8.7: Finding a Percent of a Number
Introduction
Discount Day at the Supermarket
The supermarket is having a special discount day in honor of its one year anniversary. On Saturday, discount day will begin and every customer will receive 15% off of his/her total order.
Saturday, the store is full of customers. There are free balloons being given out to the children along with coffee and donuts for the parents. Everyone is in a terrific mood and business is booming.
Many people are making large purchases. Mr. Kemp is excited to see how much business the store is getting, but at the same time he can’t help thinking about the profit he is losing by offering the 15% discount.
Mr. Kemp peers over the service desk to hear one of the girls at the cash register say, “Well, your total is $345.00 before the discount.”
Then the girl showed the customer her new total. The customer paid and left, smiling to Mr. Kemp as she passed by.
If this is the total before the discount, how much did the customer end up paying? How much money did the store lose by offering such a discount?
What You Will Learn
In this lesson you will learn to do the following:
- Find a percent of a number.
- Find prices involving discounts.
- Find prices involving sales tax.
- Estimate tips.
- Find amounts involving simple interest.
Teaching Time
Percents are found in real life all around us. We work with percents every day. In fact, they are so common that sometimes we don’t even realize that we are using them. This lesson takes what we have learned about percents and applies it in some different real world situations. Let’s begin by learning how to find the percent of a number.
I. Find a Percent of a Number
We can find the percent of another number. When we find the percent of a number, we want to figure out what part of the number is equal to the amount of the percent. Let’s look at an example.
Example
What is 10% of 25?
This is an example where we are looking for the percent of a number. We want to figure out 10% of 25. Said another way, we want to find a part of 25 that is the same as ten percent.
How can we figure out this problem?
We can figure out the percent of a number in two different ways. One is to use a proportion and one is to use key words and multiplication. Let’s look at using a proportion first.
How can we find the percent of a number using a proportion?
Remember that a proportion is created when two ratios are equal. We can compare the percent out of 100 with a part of another number.
We know that we need to find 10% of 25. The percent is out of 100 so we can write our first ratio by changing 10% into a fraction.
\begin{align*}10\% =\frac{10}{100}\end{align*}
Next, we change the 25 into a proportion. Now we are looking for what part of 25 is equal to 10%, so that is going to be what we need to find out of 25. Here is what it looks like.
\begin{align*}\frac{x}{25}\end{align*}
Our proportion is \begin{align*}\frac{10}{100}=\frac{x}{25}\end{align*}.
That’s correct! We solve the proportion to find our answer. Do you remember how to solve the proportion? We can do this by using cross products.
Next, we can write an equation.
\begin{align*}100x = 250\end{align*}
To solve this equation, we have to think "What times 100 is equal to 250?" We could also use the inverse operation of "times 100", and divide 250 by 100.
\begin{align*}x = 250 \div 100\end{align*}
\begin{align*}250 \div 100 = 2\underleftarrow{50.} = 2.5\end{align*}
Our answer is 2.5
Let’s review the steps of using a proportion!!
We can also use key words and multiplication to find the percent of a number.
Example
What is 10% of 25?
First, we look for any key words that mean an operation. The word “OF” means multiplication, so we are going to use multiplication to find an answer.
Next, we turn 10% into a decimal.
10% = .10
We are looking for 10% of 25, so we multiply the decimal .10 times 25 to find our answer.
\begin{align*}25\\ \underline{\times \ \ .10}\\ 00\\ \underline{+ \ \; 25 \ }\\ 250\end{align*}
Finally we put the decimal point into our product. We have two decimal places in .10 so we put it in two places counting from right to left.
Our answer is 2.5.
Notice that our answers are the same!! You can use either way to find the correct answer!!
It is time for you to practice a few of these on your own.
- What is 10% of 54?
- What is 25% of 80?
- What is 5% of 78?
Take a few minutes to check your work with a peer.
II. Find Prices Involving Discounts
Figuring out a discount is a time when we use a percent in a real life situation. A discount is an amount of money that is taken off of the original price. Think about shopping! We use discounts all the time when we shop. In fact, we often use mental math to figure out a discount.
We can find a discount and then a final price of the item we are purchasing. This involves two steps.
- Figure out the amount of the discount
- Subtract that amount from the original price
How do we figure out the amount of a discount?
To understand this, let’s look at an example.
Example
Tracy went shopping for a new pair of sneakers. She chose a pair of blue ones that were $58.00. The sign said that they were 15% off of the original price. What is the amount of the discount? How much did Tracy end up paying for the sneakers?
Our first step is to figure out the amount of the discount.
