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3. Percents
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Chapter 2
3. 

Percents

Navigating the world of percents can seem daunting, but this lesson simplifies the subject by teaching how to convert between different forms of numbers and percents. The lesson delves into real-life applications, such as calculating discounts during sales or determining fat content in food. Whether one is a student aiming to excel in a math test or an individual looking to understand percentages in daily life, this lesson offers invaluable insights. It covers everything from the basics of what a percent is, to more complex conversions involving mixed numbers and fractions. By the end, individuals will be equipped to handle any percent-related challenge that comes their way.

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Student Learning Objectives:
  • Convert between percents and fractions, mixed numbers, and decimals
17 Theory slides
11 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
Percents
Slide of 17
Percents are an easy way to express and relate data. This lesson will focus on converting different forms of numbers into percents and vice versa. The lesson will also show how percents appear in everyday life.

Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.

Challenge

Phone's Battery Level

Paulina realizes that she forgot to charge her phone last night. The battery is at only 14 %!

A phone with 14 percent battery level
External credits: @jannoon028
She connects her phone to a charger and starts getting ready for the day.

a

What fraction in its simplest form expresses the current battery level?

b

Paulina disconnects her phone as soon as she is ready to go to school. The battery is charged up to 56 of its capacity. What percent does Paulina see on the screen?

Discussion

Percents

A percent is the ratio of a number to 100. Percents are usually denoted by the symbol %.

p % = p/100

For example, five percent is the ratio of 5 to 100. This number can also be written as 5 %. 5 % = 5/100 Percentages are often used in real-life situations. For example, saline is a mixture of salt and water used to clean and store contact lenses. Saline contains 0.9 % salt. This means that every 100 grams of saline contains 0.9 grams of salt. 0.9 % = 0.9/100 Another situation in which percentages can be used is sales. For example, if a jacket is on sale, a store may advertise that it is 20 % off. Then, the price has been lowered by 20 %. If the jacket originally cost $100, then the price was lowered by $ 20.

20 % = 20/100
Discussion

Converting Fractions Into Percents

Fractions can be converted into percents. Consider the following fraction. 14/87 This fraction can be rewritten as a percent by following two steps.

1
Divide the Numerator by the Denominator
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First, divide the numerator by the denominator. In this case, the numerator is 14 and the denominator is 87. The procedure of long division can be used to divide up to four decimal places.

long division of 14 and 87

The quotient is 0.1609. This is the decimal equivalent of the fraction.

2
Multiply by 100
expand_more
Multiply the result by 100 to convert it to percent.

0.1609 becomes 16.09 after multiplying by 100

The result is 16.09, which can be rounded to 16.1. Finally, add a percent sign after the number. 14/87≈ 16.1 % Therefore, the fraction represents about 16.1 %.

Note that the order of these two steps can be reversed. In other words, the fraction can be multiplied by 100 first, before calculating the quotient of the numerator by the denominator.

Extra

 Converting Different Fractions
Input a fraction whose numerator and denominator are less than 100. Then, click Submit and Convert to see how it can be converted into a percent. In this applet, the fraction will be multiplied by 100 first.

The submitted fraction is converted to a percent
Example

Getting the Quiz Score

Paulina got 23 out of 25 answers correct on her math quiz.

Test sheet with a score A

Express her score as a percent.

Hint
Which number is the total and which is the part of the total? Use the definition of a percent.

Solution
We are given that Paulina answered 23 out of 25 questions correctly. Let's use this information to first write a fraction. Since the total number of questions on the quiz is 25, this is the denominator. The number of answers Paulina got correct is 23, which represents a part of 25, so it is the numerator. 23/25 Now we can convert this fraction to a percent. Start by writing it as a fraction with a denominator of 100. Note that 25 is a factor of 100. 25* 4=100 This means that multiplying the numerator and denominator by 4 will result in an equivalent fraction with a denominator of 100.

23/25
ExpandFrac

a/b=a * 4/b * 4

23 * 4/25 * 4
Multiply

Multiply

92/100

Recall the definition of a percent — the ratio of a number to 100. n % =n/100 This means that the fraction represents 92 %. 92/100=92 % Therefore, Paulina answered 92 % of the questions on her math quiz correctly.

Alternative Solution
The fraction 2325 represents the number of correct answers Paulina got on the test. This number can be rewritten as a percent by following two steps. First, let's divide the numerator by the denominator by using long division.

