body{margin:0;} noscript{z-index: 10000; position: fixed; display: block; height: 100%; width: 100%; background: #fff;} noscript .fa{font-size: 40px; display:block; color: #daa520;} noscript div{margin-top: 6%; padding-left: 20px; padding-right: 20px; text-align: center;} ! You must have JavaScript enabled to use this site.

Dashboard

Pre-Algebra View details

3. Percents

Continue to next lesson

Lesson

Exercises

Tests

eCourses /

Pre-Algebra /

Chapter 2

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.

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

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 %!

External credits: @jannoon028

She connects her phone to a charger and starts getting ready for the day.

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

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.

Divide the Numerator by the Denominator

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.

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

Multiply by 100

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.

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

Paulina answered 23 out of 25 questions correctly. To express her score as a percent, first we write it as a ratio of correct answers to total questions. 23/25 Now we can convert this fraction to a percent. Since a percent is a ratio of a number to 100, we can rewrite 23 25 as an equivalent fraction with a denominator of 100. n % =n/100 To do this, we expand the fraction by multiplying the numerator and denominator by 4.

23/25

ExpandFrac

a/b=a * 4/b * 4

23 * 4/25 * 4

Multiply

92/100

We found that 2325 = 92100, so the fraction represents 92 %. Paulina answered 92 % of the questions correctly.

Alternative Solution

Another way to write Paulina’s score of 2325 as a percent is to use division. Let’s start by dividing the numerator by the denominator.

The quotient is 0.92. To write it as a percent, we multiply it by 100 and add a percent symbol. This means moving the decimal point two places to the right, then adding the percent symbol.

Paulina got 92 % of the answers correct. This matches the answer from the main solution, so we know our calculation is correct.

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.

Write as a Fraction with a Denominator of 100

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

Simplify the Fraction

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.

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

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

How many students out of 25 would choose Option 1?

Hint

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

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

Solution

To find how many students chose Option 2, we subtract the 68 % of students who chose Option 1 from all students, which is 100 %.

100 %-68 %=32 % Next, let's write this percent as a fraction. Remember that a percent is the ratio of a number to 100. n %=n/100 ⇔ 32 %=32/100 Both 32 and 100 are divisible by 4. This means that we can simplify this fraction.

32/100

ReduceFrac

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

32 ÷ 4/100 ÷ 4

CalcQuot

Calculate quotient

8/25

Since 8 and 25 do not share any common factors, the fraction 825 is in simplest form. This means that 32 % is equal to 825.

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

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.

Convert the Fraction Part to a Decimal

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

Add the Decimal to the Integer Part

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.

Convert the Decimal to a Percent

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.

Write the Percent as a Fraction With 100 as Denominator

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

Divide the Numerator by the Denominator

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 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

Simplify the Fraction Part

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.

What mixed number corresponds to this percent?

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

Hint

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

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

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

n % =n/100 ⇕ 235 % =235/100 To convert the fraction to a mixed number, we divide the numerator by the denominator. The quotient gives us the whole-number part, the remainder gives us the numerator, and the denominator stays the same.

We found that 235100 = 2 35100. Lastly, we can simplify the fractional part by dividing both 35 and 100 by 5. 35 ÷ 5/100 ÷ 5 = 7/20 The percent 235 % is equal to the mixed number 2 720 or 2 35100.

To convert 1 710 to a percent, we first write it as a decimal. The fractional part, 710, can be written as a decimal by dividing 7 by 10. This means moving the decimal point one place to the left.

Now, we can add the integer part, 1, and the decimal part, 0.7. 1 710=1.7 Finally, we multiply the number by 100 and add the percent sign. 1.7* 100 %=170 % The mixed number 1 710 corresponds 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.

Multiply the Decimal Number by 100

The multiplication can be done by moving the decimal point two places to the right.

Add a Percent Sign

Once the decimal number has been multiplied by 100, the percent sign can be added.

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

Example

Impressive Honey Harvest

Paulina's grandfather raises bees in bee hives.

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

Express 1.58 in percent form.

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

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

Solution

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

Paulina's grandpa collected 158 % as much honey as he did in the same month last year.

To rewrite 2.8 as a percent, we proceed in the same way as in Part A. We multiply by 100 by moving the decimal point two places to the right and add the percent symbol.

The decimal number 2.8 is equivalent to 280 %. Paulina's grandpa collected 280 % of last year's harvest, so this year's harvest is almost three times as large.

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.

Divide the Percent by 100

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.

Remove the Percent Sign

Once the number has been divided by 100, the percent sign has to be removed.

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.

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?

