6.6: Applications Using Percents
Introduction
Adding it up
One afternoon while Taylor is working in the candy store, a couple decided to purchase some chocolates. They spent a lot of time looking through all of the boxes, but finally decided on a large assorted box of some very lovely dark chocolates.
“That is a good choice,” Taylor said. “That box is also on sale for 15% off.”
“Terrific,” said the man.
Taylor took the chocolates and went to the cash register. As she went to put in the pricing, she noticed that there wasn’t anything happening on the screen. She walked to the kitchen in the back where her Mom was working.
“Mom, the register won’t work,” Taylor said.
“Well I am in the middle of making taffy. You’ll have to do the math in your head,” her Mom responded.
Taylor walked back in and picked up a piece of paper and a pencil.
The original price of the chocolates was $26.50.
There is a 15% discount.
Also, there is a sales tax of 4% on all candy purchases.
Taylor began figuring out the math while the couple waited patiently.
This lesson is all about using percents. Pay close attention and at the end of the lesson you will be able to help Taylor with her sticky situation.
What You Will Learn
By the end of this lesson, you will be able to demonstrate the following skills:
- Find prices involving markups and discounts.
- Estimate appropriate tips.
- Find total bills involving tips and taxes.
- Solve real-world problems involving discounts, markups, sales tax and tips.
Teaching Time
I. Find Prices Involving Markups and Discounts
In this lesson, you will learn how to use percents in everyday life. A storeowner buys an item at a wholesale cost and marks it up by adding a percent of the price to the price he paid for it. If we buy that item when it is on sale, we can find the amount of the discount if we know the rate of discount that the store is offering. On some items that we buy, we pay a sales tax that is a percent of the price. When we eat out in a restaurant, we leave a tip that is a percent of the amount of the meal.
Let’s begin by thinking about markups and discounts.
A markup is an increase in the price of an item. A discount is a decrease in the price of an item. Both of these can be expressed as an amount of money or as a percent of the original price of the item.
Why would someone use a markup?
A markup is often how a store or a business makes a profit. They buy merchandise at one price and sell it for another price. The difference in the price they bought it at and the price they sell it at is the profit margin.
Why would someone use a discount?
Stores discount items all the time. Often they are still making a profit, but they discount an item by a specific percentage to try to sell more of that item.
To figure out a discount or a markup, we need to know the percent of change. What is a percent of change?
The percent of change is the percent that the price is changing by. If the markup is 10%, then that is the percent of change. If the discount is 10%, then that is the percent of change.
We can use the following formula:
Percent of change \begin{align*} \times \end{align*} original amount = amount of the change
Write this formula in your notebook and then continue with the lesson.
Example
A camera store buys a camera for $149 and marks up the price by 35%. What price does the camera sell for?
First, find the amount of the markup.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 35 \% \times \$ 149\\ & = 0.35 \times \$ 149\\ & = \$ 52.15\end{align*}
The markup is $52.15.
Now find the selling price by adding the markup to the wholesale price.
\begin{align*}\$ 149 + \$ 52.15 = \$ 201.15\end{align*}
The selling price is $201.15
Example
A camera that normally sells for $189 is on sale at a 20% discount. What is the sale price?
First, find the amount of the discount.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 20 \% \times \$ 189\\ & = 0.20 \times \$ 189\\ & = \$ 37.80 \end{align*}
The discount is $37.80.
Now find the sale price by subtracting the discount from the original price.
\begin{align*}\$ 189 - \$ 37.80 = \$ 151.20\end{align*}
The sale price is $151.20.
6P. Lesson Exercises
Find the markup or discount of each price.
- A camera store buys a camera for $159.00 and marks it up 10%. What is the new price of the camera?
- A chair that costs $199.00 is on sale for 15% off. What is the price of the chair?
Take a few minutes to check your work with a peer. Are your answers accurate?
II. Estimate Appropriate Tips
When you go to a restaurant, it is customary to tip the server. If you have help at the airport, it is customary to tip the baggage assistant. If you have someone carry your bags at a hotel, it is customary to tip the bellhop.
What is a tip?
A tip is an amount of money that is given to a worker such as a waiter or waitress who performs a service for you. A common tip amount is 15% of the cost of the meal or other service. Since you don’t usually calculate a tip exactly, we can estimate the amount of a tip.
Follow these steps to estimate a 15% tip
Step 1: Round the total bill to the nearest whole dollar amount.
Step 2: Find 10% of this amount by moving the decimal point one place to the left.
Step 3: Find half of the amount you found in Step 2. This will give you 5% of the total cost.
Step 4: Add the 10% amount and the 5% amount together.
Take a few minutes to write these steps in your notebook.