The long division of 23 over 25

The quotient is 0.92. Next, multiply this decimal number by 100 to write it as a percent. This can be done by moving the decimal point two places to the right.

0.92 becomes 92.0 after multiplying by 100

The product is 92.0, or 92. If we add a percent sign, we can conclude that Paulina got 92 % of the answers correct on her math quiz.

Discussion

Converting Percents Into Fractions

Percents can be converted into fractions. Consider the following percent. 48 % This percent can be written as a fraction by following two steps.

1
Write as a Fraction with a Denominator of 100
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Since a percent is the ratio of a number to 100, write it as a fraction with a numerator of the number part of the percent and a denominator denominator of 100. 48 %=48/100

2
Simplify the Fraction
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Next, check whether the fraction can be simplified. First, split the numerator and denominator into prime factors. This process is shown below for 48100. 48 & = 2* 2* 2* 2* 3 100 & = 2* 2* 5* 5 The numbers share two common factors. Their product is the greatest common factor (GCF) of 48 and 100. GCF(48,100)=2* 2=4 Finally, divide both the numerator and denominator by 4.

48/100
ReduceFrac

a/b=.a /4./.b /4.

48 ÷ 4/100 ÷ 4
CalcQuot

Calculate quotient

12/25

The fractions 1225 and 48100 are equivalent and they both correspond to 48 %.

Extra

 Converting Different Percents Between 1 and 100
Input an integer percent between 1 and 100. The process of converting this percent into a fraction will be demonstrated.

The submitted percent is converted to a fraction
Example

Choosing Between Two Lunch Options

At lunch, Paulina is trying to decide between juice and a sandwich or soda and a taco. Her friend Davontay interrupts her thoughts with an interesting fact.

Image shows two students in a school cafeteria choosing between two lunch options. Option 1 (sandwich and juice) is preferred by 68% of students. Option 2 (taco and soda) is on the right.
External credits: @pch.vector

Paulina is curious about how many students out of 25 would choose the different options if this statistic were in fact true.

a

What fraction in simplest form corresponds to the percent of students who choose Option 2?

b

How many students out of 25 would choose Option 1?


Hint
a

Calculate what percent corresponds to the number of students choosing Option 2. Use the definition of a percent.

b

Think about what the numerator and denominator of the fraction represent.

Solution
a

Davontay said that 68 % of students choose Option 1 for lunch. Let 100 % represent all students in their school. Subtract 68 % from 100 % to find how many students choose Option 2.

Option2 100 %-68 %=32 % Next, let's write this percent as a fraction. Start by recalling that a percent is the ratio of a number to 100. n %=n/100 We can apply this formula to 32 %. 32 %=32/100 This fraction can be simplified. First, split the numerator and denominator into prime factors. 32&= 2* 2* 2* 2* 2 100&= 2* 2* 5* 5 The numbers share two common factors, 2 and 2. Their product is the greates common factor (GCF) of 32 and 100. GCF(32,100)=2* 2=4 Finally, divide the numerator and denominator by their GCF, 4.

32/100
ReduceFrac

a/b=.a /4./.b /4.

32 ÷ 4/100 ÷ 4
CalcQuot

Calculate quotient

8/25

Therefore, 32 % corresponds to 825.

b

In Part A, we found that 825 students prefer Option 2 for lunch. This fraction means that 8 students out of 25 would choose the soda and taco. This leaves 25-8=17 students who prefer Option 1.

17 out of 25 choose Option 1 and 8 out of 25 choose Option 2
External credits: @pch.vector
Pop Quiz

Converting Between Percents and Simplified Fractions

Convert the given percent into the corresponding simplified fraction, or the given simplified fraction into a percent. Write each percent as an integer.

A random generator that generates different percents or fractions
Discussion

Converting Mixed Numbers Into Percents

Mixed numbers can be converted into percents. Consider the following mixed number. 7 29 This mixed number can be rewritten into a percent by following three steps.

1
Convert the Fraction Part to a Decimal
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The fraction part of 7 29 is 29. To rewrite it as a decimal, divide the numerator 2 by the denominator 9 by using long division.

The long division of 2 over 9

The quotient is approximately 0.22. Two decimal places is usually enough to convert the division to a percent.

2
Add the Decimal to the Integer Part
expand_more
The integer part of the mixed number is 7. In the previous step, the fraction part 29 was rewritten as the decimal number 0.22. Add the integer part and the decimal part. 7+0.22=7.22 The mixed number can be written as the decimal number 7.22.

