchase price, the price before tax, was $34.57.
1. 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
  \[\begin{array}{lll}Total price & = & purchase price\times (1+tax rate) \\ \text{\$}97.17 & = & purchase price\times (1+0.0425) \\ \frac{\text{\$}97.17}{1.0425} & = & purchase price \\ \text{\$}93.21 & = & purchase price\end{array}\]
  The purchase price, the price before tax, was $93.21.
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?

答えを明らかにしろ

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 $purchase price\times tax rate=\$499\times 0.0881=\text{\$}43.96$ in sales tax for the TV.
The total price that Keven will pay is the purchase price plus the sales tax, or $\text{\$}499.00+\text{\$}43.96=\text{\$}542.96$.
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?

答えを明らかにしろ

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.
\[\begin{array}{lll}total price & = & purchase price\times (1+tax rate) \\ \text{\$}467.64 & = & purchase price\times (1+0.0825) \\ \text{\$}467.64 & = & purchase price\times (1.0825) \\ \text{\$}432 & = & purchase price\end{array}\]
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.

Symbols used here

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: 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.
Original price = $75.80; percent discount is 25%
Original price = $168.90; percent discount is 30%

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

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Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

ここに Arithmetic

Order of operations Fractions Percentages Powers and roots