We need to find 15% of 58.
Good question! The original price of the sneakers is $58.00. The percent of the discount is 15%, so we can write that we need to find 15% of 58. That will give us the amount of money that Tracy did not have to pay.
Let’s multiply to find the amount of the discount.
\begin{align*}& 15\% \ \text{of} \ 58 =58\\ & \qquad \qquad \underline{\times \ \ .15}\\ & \qquad \qquad \quad \ 290\\ & \qquad \qquad \underline{+ \ \ 58 \ }\\ & \qquad \qquad \quad 8.70\end{align*}
The amount of the discount is $8.70.
Now we can subtract the amount of the discount from the original sales price and we will know how much Tracy paid for the sneakers.
\begin{align*}58.00\\ \underline{- \ \ 8.70}\\ \$49.30\end{align*}
Tracy paid $49.30 for the sneakers.
Figure these out on your own. First figure out the amount of the discount, then figure out the new price.
- If a $50.00 shirt is 25% off, how much would you pay for the shirt?
- If a video game that usually costs $45.50 is 30% off, how much would you pay for the game?
Now check your answers with a partner.
III. Find Prices Involving Sales Tax
When we figured out a price with a discount, we subtracted because a discount is an amount taken off of an original price. Sales tax is just the opposite!
What is sales tax?
Sales tax is an amount that is added to a price. Many states have sales tax. When you shop in those states, you have to add a sales tax to your total. Sales tax is a percentage.
How can we calculate a price that includes sales tax?
First, we figure out the amount of the sales tax. Then we add that to the original price to figure out the new price.
Let’s look at an example.
Example
The state of Maine has a sales tax of 5%. If you purchased a book for $25.00, how much would you pay for the book if you bought it in Maine?
First, we figure out the amount of sales tax for a $25.00 book. We need to find 5% of 25.
We change 5% to a decimal, .05, and then multiply.
\begin{align*}25\\ \underline{\times \ \ .05}\\ 1.25\end{align*}
The amount of sales tax is $1.25.
Now, because this is sales tax, we need to add this amount to our original price.
\begin{align*}\$25.00\\ \underline{+ \quad 1.25}\\ \$26.25\end{align*}
We would pay $26.25 for the book in the state of Maine.
Try a few of these on your own. Figure out the sales tax and then add it to the original price for a new total.
- What would you pay for a $35.00 book if the sales tax is 5%?
- What would you pay for a $99.00 a night hotel room if the sales tax is 7%?
Take a few minutes to check your work. Did you remember to ADD the amount of the sales tax?
IV. Estimate Tips
When you eat in a restaurant, you usually pay a tip to the server. A tip is a percent of the cost of the meal. It is also called a gratuity.
When ordering out, we can use estimation to figure out the tip for the server. It is customary to give the server 15% of the total of the meal.
We can use what we have learned about percents to estimate tips.
Let’s look at an example.
Example
John ate out for lunch. His total bill came to $12.00. How much should he tip the server?
How can we figure this out?
We know that we need to find approximately 15% of $12.00. 15% is between 10% and 20%. It is easy to find 10% because we can multiply by .10 or move the decimal point one place to the left in the total.
This would mean that 10% of $12.00 is $1.20. If 10% of $12.00 is 1.20, then 20% is 2.40.
We want to choose a number between these two amounts for 15%.
A reasonable estimate would be $1.75 or $2.00.
Sometimes people find it easier to calculate 20% using mental math so they often give the server a larger tip because it is easier to figure out!
Estimate the following tips.
- A tip on $25.00
- A tip on $18.00
Check your answers with a friend. Are your tip amounts reasonable?
V. Find Amounts Involving Simple Interest
Interest is another amount that is ADDED to a total. You hear the word “interest” when talking about borrowing money. When someone borrows money from the bank, the bank charges them a small percent of the amount that the person borrowed for each period of time that they have the money. The percent is often calculated annually or per year. In this way, the bank says, “We will loan you this money, but until you pay it back, you must pay us for each month or year that you have it.”
How can we calculate interest?
There are three main things that you need to know to calculate interest. You need to know the amount that was borrowed or the principal, the rate or the percentage the bank is charging to loan the money, and the time that you are keeping the money.
Here is a formula we can use to calculate interest.
\begin{align*}I &= prt\\ Interest &= principal \times rate \times time\end{align*}
We take the principal, multiply it with the rate, and multiply that with the length of time that the money has been borrowed, to find the interest.
Let’s look at an example.
Example
Carrie borrowed $500.00 from the bank. The bank charges 5% interest annually. If it takes Carrie 1 year to pay back the money, how much interest will she pay?