3
Convert the Decimal to a Percent
expand_more
Rewrite the decimal as a percent by multiplying it by 100 and adding a percent sign. 7.22* 100=722 % Therefore, 7 29 corresponds to approximately 722 %.
Discussion

Converting Percents Into Mixed Numbers

Percents can be converted into mixed numbers. If a percent less than 100 % is converted into a fraction, then the result is a proper fraction. This means that it only makes sense to convert percents greater than 100 % into mixed numbers. Consider the following percent. 234 % This percent can be rewritten as a mixed number by following three steps.

1
Write the Percent as a Fraction With 100 as Denominator
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A percent is the ratio of a number to 100. Start by writing the percent as a fraction with a numerator of the number part of the percent and a denominator of 100. 234 % = 234/100

2
Divide the Numerator by the Denominator
expand_more
Divide the numerator by the denominator to rewrite the improper fraction 234100 as a mixed number. Use long division to find the quotient and remainder.

The long division of 234 over 100

The quotient is 2 and the remainder is 34. Write the quotient 2 as the integer part of the mixed number and the remainder 34 as the numerator of the fraction part. Remember that the denominator of the fraction part is the same as the denominator of the improper fraction, 100. 234/100= 2 34 100

3
Simplify the Fraction Part
expand_more
The final step is to simplify the fraction part. Find the gretest common factor (GCF) of the numerator and denominator by splitting 34 and 100 into prime factors. 34 &= 2* 17 100 &= 2* 2* 5* 5 The numbers share only one common factor, 2. This means that 2 is the GCF of 34 and 100. Divide the numerator and the denominator by 2.

34/100
ReduceFrac

a/b=.a /2./.b /2.

34 ÷ 2/100 ÷ 2
CalcQuot

Calculate quotient

17/50

Finally, we can write the mixed number that corresponds to 234 %. 234 % =2 1750

Example

Learning a New Dance Routine

At the end of a dance lesson, Paulina checks her fitness tracker and discovers that she completed 235 % of her daily activity goal.

a

What mixed number corresponds to this percent?

b

Paulina's dance partner has completed 1 710 of his daily activity goal. What percent corresponds to this mixed number?

Hint
a

Rewrite the fraction as a mixed number by dividing the numerator by the denominator using long division.

b

Start by rewriting the fraction part as a decimal. Then, add the integer part and the decimal part together and then multiply by 100.

Solution
a

In order to rewrite 235 % as a mixed number, recall that a percent is the ratio of a number to 100.

n % =n/100 We can use this formula to rewrite 235 % as a fraction. 235 % =235/100 Next, divide the numerator by the denominator to convert the fraction to a mixed number. A mixed number consists of an integer part and a fraction part. Use long division to find the integer quotient and remainder.

The long division of 235 over 100

The quotient is 2, so this is the integer part of the mixed number. The remainder is 35, which makes it the numerator of the fraction part. The denominator of the fraction part of the mixed number is the same as the denominator of the fraction. 235/100= 2 35 100 Finally, simplify the fraction part. Begin by splitting the numerator and denominator into prime factors. 35&= 5* 7 100&=2* 2* 5* 5 The numbers share only one factor, 5. This means that this is their greatest common factor (GCF). Simplify the fraction by dividing its numerator and denominator by their GCF.

35/100
ReduceFrac

a/b=.a /5./.b /5.

35 ÷ 5/100 ÷ 5
CalcQuot

Calculate quotient

7/20

The percent of 235 corresponds to the mixed number of 2 720.

b

This time we want to convert the mixed number 1 710 to a percent. The fraction part is 710. Divide the numerator by the denominator to write it as a decimal. Since the denominator is 10, move the decimal point one place to the left.

7.0 after division by 10 becomes 0.7

Now add the integer part 1 and the decimal part 0.7. This gives us the decimal number 1.7. 1 710=1.7 Finally, multiply the number by 100 to rewrite it as a percent. Remember to include a percent sign! 1.7* 100=170 % Therefore, the mixed number 1 710 is equivalent to 170 %.

Pop Quiz

Converting Between Percents and Mixed Numbers

Convert the given percent into its equivalent mixed number or the given mixed number into a percent. Write the fraction part of a mixed number in simplest form. Round the percent to the nearest integer.