Now let’s apply them to a few examples.
Example
Estimate a 15% tip on a meal whose cost is $29.75.
Step 1: Round $29.75 to $30.
Step 2: 10% of \begin{align*}\$ 30 = \$ 3\end{align*}
Step 3: Half of \begin{align*}\$ 3 = \$ 1.50\end{align*}
Step 4: \begin{align*}\$ 3 + \$ 1.50 = \$ 4.50\end{align*}
A 15% tip on a meal whose cost is $29.75 is about $4.50.
Example
Estimate a 20% tip on a meal whose cost is $54.15.
Since the tip is about 20%, you can find 10% and then double it.
Step 1: Round $54.15 to $54.
Step 2: 10% of \begin{align*}\$ 54 = \$ 5.40\end{align*}
Step 3: 20% of \begin{align*}\$ 5.40 = 2 \times \$ 5.40 = \$ 10.80\end{align*}
A 20% tip on a meal whose cost is $54.15 is about $10.80.
6Q. Lesson Exercises
Find the amount of a 15% tip for each bill.
- $29.00
- $16.00
- $55.00
Take a few minutes to check your work. Did you estimate? Compare your estimates with a friend. Be sure that your estimate and your friend's estimate are close.
III. Find Total Bills Involving Tips and Taxes
You just finished figuring out a tip. When you calculate the tip, then you add this amount back to the total of the bill for the final cost of the meal. Sometimes, the tip is left in cash on the table. When this happens, you don’t have to add it to the price of the meal. However, when using a credit card, you have to add the tip to the total to be sure that the server receives his/her tip.
What about taxes?
Taxes are monies that are collected by the government to help pay for services such as schools, libraries, roads, and police and fire protection. A sales tax is a tax on something bought in a store. The rate of sales tax is given as a percent. You can find the amount of sales tax by using the following formula:
Amount of sales tax = price \begin{align*}\times\end{align*} rate of sales tax
After you calculate the amount of a tax using the percentage and the formula, you have to add it to an original amount. With sales tax, this is especially important. Let’s look at a few examples.
Example
Emily bought a jacket for $85. If the sales tax is 7.5%, find the total cost of the jacket.
First find the amount of sales tax.
7.5% of \begin{align*}\$ 85 = 0.075 \times \$ 85 = \$ 6.375\end{align*}
You need to round $6.375 to the nearest penny, which is $6.38.
Add the price of the jacket and the sales tax.
\begin{align*}\$ 85 + \$ 6.38 = \$ 91.38\end{align*}
The total cost of the jacket is $91.38.
Example
Mr. and Mrs. Green and their four children went out to dinner. The cost of the meal was $72. The restaurant added a tip of 18% to the bill. What was the total cost of the dinner?
First find the amount of the tip.
18% of \begin{align*}\$ 72 = 0.18 \times \$ 72 = \$ 12.96\end{align*}
Add the cost of the meal and the tip.
\begin{align*}\$ 72 + \$ 12.96 = \$ 84.96\end{align*}
The total cost of the dinner was $84.96.
6R. Lesson Exercises
Find the total amount of each bill.
- 5% sales tax on a $65.00 item.
- 20% tip on a bill of $30.00
- 10% sales tax on a $150.00 item.
Take a few minutes to check your work with a partner.
IV. Solve Real-World Problems Involving Discounts, Markups, Sales Tax and Tips
Now that you have learned about discounts, markups, sales tax, and tips, let's practice what you have learned to solve more real-world problems.
Example
Brandon bought a T-shirt for $8.99 and a pair of shorts for $11.99. If the rate of sales tax is 7%, how much did he spend?
First find the total cost of the items purchased.
\begin{align*}\$ 8.99 + \$ 11.99 = \$ 20.98\end{align*}
Next find the amount of sales tax.
7% of \begin{align*}\$ 20.98 = 0.07 \times \$ 20.98 = \$ 1.4686\end{align*}
You need to round $1.4686 to the nearest penny, which is $1.47.
Add the total cost of the items and the sales tax.
\begin{align*}\$ 20.98 + \$ 1.47 = \$ 22.45\end{align*}
Brandon paid $22.45.
Example
A pair of shoes with a wholesale price of $40 was marked up 50%. During a sale, the shoes were discounted 33%. What was the sale price?
First find the amount of markup.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 50 \% \times \$ 40\\ & = 0.50 \times \$ 40\\ & = \$ 20\end{align*}
The markup is $20.
Now find the selling price by adding the markup to the wholesale price.