To figure this out, let’s use our formula.
\begin{align*}I = prt\end{align*}
The principal is $500.00.
The rate is 5% \begin{align*}= .05\end{align*}
The time is 1 year.
\begin{align*}& I=(500)(.05)(1)\\ \\ & \qquad \quad \;500\\ & \qquad \underline{\times \ \ .05}\\ & \qquad \ 25.00\end{align*}
Carrie will pay $25.00 in interest.
What if it had been 3 years before she had paid back the money?
If this was the case, we would have used this formula.
\begin{align*}I &= (500)(.05)(3)\\ I &= (25.00)(3)\\ I &= \$75.00\end{align*}
Carrie would have paid $75.00 in interest.
Try a few of these on your own.
- Mark borrowed $250.00 at 4% for 3 years. How much interest did he pay?
- Kris borrowed $300.00 at 2% for 2 years. How much interest did he pay?
Check your answers with a partner.
Real Life Example Completed
Discount Day at the Supermarket
Mr. Kemp is having a difficult time thinking about all of the money that he is losing by offering discount day at the grocery store. Here is the original problem. Reread it and underline all of the important information.
The supermarket is having a special discount day in honor of its one year anniversary. On Saturday, discount day will begin and every customer will receive 15% off of his/her total order.
Saturday, the store is full of customers. There are free balloons being given out to the children along with coffee and donuts for the parents. Everyone is in a terrific mood and business is booming.
Many people are making large purchases. Mr. Kemp is excited to see how much business the store is getting, but at the same time he can’t help thinking about the profit he is losing by offering the 15% discount.
Mr. Kemp peers over the service desk to hear one of the girls at the cash register say, “Well, your total is $345.00 before the discount.”
Then the girl showed the customer her new total. The customer paid and left, smiling to Mr. Kemp as she passed by.
If this is the total before the discount, how much did the customer end up paying? How much money did the store lose by offering such a discount?
There are two questions to answer in solving this problem. First, we need to figure out the amount of the discount given the total and the 15% off.
345 \begin{align*}\times\end{align*} .15 \begin{align*}=\end{align*} $51.75
The discount is $51.75.
This is the amount that the store lost.
What did the customer end up paying?
To figure this out, we take the discount and subtract it from the original total.
345 - 51.75 = $293.25
WOW!! That customer saved a lot of money by shopping on discount day!
Vocabulary
Here are the vocabulary words that are found in this lesson.
- Percent
- a part of a whole 100, written using a % sign.
- Proportion
- two equal ratios.
- Sales Tax
- a percent added to a total
- Discount
- an amount taken off of an original price
- Tip
- 15 – 20% of a total bill paid to a server
- Gratuity
- another word for tip
- Interest
- the sum of money a person pays a bank for borrowing money
- Principal
- the amount of money borrowed
- Rate
- a percent that the bank charges for borrowing money
- Annually
- per year
Technology Integration
Khan Academy Finding a Percent
Khan Academy Finding Full Price when you know the Discounted Price
James Sousa Solving Percent Problems Using the Percent Equation
James Sousa Example of Solving Percent Problems Using the Percent Equation
James Sousa Another Example of Solving Percent Problems Using the Percent Equation
James Sousa Another Example of Solving Percent Problems Using the Percent Equation
Other Videos:
http://www.gamequarium.org/cgi-bin/search/linfo.cgi?id=7634 – This video shows how to find the percent of a number.
Time to Practice
Directions: Find the percent of each number.
1. What is 2% of 10?
2. What is 5% of 50?
3. What is 10% of 30?
4. What is 25% of 18?
5. What is 20% of 36?
6. What is 11% of 40?
7. What is 8% of 80?
8. What is 15% of 45?
9. What is 20% of 100?
10. What is 25% of 250?
11. What is 4% of 60?
12. What is 5% of 85?
Directions: Calculate each new price based on the discount and the original price.
13. Original price: $19.95, discount 15%
14. Original price: $29.95, discount 20%
15. Original price: $18.00, discount 10%
16. Original price: $47.50, discount 10%
17. Original price: $75.00, discount 30%
18. Original price: $125.00, discount 20%
19. Original price: $225.50, discount 10%
20. Original price: $456.00, discount 25%
Directions: Calculate the total amount paid including sales tax if the sales tax is 4%.
21. Total: $56.75
22. Total: $43.25
23. Total: $65.00
24. Total: $25.50
25. Total: $18.75
26. Total: $59.00
27. Total: $21.50
28. Total: $44.50
29. Total: $125.50
30. Total: $430.00