# 1.6: Solve Sales Tax, Commission, and Discount Applications

- Page ID
- 152026

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\id}{\mathrm{id}}\)

\( \newcommand{\Span}{\mathrm{span}}\)

\( \newcommand{\kernel}{\mathrm{null}\,}\)

\( \newcommand{\range}{\mathrm{range}\,}\)

\( \newcommand{\RealPart}{\mathrm{Re}}\)

\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

\( \newcommand{\Argument}{\mathrm{Arg}}\)

\( \newcommand{\norm}[1]{\| #1 \|}\)

\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)

\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)

\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vectorC}[1]{\textbf{#1}} \)

\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)After completing this section, you should be able to:

- Solve sales tax applications
- Solve commission applications
- Solve discount applications
- Solve mark-up applications

Sales tax and commissions are applications of percent in our everyday lives.

We will continue working with the percent equation that we learned in Section 1.5, while learning about new applications. Many of these applications should be familiar to you from your everyday life, so remember to always check that the answer makes sense.

Remember that whatever the application, once we write the sentence with the given information (Step 2 in our How To Solve an Application Problem), we can translate it to a percent equation and then solve it.

### Solve Sales Tax Applications

Do you pay a tax when you shop in your city or state? In many parts of the United States, sales tax is added to the purchase price of an item. See Figure 1.6.1. The sales tax is determined by computing a percent of the purchase price.

To find the sales tax multiply the purchase price by the sales tax rate. Remember to convert the sales tax rate from a percent to a decimal number. Once the sales tax is calculated, it is added to the purchase price. The result is the total cost—this is what the customer pays.

### Sales Tax

The sales tax is a percent of the purchase price.

**Sales Tax = Tax Rate (as a decimal number) x Purchase Price**

**Total Cost = Purchase Price + Sales Tax**

### Example 1.6.1

Cathy bought a bicycle in Washington, where the sales tax rate was $\text{6.5\%}$ of the purchase price. What was

ⓐ the sales tax and

ⓑ the total cost of a bicycle if the purchase price of the bicycle was $\text{\$392}?$

**Answer**-
ⓐ Identify what you are asked to find. What is the sales tax? Choose a variable to represent it. Let $t=$ sales tax. Write a sentence that gives the information to find it. The sales tax is 6.5% of the purchase price. Translate into an equation. (Remember to change the percent to a decimal). Solve the equation by simplifying. Check: Is this answer reasonable? Yes, because the sales tax is less than 10% of the purchase price. Write a complete sentence that answers the question. The sales tax is $25.48. ⓑ Identify what you are asked to find. What is the total cost of the bicycle? Choose a variable to represent it. Let $c=$ total cost of bicycle. Write a sentence that gives the information to find it. The total cost is the purchase price plus the sales tax. Translate into an equation. Solve by simplifying. Check: Is this answer reasonable? Yes, because the total cost is a little more than the purchase price. Write a complete sentence that answers the question. The total cost of the bicycle is $417.48.

### Your Turn 1.6.1

Find ⓐ the sales tax and ⓑ the total cost: Alexandra bought a television set for $\text{\$724}$ in Boston, where the sales tax rate was $\text{6.25\%}$ of the purchase price.

### Example 1.6.2

Evelyn bought a new smartphone for $\text{\$499}$ plus tax. She was surprised when she got the receipt and saw that the tax was $\text{\$42.42}.$ What was the sales tax rate for this purchase?

**Answer**-
Identify what you are asked to find. What is the sales tax rate? Choose a variable to represent it. Let $r=$ sales tax. Write a sentence that gives the information to find it. What percent of the price is the sales tax? Translate into an equation. Solve by dividing. Simplify. Check. Is this answer reasonable? Yes, because 8.5% is close to 10%. 10% of $500 is $50, which is close to $42.42. Write a complete sentence that answers the question. The sales tax rate is 8.5%.

### Your Turn 1.6.2

Diego bought a new car for $\text{\$26,525}.$ He was surprised that the dealer then added$\text{\$2,387.25}.$ What was the sales tax rate for this purchase?

### Solve Commission Applications

Sales people often receive a commission, or percent of total sales, for their sales. Their income may be just the commission they earn, or it may be their commission added to their hourly wages or salary. The commission they earn is calculated as a certain percent of the price of each item they sell. That percent is called the **rate of commission**.