The selling price is \begin{align*}\$ 40 + \$ 20 = \$ 60.\end{align*}
Now find the amount of discount.
\begin{align*}\text{Amount of change} & = \text{percent of change} \times \text{original amount}\\ & = 33 \% \times \$ 60\\ & = 0.33 \times \$ 60\\ & = \$ 19.80\end{align*}
The discount is $19.80.
Now find the sale price by subtracting the discount from the selling price.
\begin{align*}\$ 60 - \$ 19.80 = \$ 40.20\end{align*}
The sale price is $40.20.
Remember the problem in the introduction about the candy store? Well, now that you know all about tips, markups, discounts and sales tax, let’s go back and figure out that problem.
Real Life Example Completed
Adding it up
Here is the original problem once again. Reread it and underline any important information. Then figure out the total of the purchase.
One afternoon while Taylor is working in the candy store, a couple decided to purchase some chocolates. They spent a lot of time looking through all of the boxes, but finally decided on a large assorted box of some very lovely dark chocolates.
“That is a good choice,” Taylor said. “That box is also on sale for 15% off.”
“Terrific,” said the man.
Taylor took the chocolates and went to the cash register. As she went to put in the pricing, she noticed that there wasn’t anything happening on the screen. She walked to the kitchen in the back where her Mom was working.
“Mom, the register won’t work,” Taylor said.
“Well I am in the middle of making taffy. You’ll have to do the math in your head,” her Mom responded.
Taylor walked back in and picked up a piece of paper and a pencil.
The original price of the chocolates was $26.50.
There is a 15% discount.
Also, there is a sales tax of 4% on all candy purchases.
Taylor began figuring out the math while the couple waited patiently.
First, figure out the discount.
\begin{align*}26.50 \times .15 = 3.975\end{align*}
Round up to 3.98
Next, subtract this discount from the original price.
\begin{align*}26.50 - 3.98 = \$ 22.52\end{align*}
Now figure out the sales tax.
\begin{align*}22.52 \times .04 = .90\end{align*}
Finally, add the sales tax to the discounted price.
\begin{align*}22.52 + .90 = \$ 23.42\end{align*}
The couple will pay $23.42 for the box of chocolates.
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Markup
- the amount that a store marks up an item that has been purchased at a wholesale cost.It is how stores make money on the items that they sell. It is a percent added to the wholesale cost.
- Discount
- a percentage off of a selling price. It is an amount that is subtracted from the original price.
- Percent of Change
- the percentage amount that a price is being reduced or increased by.
- Tip
- a percent that is added to a price or given for a service.
- Estimate
- an approximate calculation. It is not exact.
- Taxes
- a percent added to a bill as in a sales tax. It is an amount of money collected by the government.
Technology Integration
James Sousa, Example of a Percent Application Problem
James Sousa, Another Example of a Percent Application Problem
James Sousa, Another Example of a Percent Application Problem
James Sousa, Another Example of a Percent Application Problem
Other Videos:
- http://www.mathplayground.com/howto_findsaleprice.html – This is a video on calculating a sales price.
- http://www.mathplayground.com/howto_percentwp.html – This is a video on how to calculate a tip.
Time to Practice
Directions: Here is a list of wholesale prices. Figure out each sale price if the markup is 20%.
1. $19.00
2. $12.00
3. $18.00
4. $25.00
5. $13.50
6. $9.95
7. $45.00
8. $90.00
9. $85.00
10. $17.00
Directions: Each item is discounted 15%. Figure out each new sale price; the original prices are listed below.
11. $55.00
12. $35.50
13. $18.00
14. $8.75
15. $25.00
16. $40.00
17. $35.00
18. $12.50
19. $38.90
20. $16.75
Directions: Solve each word problem.
21. At an end of summer sale, an air conditioner that originally sold for $350 was on sale at a discount of 25%. What was the sale price?
22. Leah is buying a comforter at a department store. The regular price is $89 and it is on sale at a 20% discount. In addition, Leah has a coupon that will give her an additional 10% off the discounted price. How much will Leah pay for the comforter?
23. Mr. Smith owns a hardware store. He marks up merchandise by 32% over the wholesale cost. How much would he charge for a ladder that costs him $62.50?
24. Estimate a 15% tip on a restaurant bill of $16.60.
25. Estimate a 20% tip on a restaurant bill of $38.95.
26. Alex bought a CD for $13.99 and a book for $6.99. The sales tax rate is 5.5% in the state where Alex lives. He gave the clerk $25. How much change did he receive?
27. Find the total cost of a meal including 7% sales tax and a 15% tip on a meal whose cost is $25.75.
28. This year the base price of a new car is marked up 4% over the same model from last year. The base price last year was $14,500. The sales tax rate is 8%. How much will it cost to buy the base model this year?