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4.01 Converting between percentages, fractions, and decimals

Lesson

Introduction

We use percentages everywhere in our daily lives, from taxes to discounts to nutritional information. We know that a 50\% discount will save us money, and 95\% sugar-free drink is healthier than a 20\% sugar-free drink, but what do these numbers really mean?

A percentage is an amount out of 100, denoted by the symbol \%.

What is one percent?

We know from the definition that 1\% represents 1 out of 100. In other words, 1\% is equal to one hundredth.

We have encountered hundredths before when looking at place values and there were two different ways that we represented them: \dfrac{1}{100} and 0.01.

Both of these ways to write one hundredth are also ways to write 1\% and we will be using these ways to helps us convert between percentages, fractions and decimals.

Examples

Example 1

Consider the grid below.

A 10 by 10 square grid with 50 squares shaded.
a

How many squares are shaded?

Worked Solution
Create a strategy

Count the number of squares shaded.

Apply the idea

There are 5 rows of 10 squares shaded so there are 50 squares shaded.

b

What percentage of the grid is shaded?

Worked Solution
Create a strategy

Find the fraction shaded then convert to a percentage.

Apply the idea

We have 50 out of 100\ squares in the grid are shaded. We can express this as fraction \dfrac{50}{100}.

\displaystyle \text{Percent}\displaystyle =\displaystyle \frac{50}{100} \times 100 \% Multiply by 100\%
\displaystyle =\displaystyle 50\% Evaluate
c

What fraction of the grid does this percentage represent?

A
One quarter
B
One tenth
C
One half
D
One fifth
Worked Solution
Create a strategy

Simplify the fraction from part (a).

Apply the idea
\displaystyle \frac{50}{100}\displaystyle =\displaystyle \frac{1}{2}Simplify the fraction

The correct option is C.

Idea summary

1\% is equal to one hundredth. 1\% = \dfrac{1}{100} = 0.01

Convert percentages to fractions

We can convert quite easily between fractions and percentages by remembering what percentages represent. Percentages represent a value out of 100, and any value out of 100 can be written as a fraction with the denominator 100.

For example, 33\% represents "33 out of 100" which can be written as the fraction \dfrac{33}{100}.

As we can see, using a denominator of 100 can help us convert between these two types of values.

Examples

Example 2

Write 55\% as a fraction in its simplest form.

Worked Solution
Create a strategy

To write a percentage as fraction divide by 100.

Apply the idea
\displaystyle 55\%\displaystyle =\displaystyle \frac{55 }{100 }Divide by 100
\displaystyle =\displaystyle \frac{55\div 5 }{100 \div 5 }Divide the numerator and denominator by 5
\displaystyle =\displaystyle \frac{11 }{20 }Simplify the fraction

Example 3

Write 107\% as a mixed number in its simplest form.

Worked Solution
Create a strategy

To write a percentage as a mixed number divide by 100.

Apply the idea
\displaystyle 107\%\displaystyle =\displaystyle \frac{107 }{100 }Divide by 100
\displaystyle =\displaystyle 1\frac{7 }{100 }Convert to a mixed number
Idea summary

We can convert any percentage into a fraction by writing the percentage value as the numerator and 100 as the denominator. After doing this, we can simplify the fraction to get it into its simplest form.

Convert fractions to percentages

To convert a fraction into a percentage, we can just reverse the above steps:

We can convert any fraction into a percentage by finding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the \% symbol to represent the percentage.

Examples

Example 4

Write \dfrac{3}{4} as a percentage.

Worked Solution
Create a strategy

To write as a percentage multiply by 100\%.

Apply the idea
\displaystyle \frac{3}{4}\displaystyle =\displaystyle \frac{3}{4}\times 100\%Multiply by 100\%
\displaystyle =\displaystyle \frac{300}{4}\%Perform the multiplication
\displaystyle =\displaystyle 75\%Simplify the fraction

Example 5

Write 3 \dfrac{7}{10} as a percentage.

Worked Solution
Create a strategy

To write a mixed number as a percentage multiply by 100\%.

Apply the idea
\displaystyle 3 \dfrac{7}{10} \displaystyle =\displaystyle 3 \dfrac{7}{10}\times 100\%Multiply by 100\%
\displaystyle =\displaystyle \frac{37 }{10 }\times 100Convert to a improper fraction
\displaystyle =\displaystyle \frac{3700 }{10 }\%Evaluate
\displaystyle =\displaystyle 370\%Simplify the fraction
Idea summary

We can convert any fraction into a percentage by finding its equivalent fraction that has a denominator of 100. After this, we can write the value in the numerator followed by the \% symbol to represent the percentage.

Convert percentages to decimals

We can convert between decimals and percentages by taking advantage of the hundredths place value. We know that 1\% represents 1 hundredth which we can write as 0.01 as a decimal. Using the same logic, we can convert larger percentages.

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the \% symbol.

For example, 83\% represents 83 hundredths. This is 0.83 when written as a decimal.

As we can see, We can convert from a percentage into a decimal by thinking of the percentage value as a number of hundredths.

Examples

Example 6

Write 54\% as a decimal.

Worked Solution
Create a strategy

To write a percentage as a decimal divide by 100.

