Solve Sales Tax, Commission, and Discount Applications

6.3 Solve Sales Tax, Commission, and Discount Applications

The topics covered in this section are:

  1. Solve sales tax applications
  2. Solve commission applications
  3. Solve discount applications
  4. Solve mark-up applications

6.3.1 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.

HOW TO: Solve an application

  1. Identify what you are asked to find and choose a variable to represent it.
  2. Write a sentence that gives the information to find it.
  3. Translate the sentence into an equation.
  4. Solve the equation using good algebra techniques.
  5. Check the answer in the problem and make sure it makes sense.
  6. Write a complete sentence that answers the question.

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.

The figure shows a restaurant check with sales tax
Figure 6.7 The sales tax is calculated as a percent of the purchase price.

SALES TAX

The sales tax is a percent of the purchase price.

$\mathrm{Sales\ Tax} = \mathrm{Tax\ Rate} \cdot \mathrm{Purchase\ Price}$

$\mathrm{Total\ Cost} = \mathrm{Purchase\ Price} + \mathrm{Sales\ Tax}$

Example 1

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

  1. the sales tax and
  2. the total cost of a bicycle if the purchase price of the bicycle was $ \$ 392$?
Solution

Part 1.
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)..
Simplify.$t=25.48$
Check: Is this answer reasonable?
Yes, because the sales tax amount is less than $10 \%$ of the purchase price.
Write a complete sentence that answers the question.The sales tax is $ \$25.48$.
Part 2.
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 the 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..
Simplify.$c=417.48$
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$.

Example 2

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

Solution

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..
Divide.$\large \frac{499r}{499} = \frac{42.42}{499}$
Simplify.$r=0.085$
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 $8.5 \%$

6.3.2 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.

$\mathrm{commission} = \mathrm{rate\ of\ commission} \cdot \mathrm{total\ sales}$

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 3

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

Solution

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..
Simplify.
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$.

Example 4

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

Solution

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..
Divide.$\large \frac{87}{1450} = \frac{1450r}{1450}$
Simplify.$0.06=r$
Change to percent form.$r=6 \%$
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.

6.3.3 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 figure shows a sale sign with a discount rate
Figure 6.8 Applications of discounts are common in everyday life. (credit: Charleston’s TheDigitel, Flickr)

DISCOUNT

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

$\mathrm{amount\ of\ discount} = \mathrm{discount\ rate} \cdot \mathrm{original\ price}$

$\mathrm{sale\ price} = \mathrm{original\ price} – \mathrm{discount}$

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 5

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

Solution

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.$s=29$
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$.

In Example 5, the amount of discount was a set amount, $ \$10$. In Example 6 the discount is given as a percent of the original price.

Example 6

Elise bought a dress that was discounted $35 \%$ off of the original price of $ \$140$. What was Part 1. the amount of discount and Part 2. the sale price of the dress?

Solution

Part 1. 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..
Simplify.$d=49$
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$.

Part 2.

Original price $= \$ 140$

Amount of discount $= \$ 49$

Sale price $=$?

Identify what you are asked to find.What is the amount of discount?
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.$s=91$
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$.

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 7

Jeannette bought a swimsuit at a sale price of $ \$ 13.95$. The original price of the swimsuit was $ \$31$. Find the Part 1. amount of discount and Part 2. discount rate.

Solution

Part 1. Before beginning, you may find it helpful to organize the information in a list.

Original price $= $ \$31$

Amount of discount $=$?

Sale price $= \$ 13.95$

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 the original price minus the sale price.
Translate into an equation..
Simplify.$d=17.05$
Check if this answer is reasonable.
Yes. A $ \$17.05$ discount is rless than the original price.
Write a complete sentence that answers the question.The amount of discount was $ \$17.05$.

Part 2. 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..
Divide.$\large \frac{17.05}{31} = \frac{r(31)}{31}$
Simplify.$0.55=r$
Check if this answer is reasonable.
The rate of discount was a little more than $50 \%$ and the amount of discount is a little more that half of $ \$31$.
Write a complete sentence that answers the question.The rate of discount was $55 \%$

6.3.4 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.

$\mathrm{amount\ of\ mark-up} = \mathrm{mark-up\ rate} \cdot \mathrm{wholesale\ price}$

$\mathrm{list\ price} = \mathrm{wholesale\ price} + \mathrm{mark\ up}$

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

Example 8

Adam’s art gallery bought a photograph at the wholesale price of $ \$250$. Adam marked the price up $40 \%$. Find the Part 1. amount of mark-up and Part 2. the list price of the photograph.

Solution

Part 1.
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..
Simplify.$m=100$
Check if this answer is reasonable.
Yes. The markup 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$.
Part 2.
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..
Simplify.$p=350$
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$.
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