6.3 Solve Sales Tax, Commission, and Discount Applications

Learning objectives

Learning objectives

• Solve sales tax applications

• Solve commission applications

• Solve discount applications

• Solve mark-up applications

Solve Sales Tax Applications

Sales tax and commissions are applications of percent in our everyday lives. To solve these applications, we will follow the same strategy we used in the section on decimal operations. We show it again here for easy reference.

Remember that whatever the application, once we write the sentence with the given information (Step 2), we can translate it to a percent equation and then solve it.

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 6.7. 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.

Example 6.25

Solution

1. Write the sales-tax equation. Let $t$ be the tax on the 392-dollar purchase. The first figure translates $6.5\%$ of the price as $t=0.065(392)$.
2. Calculate the tax. Multiplying gives $t=25.48$, which the second figure records.
3. Add tax to the purchase price. The third figure forms $c=392+25.48$, and the last figure shows $c=417.48$. Thus, the sales tax was $25.48$ and the total bicycle cost was $417.48$.

Takeaway: Find sales tax as a percent of the price first, and only then add it to obtain the total cost.

Example 6.26

Solution

1. Form the percent equation. Let $r$ be the decimal tax rate. The first figure shows $r(499)=42.42$, because the $42.42$ tax is $r$ of the $499$ price.
2. Divide both sides by $499$. The second figure gives $\frac{499r}{499}=\frac{42.42}{499}$, so $r=\frac{42.42}{499}$.
3. Convert to a percent. The last figure displays $r=0.085$, and $0.085=8.5\%$. Therefore, the sales tax rate was $8.5\%$.

Takeaway: The tax rate is the tax amount divided by the pre-tax price, expressed as a percent.

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.

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 6.27

Solution

1. Write the commission equation. Let $c$ be the commission. The first figure translates $3\%$ of the $260,000$ price as $c=0.03(260,000)$.
2. Evaluate the product. Multiplying gives $c=7,800$, as shown in the second figure.
3. State the result. Helene will receive a commission of $7,800$ on the $260,000$ house sale.

Takeaway: Convert the commission percent to a decimal before multiplying it by the full sale price.

Example 6.28

Solution

1. Form the percent equation. Let $r$ be the decimal commission rate. The first figure matches the $87$ commission with $r$ times the $1,450$ sale price, giving $87=1450r$.
2. Divide both sides by $1,450$. The second figure shows $\frac{1450r}{1450}=\frac{87}{1450}$, so $r=\frac{87}{1450}$.
3. Convert to a percent. The last two figures show $r=0.06=6\%$. Therefore, Rikki received a $6\%$ commission rate.

Takeaway: A commission rate is found by dividing the commission amount by the total sale price.

Solve Discount Applications

Applications of discount are very common in retail settings Figure 6.8. 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.

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 6.29

Solution

1. Identify the original price and discount. The sunglasses originally cost $39$, and the store takes $10$ off that price.
2. Form the sale-price equation. Let $s$ be the sale price. The first figure shows $s=39-10$, because subtracting the discount from the original price gives the amount paid.
3. Evaluate the subtraction. The second figure gives $s=29$. The sale price of the sunglasses was therefore $29$.

Takeaway: For a discount stated as a set dollar amount, subtract that amount directly from the original price.

In Example 6.29, the amount of discount was a set amount, $\text{\textdollar 10}.$ In Example 6.30 the discount is given as a percent of the original price.

Example 6.30

Solution

1. Identify the original price and rate. The dress originally costs $140$, and the discount rate is $35\%$, which has decimal form $0.35$.
2. Form the discount equation. Let $d$ be the discount amount. Since the discount is $35\%$ of the original price, $d=0.35(140)$.
3. Calculate the discount. Multiplying gives $d=49$, so the amount removed from the original price is $49$ dollars.
4. Calculate the sale price. Subtract the discount from the original price with $140-49=91$, so the dress sells for $91$ dollars.

Takeaway: The $35\%$ discount on the $140$ dress in Example 6.30 removes $49$ dollars and leaves a sale price of $91$ dollars.