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

Hint

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

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

Solution

Remember that a percent is a part of 100. This is why, to write 3.2 % as a decimal, we remove the percent sign and divide 3.2 by 100. We can do this by moving the decimal point two places to the left.

We found that 3.2 % written as a decimal is 0.032.

To write 17 % as a decimal, we do the same as in Part A. This means that we remove the percent sign and move the decimal point in 17.0 two places to the left.

The corresponding decimal is 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 %.

External credits: @jannoon028

She connected her phone to a charger and started getting ready.

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

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

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

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

Solution

To express the 14 % battery level of Paulina's phone as a fraction, we start by recalling the definition of a percent.

n %=n/100 ⇕ 14 % = 14/100 Next, we can simplify this fraction. We can do this by dividing both the numerator and denominator by 2. 14 ÷ 2/100 ÷ 2 = 7/50 We found that 14 % corresponds to 750. The numbers 7 and 50 do not share any more common factors, so the fraction is in simplest form.

To determine the percentage Paulina's phone charged to, we multiply the full capacity of 100 % by 56.

5/6* 100 %

NumToDivByOne

a=a/1

5/6*100/1 %

MultFrac

Multiply fractions

5* 100/6* 1 %

Multiply

500/6 %

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

Paulina's phone charged to about 83.3 %. Since phones usually do not show fractional percentages of charge, she saw 83 % on her phone screen.

Level 1

Level 2

Level 3

Percents

Exercise 2.1

Write the mixed number as a percent in integer form. 34 57

Write the percent as a mixed number. 482 %

We are asked to rewrite the mixed number of 34 57 as a percent. The fraction part is 57. We can write it as a decimal by dividing the numerator by the denominator. Let's use long division to divide to three decimal places.

The quotient is 0.714. Since the last digit is less than 5, we can round this number to 0.71. Now let's add the integer part 34 to the decimal part 0.71. This results in the number 34.71. 34 57≈ 34.71 Finally, we will multiply this number by 100 and add a percent sign to rewrite it as a percent. 34.71* 100=3471 % Therefore, the mixed number 34 57 corresponds to 3471 %.

We want to rewrite 482 % into a mixed number. Let's start by recalling the definition of a percent. It is the ratio of a number to 100. n % =n/100 We can use this formula to rewrite 482 % as a fraction. 482 % =482/100 Next, divide the numerator by the denominator to rewrite this improper fraction as a mixed number. Recall that a mixed number consists of an integer part and a fraction part. Use long division to find the integer quotient and remainder.

The quotient is 4. This is the integer part of the mixed number. The remainder is 82, so this will be 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. 482/100= 4 82 100 Finally, let's simplify the fraction part. We will begin by splitting the numerator and denominator into prime factors. 82&= 2* 41 100&= 2* 2* 5* 5 The numbers share only one common factor, 2, which makes it their GCF. We can simplify the fraction by dividing its numerator and denominator by their GCF.

We found that 482 % is equivalent to the mixed number 4 4150.

Dylan and Emily started putting together a puzzle that has 250 pieces.

After half an hour of working on the puzzle, Emily noted that they already completed 34 pieces of the puzzle. What percent of the whole puzzle is that?

Later that evening, they decided to stop and continue working the next day. If they completed 60 % of the puzzle, how many pieces did they put together?

We know that after working for half an hour, Emily and Dylan correctly placed 34 puzzle pieces. There are 250 pieces in total. This means that they put together 34 250 of the whole puzzle. 34/250 Let's convert this fraction into a percent. First, let's divide the numerator by the denominator by using long division.

The quotient is 0.136. Next, multiply the result by 100 to rewrite the decimal number as a percent. We can do this by moving the decimal point two places to the right. 0.136* 100=13.6 The result is 13.6. This means that Emily and Dylan completed 13.6 % of the puzzle after half an hour of working on it.

We are asked to determine how many pieces Emily and Dylan successfully put together if they completed 60 % of the puzzle. We can do this by writing the corresponding fraction with the denominator of 250. The numerator will indicate the number of placed pieces. 60 %=?/250 Let's begin by recalling the definition of a percent. It is a ratio of a number to 100. n %=n/100 We can apply this formula to 60 %. 60 %=60/100 The next step would be to simplify the fraction. However, we want to write it with a denominator of 250, so instead of simplying, we will expand the fraction. We can find the factor we want to expand the fraction by if we divide 250 by 100. To do so, move the decimal point two places to the left. 250÷ 100=2.5 Finally, let's multiply both the numerator and denominator by 2.5.

Therefore, Emily and Dylan put together 150 pieces after completing 60 % of the puzzle.

Pair the equivalent numbers.