A random generator that generates different percents or mixed numbers
Discussion

Converting Decimal Numbers Into Percents

Decimal numbers can be converted into a percent. Consider the following decimal number. 5.71 This decimal can be rewritten as a percent by multiplying it by 100 and adding the percent sign.

1
Multiply the Decimal Number by 100
expand_more
The multiplication can be done by moving the decimal point two places to the right.

5.71 becomes 571.0

2
Add a Percent Sign
expand_more
Once the decimal number has been multiplied by 100, the percent sign can be added.

5.71 times 100 =571%

Therefore, the decimal number 5.71 is equivalent to 571 %.

Example

Impressive Honey Harvest

Paulina's grandfather raises bees in bee hives.

Bee flying over honeycombs
He tells Paulina that he has already collected 1.58 times more honey than in the same month last year.

a

Express 1.58 in percent form.

b

When the year comes to an end, grandpa calculates that in total he collected 2.8 times more honey this year compared to the last year. What percent does this number correspond to?

Hint
a

Multiply the decimal number by 100 and add a percent sign.

b

Multiply the decimal number by 100 and add a percent sign.

Solution
a

To rewrite the decimal number 1.58 as a percent, multiply the number by 100 and add a percent sign.

1.58 times 100 =158%

The decimal number 1.58 corresponds to 158 %. This number implies that Paulina's grandpa collected 158 % of the amount of honey he harvested in the same month last year.

b

Now the decimal number 2.8 will be rewritten as a percent. To do so, multiply by 100 and add a percent sign afterwards. An easy way to do this is by moving the decimal point two places to the right.

2.8 becomes 280.0 after multiplying by 100

Therefore, the product is 280.0, or just 280.

2.8 times 100 = 280%

As we can see, 2.8 is equivalent to 280 %. This number means that Paulina's grandpa collected 280 % of last year's harvest. Since last year's harvest is represented by 100 %, this means that this year the harvest almost tripled.

Discussion

Converting Percents Into Decimal Numbers

Percents can be converted into decimal numbers. Consider the following percent. 93 % This percent can be rewritten as a decimal number by dividing it by 100 and removing the percent sign.

1
Divide the Percent by 100
expand_more
To divide by 100, the decimal point can be moved two places to the left. Do not forget to write a 0 to the left of the decimal point if there is no number to its left after moving it.

93.0 becomes 0.93

2
Remove the Percent Sign
expand_more
Once the number has been divided by 100, the percent sign has to be removed.

93 divided by 100 = 0.93

Therefore, 93 % is equivalent to 0.93.

Example

Foods Containing Percents

Paulina notices that some of the foods in her refrigerator have percents written on them.

A fridge on the left shows eggs, milk, chicken, and vegetables inside. The fridge door is open. On the right, there is a small table.
External credits: @brgfx
a

The first thing that Paulina sees is a bottle of milk. The label says it is 3.2 % fat. What is the decimal form of this percent?

b

Her bread is 17 % fat. What decimal is equivalent to this percent?

Hint
a

Divide the percent by 100 to write it as a decimal.

b

To divide an integer by 100, move its decimal point two places to the left.

Solution
a

The bottle of milk in the fridge is 3.2 % fat. We can write this number as a decimal by dividing it by 100. To divide a number by 100, we can move the decimal point two places to the left.

3.2 after division by 100 becomes 0.032

Remember to remove the percent sign.

3.2 divided by 0.032 = 0.032

b

The bread is 17 % fat. To write it as a decimal, we divide the number by 100 by moving its decimal point two places to the left and remove the percent sign.

17.0 after division by 100 becomes 0.17

The corresponding decimal is 0.17.

17 divided by 100 = 0.17
Closure

Calculating the Phone's Battery Level

When Paulina looked at her phone this morning, she noticed that the battery was only at 14 %.

A phone with 14 percent battery level
External credits: @jannoon028
She connected her phone to a charger and started getting ready.

a

What fraction in its simplest form expresses the current battery level?

b

Paulina disconnected her phone as soon as she was ready to go to school. The battery was charged up to 56 of its capacity. What percent did Paulina see on the screen?

Hint
a

Use the definition of a percent. Next, simplify the fraction by dividing its numerator and denominator by their greatest common factor (GCF).

b

Multiply the fraction by 100 and then divide the numerator by the denominator by long division.