Apply the idea
\displaystyle 54\% \displaystyle =\displaystyle \frac{54}{100} Divide by 100
\displaystyle =\displaystyle 0.54Write as a decimal
Reflect and check

We wrote the percentage as a decimal representing the number of hundredths indicated by the percentage.

Notice that converting the percentages into decimals had the same effect as decreasing the place value of the digits in the percentage by two places, then removing the \% symbol. This is equivalent to dividing by 100 and removing the \% symbol.

Idea summary

We can convert any percentage into a decimal by dividing the percentage value by 100, which is equivalent to decreasing the place value of each digit by two places, and removing the \% symbol.

Convert decimals to percentages

To convert from a decimal into a percentage, we can just reverse the above steps. We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a \% symbol.

A percentage is limited to representing hundredths, so smaller units like thousandths cannot be represented by whole number percentages.

Remember to attach the \% symbol to decimal at the same time as increasing the place values. This also applies for when we are reversing these steps to convert percentages into decimals.

Exploration

The applet below uses area model to represent conversion between fractions, decimals and percentages.

Loading interactive...

When we increase the number of shaded parts, the value of the fraction, decimal and percentage changes accordingly.

As with any skill in mathematics, there is always another way. While the methods above use the meaning of percentages, fractions and decimals to make the conversions, another way to convert between them is to treat the \% symbol like a unit.

Remember that 100\% is equal to one whole. This means that we can convert from percentages by dividing by 100\% and convert into percentages by multiplying by 100\%.

Although we are treating the \% symbol like a unit, it is not a unit. This is because it represents "out of 100" which is not a unit of measurement.

There are some common conversions that we can remember to help us convert between percentages, fractions and decimals.

A table of benchmark fraction, decimal and percentage conversions. Ask your teacher for more information.

Examples

Example 7

Convert between percentages, fractions and decimals to complete the table below.

FractionDecimalPercentage
\dfrac{11}{100}
1.83
\dfrac{5}{8}

Write the answers as mixed number percentages and simplified mixed numbers where necessary.

Worked Solution
Create a strategy

To convert fractions and decimals to percentages multiply by 100\%. To convert percentages to fractions or decimals, divide by 100.

Apply the idea
\displaystyle \dfrac{11}{100}\displaystyle =\displaystyle \dfrac{11}{100} \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 11\%Evaluate
\displaystyle =\displaystyle 0.11Convert to a decimal
\displaystyle 1.83\displaystyle =\displaystyle 1.83 \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 183\%Evaluate
\displaystyle =\displaystyle \dfrac{183}{100}Convert to a fraction
\displaystyle \dfrac{5}{8}\displaystyle =\displaystyle \dfrac{5}{8} \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 62.5\%Evaluate
\displaystyle =\displaystyle 0.625Convert to a decimal
FractionDecimalPercentage
\dfrac{11}{100}0.1111\%
\dfrac{183}{100}1.83183\%
\dfrac{5}{8}0.62562\dfrac{1}{2}\%
Idea summary

We can convert any decimal into a percentage by multiplying the decimal by 100, which is equivalent to increasing the place value of each digit by two places, and attaching a \% symbol.

Compare fractions, decimals, percents

Now that we are able to convert between these three ways to represent values, we can compare them.

When comparing any values, we always want to have them all in the same form. This means that when we are comparing percentages, fractions and decimals, we want to convert them so that they are all of one type.

Examples

Example 8

In this question we will be working with the numbers \dfrac{1}{4}, 60\% and 0.3.

a

Convert \dfrac{1}{4} into a percentage. Do not round your answer.

Worked Solution
Create a strategy

Multiply \dfrac{1}{4} by 100\% to convert.

Apply the idea
\displaystyle \dfrac{1}{4}\displaystyle =\displaystyle \dfrac{1}{4}\times 100\%Multiply by 100\%
\displaystyle =\displaystyle \dfrac{1}{4} \times \dfrac{100\%}{1}Write as a fraction
\displaystyle =\displaystyle \dfrac{100\%}{4}Multiply
\displaystyle =\displaystyle 25\%Evaluate the division
b

Convert 0.3 into a percentage.

Worked Solution
Create a strategy

Multiply the decimal by 100\%.

Apply the idea
\displaystyle 0.3\displaystyle =\displaystyle 0.3\times100\%Multiply by 100\%
\displaystyle =\displaystyle 30\%Evaluate
c

Which of the following arranges \dfrac{1}{4}, 60\% and 0.3 from largest to smallest?

A
60\%, \dfrac{1}{4}, 0.3
B
60\%, 0.3, \dfrac{1}{4}
C
\dfrac{1}{4}, 60\%, 0.3
D
0.3, \dfrac{1}{4}, 60\%
Worked Solution
Create a strategy

Compare the percentages.

Apply the idea

The percentages in order are: 60\%,\,30\%,\,25\%

So the original numbers in order are: 60\%,\,0.3,\,\dfrac{1}{4}

The answer is option B.

Idea summary

Ascending order means smallest to largest.

Descending order means largest to smallest.

To compare decimals, percentages and fractions we need to convert them so that they are all the same type before we can compare them.

Outcomes

VCMNA247

Connect fractions, decimals and percentages and carry out simple conversions

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