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Percents can be seen in numerous daily situations all over the world because they 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

A sudden loud sound coming from Paulina's phone wakes her up. It iss

6 : 00 AM

and time to start getting ready for school. When she looks at her phone, Paulina realizes that she forgot to charge it the night before. 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.

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

\frac{5}{6}

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 $% .$

$p % = \frac{p}{1 0 0}$

For example, five percent, or

5 %,

is the ratio of

5

to

100 .

5 % = \frac{5}{1 0 0}

Percentages are often used in real-life situations. For example, saline is a mixture of salt and water often used to help with dry eyes or to clean and store contact lenses, as it does not irritate the eyes. Saline contains

0.9 %

salt. This means that every

100

grams of saline contains

0.9

grams of salt.

0.9 % = \frac{0 . 9}{1 0 0}

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 % = \frac{2 0}{1 0 0}

Discussion

Converting Fractions Into Percents

It is possible to convert a fraction into a percent and vice versa. Consider the following fraction.

\frac{1 4}{8 7}

This fraction can be rewritten as a percent by following two steps.

1

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 long division of 14 over 87

The quotient is

0.1609 .

This is the decimal equivalent of the fraction.

2

Multiply by

100

Next, multiply the result by

100

to convert it to percent. Note that multiplying a decimal number by

100

comes down to moving the decimal point two places to the right.

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.

\frac{1 4}{8 7} \approx 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 Test Score

Paulina's first class is math. The teacher gives everyone back their graded papers from last Thursday's test. Paulina is thrilled to see that she got $23$ answers correctly.

Test sheet with a score A

There were

25

questions in total. Paulina wonders if she managed to answer

90 %

of the questions correctly. Find the fraction that describes Paulina's score and then express it as a percent.

Hint

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

Solution

It is given that Paulina answered

23

out of

25

questions correctly. This information will be used to first write a fraction. Since the total number of questions 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.

\frac{2 3}{2 5}

Now this fraction can be expressed as a percent. Start by writing it as a fraction with a denominator of

100 .

Note that

25

is a factor of

100 .

25 \cdot 4 = 100

This means that multiplying the numerator and denominator by

4

will result in an equivalent fraction with a denominator of

100 .

\frac{2 3}{2 5}

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

\frac{2 3 \cdot 4}{2 5 \cdot 4}

Multiply

\frac{9 2}{1 0 0}

Recall the definition of a percent — the ratio of a number to

100 .

n % = \frac{n}{1 0 0}

This means that the fraction represents

92 % .

\frac{9 2}{1 0 0} = 92 %

Therefore, Paulina did score more than

90 %

on the math test. Great job!

Alternative Solution

Fraction

\frac{2 3}{2 5}

represents the number of correct answers Paulina got on the test. This number can be rewritten as a percent by following two steps. First, divide the numerator by the denominator by using long division.

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.

The product is

92.0,

or

92 .

By adding a percent sign, it can be concluded that Paulina got

92 %

of the answers correct on her math test.

Discussion

Converting Percents Into Fractions

It is possible to convert a percent into a fraction and the other way around. Consider the following percent.

48 %

This percent can be written as a fraction by following two steps.

1

Write as a Fraction Whose Denominator Is

100

Since a percent is the ratio of a number to

100,

write it as a fraction whose numerator is the number part of the percent and whose denominator is

100 .

48 % = \frac{4 8}{1 0 0}

2

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

\frac{4 8}{1 0 0} .

48100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 = 2 \cdot 2 \cdot 5 \cdot 5

The numbers share two common factors. Their product is the greatest common factor (GCF) of

48

and

100 .

GCF (48, 100) = 2 \cdot 2 = 4

Finally, divide both the numerator and denominator by

4 .

\frac{4 8}{1 0 0}

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{4 8 \div 4}{1 0 0 \div 4}

Calculate quotient

\frac{1 2}{2 5}

The fractions

\frac{1 2}{2 5}

and

\frac{4 8}{1 0 0}

are equivalent and they both correspond to

48 % .

Extra

Converting Different Percents Between

1

and

100

Input a percent that is an integer 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

It is lunch time! Paulina and her friends go to the school cafeteria. She is trying to decide between juice and a sandwich or soda and a taco. What a hard decision! Her good know-it-all 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 different options, if this statistic were in fact true.

a What fraction in simplest form corresponds to the percent of students choosing 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 .

Option 2 100 % - 68 % = 32 %

Next, this percent should be written as a fraction. Start by recalling that a percent is the ratio of a number to

100 .

n % = \frac{n}{1 0 0}

Apply this formula to

32 % .

32 % = \frac{3 2}{1 0 0}

This fraction should also be simplified. First, split the numerator and denominator into prime factors.

32100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2 \cdot 2 \cdot 5 \cdot 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 \cdot 2 = 4

Finally, divide the numerator and denominator by their GCF,

4 .

\frac{3 2}{1 0 0}

$\frac{a}{b} = \frac{a / 4}{b / 4}$

\frac{3 2 \div 4}{1 0 0 \div 4}

Calculate quotient

\frac{8}{2 5}

Therefore,

32 %

corresponds to

\frac{8}{2 5} .

b In Part A, it was found that

\frac{8}{2 5}

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 .

External credits: @pch.vector

After realizing this, Paulina decides to go for Option $2 .$ Gotta show some love to the soda and taco! 😄

Pop Quiz

Converting Between Percents and Simplified Fractions

Convert the given percent into the corresponding simplified fraction or the other way around — 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

It is possible to convert a mixed number into a percent and the other way around. Consider the following mixed number.

7 \frac{2}{9}

This mixed number can be rewritten into a percent by following three steps.

1

Convert the Fraction Part to a Decimal

The fraction part of

7 \frac{2}{9}

is

\frac{2}{9} .

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 .

Note that it is often enough to divide to two decimal places.

2

Add the Decimal to the Integer Part

The integer part of the mixed number is

7 .

In the previous step, the fraction part

\frac{2}{9}

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

Rewrite the decimal as a percent by multiplying it by

100

and adding a percent sign.

7.22 \cdot 100 = 722 %

Therefore,

7 \frac{2}{9}

corresponds to

722 % .

Discussion

Converting Percents Into Mixed Numbers

It is possible to convert a percent into a mixed number and the other way around. Keep in mind that 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 into a mixed number by following three steps.

1

Write the Percent as a Fraction With

100

as Denominator

Recall that a percent is the ratio of a number to

100 .

Start by writing the percent as a fraction whose numerator is the number part of the percent and whose denominator is

100 .

234 % = \frac{2 3 4}{1 0 0}

2

Divide the Numerator by the Denominator

Divide the numerator by the denominator to rewrite the improper fraction

\frac{2 3 4}{1 0 0}

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 .

\frac{2 3 4}{1 0 0} = 2 \frac{3 4}{1 0 0}

3

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.

34100 = 2 \cdot 17 = 2 \cdot 2 \cdot 5 \cdot 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 .

\frac{3 4}{1 0 0}

$\frac{a}{b} = \frac{a / 2}{b / 2}$