Solution
a

Paulina's phone was at 14 % battery level when she woke up. To express this number as a fraction, recall the definition of a percent.

n %=n/100 Write 14 as a numerator with a denominator of 100. 14 %=14/100 Next, simplify the fraction. Start by splitting 14 and 100 into prime factors. 14 &= 2* 7 100 &= 2* 2* 5* 5 The numbers share only one common factor, 2, so it is their greatest common factor (GCF). Divide the numerator and denominator of the fraction by their GCF to simplify it.

14/100
ReduceFrac

a/b=.a /2./.b /2.

14 ÷ 2/100 ÷ 2
CalcQuot

Calculate quotient

7/50

Therefore, 14 % corresponds to 750.

b

Paulina's phone charged to 56 of its capacity. To determine what percent Paulina sees on the screen, express this fraction as a percent. First, multiply the fraction by 100.

5/6* 100
NumToDivByOne

a=a/1

5/6*100/1
MultFrac

Multiply fractions

5* 100/6* 1
Multiply

Multiply

500/6

Next, divide the numerator of this fraction by the denominator by using long division.

The long division of 500 over 6

The result is 83.3, which can be rounded to 83. This means that 56 corresponds to about 83 %.



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Percents
Exercise 3.1
Paulina really likes long morning walks with her dog.

Paulina walking her dog on a hill

She recently discovered that people who own a dog are 16 % more active than people who do not own a dog.

 What decimal number corresponds to 16 %?

What fraction in simplest form corresponds to 16 %?

If 100 % is the activity level of a person without a dog, then 116 % is the activity level of a person with a dog. What mixed number corresponds to 116 %?

We are asked to find a decimal number that corresponds to 16 %. We can do this by dividing the percent by 100. An easy way to perform this particular division is to move the decimal point two places to the left.

Therefore, 0.16 is the decimal number that corresponds to 16 %.


Let's rewrite 16 % as a fraction in simplest form. We will start by recalling the definition of a percent. It is the ratio of a number to 100. n %=n/100 Let's apply this formula to 16 %. 16 %=16/100 Next, we need to simplify the fraction. Let's find the GCF of 16 and 100 by splitting the numbers into prime factors. 16 &= 2* 2* 2* 2 100 &= 2* 2* 5* 5 The numbers share two common factors, 2 and 2. Their product is the GCF of 16 and 100. GCF(16,100)=2* 2 =4 Now we will divide the numerator and denominator by 4.

The fraction equivalent of 16 % is 425.

We need to rewrite 116 % as a mixed number. Let's begin by using the definition of a percent to write this percent as an improper fraction. 116 %=116/100 Next, we will divide the numerator by the denominator using long division to find the integer part and the fraction part of the mixed number.

The quotient 1 is the integer part and the remainder 16 is the numerator of the fraction part. The denominator of the fraction part is the same as the denominator of the improper fraction. 116/100= 1 16 100 Finally, we need to simplify the fraction part. From Part B, we know that it simplifies to 425. Therefore, 116 % written as a mixed number is 1 425. 116 % → 1 425

E
Hint & Answer
S
Solution

Consider a percent, a decimal number, a mixed number, and a fraction. 10 %, 2.64, 4 23, 5/6 What are some real-life situations that each number could represent?

Let's consider each number one at a time.

Percent

Let's think about a real-life situation where 10 % could appear. One of the possible situations is a 10 % discount on some item — for example, a pair of jeans.

There are many more possible places we might see percents. They appear in weather reports, sports statistics, grades, taxes, tips, and many more parts of our life.

Decimal Number

Now let's try to find a situation where we might find the decimal number 2.64. Decimal numbers are very useful because their form allows us to easily understand the value and compare numbers. For example, we could buy 2.64 pounds of apples.

Other possible examples include the length of an object, the speed of a car, the temperature of a beverage, or even the amount of money in a bank account. We could continue this list for a very long time!

Mixed Number

Now we will consider 4 23. Mixed numbers are useful when there is a need to express an exact value in a concise and precise way. Imagine that the instructions on the packaging of soil nutrients says to use 2 13 cups per plant.

Then, we would need 2 * 2 13=4 23 cups of soil nutrients for two plants.

Fraction

Finally, let's consider the fraction 56. Fractions are one of the most-used forms of numbers. They might appear behind the scenes calculations or in a more direct fashion. As an example, time is commonly measured in fractions.

Half an hour, or 12h, and a quater of an hour, or 14h, are phrases that are often used when telling time. In our case, 56, or 5060, would be equivalent to 50 minutes.

E
Hint & Answer
S
Solution

Percents

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