Let's consider each percent one at a time.

First Percent: 715 %

The first percent is 715 %. This percent is greater than 100 %, which indicates that it might correspond to a mixed number or an improper fraction. Let's first rewrite it as a mixed number. Recall the definition of a percent. n % =n/100 We can use this formula to rewrite 715 % as a fraction. 715 % =715/100 Next, we need to rewrite this improper fraction as a mixed number. Recall that a mixed number consists of an integer part and a fraction part. We can find these parts by dividing the numerator by the denominator using long division.

The quotient is 7, which makes it the integer part of the mixed number. The remainder is 15, so 15 is 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. 715/100= 7 15 100 Finally, let's simplify the fraction part. We can start by splitting the numerator and denominator into prime factors. 15&=3* 5 100&=2* 2* 5* 5 The numbers share only one common factor, 5, which makes it their greatest common factor (GCF). We can simplify the fraction by dividing its numerator and denominator by 5.

If we add the integer part to the fraction part, we find that 715 % corresponds to the mixed number 7 320.

Second Percent: 69.2 %

The second percent is 69.2 %. Since it is less than 100 %, it cannot correspond to a mixed number or an improper fraction. This leaves only decimal numbers. We can rewrite the percent as a decimal number by dividing the it by 100. An easy way to divide by 100 is to move the decimal point two places to the left.

Therefore, 69.2 % is equivalent to the decimal number 0.692.

Third Percent: 83.1 %

Now let's consider 83.1 %. This number is also less than 100 %, so it looks like it might correspond to the remaining decimal number 0.831. Let's confirm this by converting 0.831 back to a percent. 0.831 → ? % First, we multiply the number by 100. We can do this easily by moving the decimal point two places to the right.

Do not forget to add a percent sign to the product.

We confirmed that 0.831 does indeed correspond to 83.1 %.

Fourth Percent: 212.5 %

Finally, let's consider 212.5 %. The only remaining number in the right column is the improper fraction 11956. Let's rewrite it as a percent to see if the numbers are, in fact, equivalent. 119/56 → ? % First, divide the numerator by the denominator. Let's use long division.

The quotient is 2.125. Now, we will rewrite this decimal as a percent by multiplying it by 100 %. To multiply by 100, move the decimal point two places to the right. 2.125* 100 %=212.5 % As we can see, 11956 does indeed correspond to 212.5 %. We have successfully matched all the percents!

Dylan and Emily are working on their math homework. They need to rewrite a mixed number as a percent.

However, their solutions are different. Who is correct?

Both Dylan and Emily are working with the mixed number 9 34. Let's analyze each of their solutions step by step.

Dylan's Solution

First, Dylan identifies that 34 is the fraction part of the mixed number. Step1 Fraction part: 3/4 ✓ Next, he rewrites the fraction as a decimal. Let's use long division to check whether his result is correct.

Dylan found the corresponding decimal correctly. Step 2 3/4=0.75 ✓ Finally, he adds the integer part 9 to the decimal part 0.75, getting the sum 9.75. This is the decimal corresponding to the mixed number. However, Dylan ended his process there by adding the percent sign. This is incorrect. Step 3 9+0.75=9.75 % * Instead, he needed to first multiply the decimal by 100 and only then add a percent sign. 9.75* 100=975 % ✓ Therefore, Dylan's solution is incorrect.

Emily's Solution

Let's now take a look at Emily's solution. She starts by applying the formula for rewriting a mixed number as an improper fraction. This step is not strictly necessary for converting a mixed number into a percent. However, we can solve math problems in different ways, so it does not immediately mean that the solution is incorrect. Step1 9 34=9* 4+3/4 ✓ Emily applied the formula for converting a mixed number to an improper fraction correctly. She moves on to evaluate the improper fraction by simplifying the numerator. Let's see how she did!

She simplified the fraction properly. Good job! Step2 9 34=39/4 ✓ Next, Emily calculates the quotient of 39 and 4 to rewrite the number as a decimal. Let's do this, too. We will divide using long division.

Once again, Emily's calculation is correct. The improper fraction 394 corresponds to 9.75. Step3 39/4=9.75 ✓ Finally, Emily multiplies the decimal by 100 to rewrite it as a percent. She gets 975 %, which is the correct result. Step4 9.75* 100=975 % ✓ Therefore, Emily's solution is correct even though she did not use the most common steps for the conversion.

Percents

Recommended exercises

To get personalized recommended exercises, please login first.

Return to lesson.

Edit Lesson

≥

■2

■▢

e▢

÷■1

=

√▢

▢√

∫▢▢

−

≤

log

log▢

()

∣

sin

cos

tan