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 6.31

Solution

1. Identify the two prices. The original swimsuit price is $31$, and the sale price is $13.95$, so the discount is the difference between them.
2. Find the discount amount. Subtracting gives $31-13.95=17.05$, so $17.05$ dollars was removed from the original price.
3. Form the discount-rate equation. Let $r$ be the discount rate. The discount is $r$ times the original price, so $17.05=31r$.
4. Solve for the rate. Dividing by $31$ gives $r=\frac{17.05}{31}=0.55$.
5. Convert and check. Multiplying $0.55$ by $100$ gives $55\%$, and $0.55(31)=17.05$ confirms the original discount amount.

Takeaway: In Example 6.31, the swimsuit discount is $17.05$, and dividing it by the $31$ original price gives a $55\%$ discount rate.

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.

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

Example 6.32

Solution

1. Write the mark-up equation. Let $m$ be the amount of the mark-up. The first figure shows $m=0.40(250)$, because the mark-up is $40\%$ of the $250$ wholesale price.
2. Calculate the mark-up. Multiplying gives $m=100$, as displayed in the second figure.
3. Add the mark-up to the wholesale price. The third figure forms $p=250+100$, and the fourth shows $p=350$. Thus, the mark-up was $100$ and the list price was $350$.

Takeaway: When a mark-up is based on the wholesale price, the list price equals wholesale price plus the calculated mark-up.

Section 6.3 Exercises

Section 6.3 Exercises

Solve Sales Tax Applications

In the following exercises, find ⓐ the sales tax and ⓑ the total cost.

In the following exercises, find the sales tax rate.

Solve Commission Applications

In the following exercises, find the commission.

In the following exercises, find the rate of commission.

Solve Discount Applications

In the following exercises, find the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

Solve Mark-up Applications

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

1. What is the sales tax rate if a $\text{\textdollar 7,594}$ purchase will have $\text{\textdollar 569.55}$ of sales tax added to it?

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

3. Fernando receives $\mathrm{18\%}$ commission when he makes a computer sale. How much commission will he receive for selling a computer for $\text{\textdollar 2,190}?$

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

2. Bernice earned $\text{\textdollar 451}$ commission when she sold an $\text{\textdollar 8,200}$ living room set. What rate of commission did she get?

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

1. Orlando bought a pair of shoes that was on sale for $\text{\textdollar 30}$ off. The original price of the shoes was $\text{\textdollar 112}.$ What was the sale price of the shoes?

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

3. Find ⓐ the amount of discount and ⓑ the sale price: Oscar bought a barbecue grill that was discounted $\mathrm{65\%}$ from an original price of $\text{\textdollar 395}.$

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

2. Find ⓐ the amount of discount and ⓑ the discount rate: Nick bought a multi-room air conditioner at a sale price of $\text{\textdollar 340}.$ The original price of the air conditioner was $\text{\textdollar 400}.$

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

1. The Auto Resale Store bought Pablo's Toyota for $\text{\textdollar 8,500}.$ They marked the price up $\mathrm{35\%}.$ Find the ⓐ amount of mark-up and ⓑ the list price.

2. The cost of a pair of boots was $\text{\textdollar 84}.$ The sales tax rate is $\mathrm{5\%}$ of the purchase price.

3. The cost of a microwave oven was $\text{\textdollar 129}.$ The sales tax rate is $\mathrm{7.5\%}$ of the purchase price.

1. The cost of a file cabinet is $\text{\textdollar 250}.$ The sales tax rate is $\mathrm{6.85\%}$ of the purchase price.

2. The cost of a $\mathrm{6-drawer}$ dresser $\text{\textdollar 1,199}.$ The sales tax rate is $\mathrm{5.125\%}$ of the purchase price.

3. Shawna bought a mixer for $\text{\textdollar 300}.$ The sales tax on the purchase was $\text{\textdollar 19.50}.$

1. Bopha bought a bedroom set for $\text{\textdollar 3,600}.$ The sales tax on the purchase was $\text{\textdollar 246.60}.$

2. Christopher sold his dinette set for $\text{\textdollar 225}$ through an online site, which charged him $\mathrm{9\%}$ of the selling price as commission. How much was the commission?