### Commission

A commission is a percentage of total sales as determined by the rate of commission.

**commission = rate of commission (as a decimal number) x total sales**

To find the commission on a sale, multiply the rate of commission by the total sales. Just as we did for computing sales tax, remember to first convert the rate of commission from a percent to a decimal.

### Example 1.6.3

Helene is a realtor. She receives $\text{3\%}$ commission when she sells a house. How much commission will she receive for selling a house that costs $\text{\$260,000}?$

**Answer**-
Identify what you are asked to find. What is the commission? Choose a variable to represent it. Let $c=$ the commission. Write a sentence that gives the information to find it. The commission is 3% of the price. Translate into an equation. Solve by simplifying. Check. Is this answer reasonable? Yes. 1% of $260,000 is $2,600. And $7,800 is three times $2,600. Write a complete sentence that answers the question. Helene will receive a commission of $7,800.

### Your Turn 1.6.3

Bob is a travel agent. He receives $\text{7\%}$ commission when he books a cruise for a customer. How much commission will he receive for booking a $\text{\$3,900}$ cruise?

### Example 1.6.4

Rikki earned $\text{\$87}$ commission when she sold a $\text{\$1,450}$ stove. What rate of commission did she get?

**Answer**-
Identify what you are asked to find. What is the rate of commission? Choose a variable to represent it. Let $r=$the rate of commission. Write a sentence that gives the information to find it. The commission is what percent of the sale? Translate into an equation. Solve by dividing. Simplify. Change to percent form. Check if this answer is reasonable. Yes. A 10% commission would have been $145. The 6% commission, $87, is a little more than half of that. Write a complete sentence that answers the question. The commission was 6% of the price of the stove.

### Your Turn 1.6.4

Homer received $\text{\$1,140}$ commission when he sold a car for $\text{\$28,500}.$ What rate of commission did he get?

### Solve Discount Applications

Applications of discount are very common in retail settings. When you buy an item on sale, the **original price** of the item has been reduced by some dollar amount. The **discount rate**, usually given as a percent, is used to determine the amount of the discount. To determine the **amount of discount**, we multiply the discount rate by the original price. We summarize the discount model in the box below.

### Discount

An amount of discount is a percent off the original price.

**Amount of Discount = Discount Rate (as a decimal number) x Original Price**

**Sale Price = Original Price - Amount of Discount**

The sale price should always be less than the original price. In some cases, the amount of discount is a fixed dollar amount. Then we just find the sale price by subtracting the amount of discount from the original price.

### Example 1.6.5

Jason bought a pair of sunglasses that were on sale for $\text{\$10}$ off. The original price of the sunglasses was $\text{\$39}.$ What was the sale price of the sunglasses?

**Answer**-
Identify what you are asked to find. What is the sale price? Choose a variable to represent it. Let $s=$ the sale price. Write a sentence that gives the information to find it. The sale price is the original price minus the discount. Translate into an equation. Simplify. Check if this answer is reasonable. Yes. The sale price, $29, is less than the original price, $39. Write a complete sentence that answers the question. The sale price of the sunglasses was $29.

### Your Turn 1.6.5

Marta bought a dishwasher that was on sale for $\text{\$75}$ off. The original price of the dishwasher was $\text{\$525}.$ What was the sale price of the dishwasher?

In Example 1.6.5, the amount of discount was a set amount. Let's look at an example now where the amount of discount is a percent of the original price.

### Example 1.6.6

Elise bought a dress that was discounted $\text{35\%}$ off of the original price of $\text{\$140}.$ What was ⓐ the amount of discount and ⓑ the sale price of the dress?

**Answer**-
ⓐ Before beginning, you may find it helpful to organize the information in a list.

Original price = $140

Discount rate = 35%

Amount of discount = ?Identify what you are asked to find. What is the amount of discount? Choose a variable to represent it. Let $d=$ the amount of discount. Write a sentence that gives the information to find it. The discount is 35% of the original price. Translate into an equation. Solve by simplifying. Check if this answer is reasonable. Yes. A $49 discount is reasonable for a $140 dress. Write a complete sentence that answers the question. The amount of discount was $49. ⓑ

Original price = $140

Amount of discount = $49

Sale price = ?Identify what you are asked to find. What is the sale price of the dress? Choose a variable to represent it. Let $s=$ the sale price. Write a sentence that gives the information to find it. The sale price is the original price minus the discount. Translate into an equation. Solve by simplifying. Check if this answer is reasonable. Yes. The sale price, $91, is less than the original price, $140. Write a complete sentence that answers the question. The sale price of the dress was $91.