3. Farrah works in a jewelry store and receives $\mathrm{12\%}$ commission when she makes a sale. How much commission will she receive for selling a $\text{\textdollar 8,125}$ ring?

1. Hector receives $\mathrm{17.5\%}$ commission when he sells an insurance policy. How much commission will he receive for selling a policy for $\text{\textdollar 4,910}?$

2. Dontay is a realtor and earned $\text{\textdollar 11,250}$ commission on the sale of a $\text{\textdollar 375,000}$ house. What is his rate of commission?

3. As a waitress, Emily earned $\text{\textdollar 420}$ in tips on sales of $\text{\textdollar 2,625}$ last Saturday night. What was her rate of commission?

1. Maureen earned $\text{\textdollar 7,052.50}$ commission when she sold a $\text{\textdollar 45,500}$ car. What was the rate of commission?

2. Perla bought a cellphone that was on sale for $\text{\textdollar 50}$ off. The original price of the cellphone was $\text{\textdollar 189}.$

3. Sophie saw a dress she liked on sale for $\text{\textdollar 15}$ off. The original price of the dress was $\text{\textdollar 96}.$

1. Rick wants to buy a tool set with original price $\text{\textdollar 165}.$ Next week the tool set will be on sale for $\text{\textdollar 40}$ off.

2. Janelle bought a beach chair on sale at $\mathrm{60\%}$ off. The original price was $\text{\textdollar 44.95}$

3. Kathy wants to buy a camera that lists for $\text{\textdollar 389}.$ The camera is on sale with a $\mathrm{33\%}$ discount.

1. Erys bought a treadmill on sale at $\mathrm{35\%}$ off. The original price was $\text{\textdollar 949.95}.$

2. Jay bought a guitar on sale at $\mathrm{45\%}$ off. The original price was $\text{\textdollar 514.75}.$

3. Larry and Donna bought a sofa at the sale price of $\text{\textdollar 1,344}.$ The original price of the sofa was $\text{\textdollar 1,920}.$

1. Patty bought a baby stroller on sale for $\text{\textdollar 301.75}.$ The original price of the stroller was $\text{\textdollar 355.}$

2. Nikki bought a patio set on sale for $\text{\textdollar 480}.$ The original price was $\text{\textdollar 850}.$

3. Daria bought a bracelet at wholesale cost $\text{\textdollar 16}$ to sell in her handicraft store. She marked the price up $\mathrm{45\%}.$

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

2. Find ⓐ the sales tax and ⓑ the total cost: Kim bought a winter coat for $\text{\textdollar 250}$ in St. Louis, where the sales tax rate was $\mathrm{8.2\%}$ of the purchase price.

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

1. Tom paid $\text{\textdollar 0.60}$ a pound for tomatoes to sell at his produce store. He added a $\mathrm{33\%}$ mark-up.

2. Alan bought a used bicycle for $\text{\textdollar 115}.$ After re-conditioning it, he added $\mathrm{225\%}$ mark-up and then advertised it for sale.

3. Michael bought a classic car for $\text{\textdollar 8,500}.$ He restored it, then added $\mathrm{150\%}$ mark-up before advertising it for sale.

1. Coupons Yvonne can use two coupons for the same purchase at her favorite department store. One coupon gives her $\text{\textdollar 20}$ off and the other gives her $\mathrm{25\%}$ off. She wants to buy a bedspread that sells for $\text{\textdollar 195}.$

1. ⓐ Calculate the discount price if Yvonne uses the $\text{\textdollar 20}$ coupon first and then takes $\mathrm{25\%}$ off.

2. ⓑ Calculate the discount price if Yvonne uses the $\mathrm{25\%}$ off coupon first and then uses the $\text{\textdollar 20}$ coupon.

3. ⓒ In which order should Yvonne use the coupons?

2. Priam bought a jacket that was on sale for $\mathrm{40\%}$ off. The original price of the jacket was $\text{\textdollar 150}.$ While the sales clerk figured the price by calculating the amount of discount and then subtracting that amount from $\text{\textdollar 150},$ Priam found the price faster by calculating $\mathrm{60\%}$ of $\text{\textdollar 150}.$

1. ⓐ Explain why Priam was correct.

2. ⓑ Will Priam's method work for any original price?