### Your Turn 1.6.6

Find ⓐ the amount of discount and ⓑ the sale price: Sergio bought a belt that was discounted $\text{40\%}$ from an original price of $\text{\$29}.$

There may be times when you buy something on sale and want to know the discount rate. The next example will show this case.

### Example 1.6.7

Jeannette bought a swimsuit at a sale price of $\text{\$13.95}.$ The original price of the swimsuit was $\text{\$31}.$ Find the ⓐ amount of discount and ⓑ discount rate.

**Answer**-
ⓐ Before beginning, you may find it helpful to organize the information in a list.

Original price = $31

Amount of discount = ?

Sale price = $13.95Identify what you are asked to find. What is the amount of discount? Choose a variable to represent it. Let $d=$ the amount of discount. Write a sentence that gives the information to find it. The discount is the original price minus the sale price. Translate into an equation. Simplify. Check if this answer is reasonable. Yes. The $17.05 discount is less than the original price of $31. Write a complete sentence that answers the question. The amount of discount was $17.05. ⓑ Before beginning, you may find it helpful to organize the information in a list.

Original price = $31

Amount of discount = $17.05

Discount rate = ?Identify what you are asked to find. What is the discount rate? Choose a variable to represent it. Let $r=$ the discount rate. Write a sentence that gives the information to find it. The discount is what percent of the original price? Translate into an equation. Solve by dividing. Simplify. Check if this answer is reasonable. Yes. The rate of discount was a little more than 50% and the amount of discount is a little more than half of $31. Write a complete sentence that answers the question. The rate of discount was 55%.

### Your Turn 1.6.7

Find ⓐ the amount of discount and ⓑ the discount rate: Lena bought a kitchen table at the sale price of $\text{\$375.20}.$ The original price of the table was $\text{\$560}.$

### Solve Mark-up Applications

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the **wholesale price**. The retailer then adds a **mark-up** to the wholesale price to get the **list price**, the price he sells the item for. The mark-up is usually calculated as a percent of the wholesale price. The percent is called the **mark-up rate**. To determine the amount of mark-up, multiply the mark-up rate by the wholesale price. We summarize the mark-up model in the box below.

### Mark-up

The mark-up is the amount added to the wholesale price.

**Amount of Mark-up = Mark-up Rate (as a decimal number) x Wholesale Price**

**List Price = Wholesale Price + Amount of Mark-Up**

The list price should always be more than the wholesale price.

### Example 1.6.8

Adam's art gallery bought a photograph at the wholesale price of$\text{\$250}.$ Adam marked the price up$\text{40}$$\text{\%}.$ Find the ⓐ amount of mark-up and ⓑ the list price of the photograph.

**Answer**-
ⓐ Identify what you are asked to find. What is the amount of mark-up? Choose a variable to represent it. Let $m=$the amount of each mark-up. Write a sentence that gives the information to find it. The mark-up is 40% of the wholesale price. Translate into an equation. Solve by simplifying. Check if this answer is reasonable. Yes. The mark-up rate is less than 50% and $100 is less than half of $250. Write a complete sentence that answers the question. The mark-up on the photograph was $100. ⓑ Identify what you are asked to find. What is the list price? Choose a variable to represent it. Let $p=$ the list price. Write a sentence that gives the information to find it. The list price is the wholesale price plus the mark-up. Translate into an equation. Solve by simplifying. Check if this answer is reasonable. Yes. The list price, $350, is more than the wholesale price, $250. Write a complete sentence that answers the question. The list price of the photograph was $350.

### Your Turn 1.6.8

Jim's music store bought a guitar at wholesale price $\text{\$1,200}.$ Jim marked the price up $\text{50\%}.$ Find the ⓐ amount of mark-up and ⓑ the list price.