Fractions, Decimals and Percentages

What you'll learn

How fractions, decimals and percentages describe the same value in different ways.
How to convert between decimals, percentages and fractions.
How to simplify fractions after converting.
How to compare and order mixed numbers like 60%, $35\frac{3}{5}$ and 0.58.

The big idea: three ways to say the same thing

A fraction, decimal and percentage can all represent the same part of a whole.

For example, half of something can be written as:

the fraction $12\frac{1}{2}$
the decimal 0.5
the percentage 50%

$Number line showing equivalent fractions, decimals and percentages from 0 to 1$

Definition

Key words

A fraction shows a part of a whole. In $35\frac{3}{5}$ , the top number 3 is the numerator and the bottom number 5 is the denominator.
A decimal is a number written using place value after a decimal point, such as 0.72.
A percentage means “out of 100”. So 23% means 23 out of 100.
Equivalent means “equal in value”, even if the numbers look different.

Key Idea

Same value, different form

When you are converting, you are not changing the amount. You are only changing how it is written.

Place value in decimals

Decimals are based on tenths, hundredths, thousandths, and so on.

0.3 means 3 tenths.
0.03 means 3 hundredths.
0.003 means 3 thousandths.

This matters because decimals often convert directly into fractions.

Example

Writing a decimal as a fraction

Write 0.06 as a fraction.

Read the place value. The 6 is in the hundredths column, so 0.06 means 6 hundredths.
Write this as a fraction.
$6100\frac{6}{100}$
Simplify by dividing the numerator and denominator by 2.
$6100=350\frac{6}{100}=\frac{3}{50}$

Common Mistake

Forgetting the zero

0.06 is not the same as 0.6. The zero after the decimal point changes the place value: 0.06 means 6 hundredths, but 0.6 means 6 tenths.

Percentages mean “out of 100”

The word percent comes from “per cent”, meaning “out of 100”.

So:

18% means $18100\frac{18}{100}$
4% means $4100\frac{4}{100}$
100% means one whole

This makes percentage-to-fraction questions very direct.

Example

Writing a percentage as a fraction

Write 17% as a fraction.

Percent means out of 100, so write 17 over 100.
$17%=1710017\%=\frac{17}{100}$
Check whether the fraction simplifies. 17 and 100 have no common factor other than 1.
The final answer is $17100\frac{17}{100}$ .

Example

Writing a small percentage as a decimal

Write 4% as a decimal.

Write 4% as a fraction out of 100.
$4%=41004\%=\frac{4}{100}$
Divide by 100 to convert to a decimal.
$4100=0.04\frac{4}{100}=0.04$

Converting decimals to percentages

To change a decimal to a percentage, multiply by 100.

This works because a percentage is the number of hundredths.

Tip

Decimal to percentage shortcut

To multiply a decimal by 100, move the decimal point two places to the right. For example, 0.29 becomes 29%.

Example

Writing a decimal as a percentage

Write 0.79 as a percentage.

Multiply the decimal by 100.
$0.79 \times 100 = 79$
Add the percentage sign.
$79\%$

Example

Another decimal to percentage example

Write 0.06 as a percentage.

Multiply by 100.
$0.06 \times 100 = 6$
Add the percentage sign.
$6\%$

Converting percentages to decimals

To change a percentage to a decimal, divide by 100.

This is the reverse of changing a decimal to a percentage.

Example

Writing a percentage as a decimal

Write 18% as a decimal.

Divide 18 by 100.
$18 \div 100 = 0.18$
So 18% as a decimal is 0.18.

Common Mistake

One-place percentages

30% is 0.3, not 0.30%. Once you have converted to a decimal, remove the percentage sign.

Converting fractions to decimals

A fraction bar means divide. So to change a fraction to a decimal, divide the numerator by the denominator.

For common fractions, you can also use equivalent fractions with denominator 10 or 100.

Example

Writing a fraction as a decimal

Write $710\frac{7}{10}$ as a decimal.

The denominator is 10, so the fraction is 7 tenths.
Write 7 tenths as a decimal.
$710=0.7\frac{7}{10}=0.7$

Example

Converting fifths to decimals

Write $25\frac{2}{5}$ as a decimal.

Make an equivalent fraction with denominator 10. Multiply the numerator and denominator by 2.
$25=410\frac{2}{5}=\frac{4}{10}$
Write 4 tenths as a decimal.
$410=0.4\frac{4}{10}=0.4$

Converting fractions to percentages

To change a fraction to a percentage, you can either:

change the fraction to a decimal, then multiply by 100
make an equivalent fraction out of 100, if possible

Example

Writing a fraction as a percentage

Write $350\frac{3}{50}$ as a percentage.

Percent means out of 100, so make the denominator 100.
Multiply the numerator and denominator by 2.
$350=6100\frac{3}{50}=\frac{6}{100}$
Since $6100\frac{6}{100}$ means 6 out of 100, the percentage is 6%.

Example

Using a decimal first

Write $920\frac{9}{20}$ as a percentage.

Make the denominator 100. Since 20 multiplied by 5 gives 100, multiply the numerator by 5 as well.
$920=45100\frac{9}{20}=\frac{45}{100}$
So the percentage is 45%.

Simplifying fractions

A fraction is in its simplest form when the numerator and denominator have no common factor except 1.

For example, $25100\frac{25}{100}$ can be simplified because 25 divides into both 25 and 100.

Example

Writing a decimal as a simplified fraction

Write 0.25 as a fraction.

0.25 means 25 hundredths.
$0.25=251000.25=\frac{25}{100}$
Divide the numerator and denominator by 25.
$25100=14\frac{25}{100}=\frac{1}{4}$

Tip

Useful fraction facts

Try to remember these common equivalents: $12=0.5=50%\frac{1}{2}=0.5=50\%$ , $14=0.25=25%\frac{1}{4}=0.25=25\%$ , and $34=0.75=75%\frac{3}{4}=0.75=75\%$ .

Comparing fractions, decimals and percentages

When numbers are written in different forms, convert them all into the same form before comparing.

Decimals are often the easiest form for ordering.

Example

Checking a statement

A student says 16% is greater than 0.2. Is the student correct?

Convert 16% to a decimal by dividing by 100.
$16%=0.1616\%=0.16$
Compare 0.16 and 0.2. You can write 0.2 as 0.20 to compare the digits.
Since 0.16 is less than 0.20, the student is not correct.

Example

Comparing two people’s savings

Two people earn the same monthly salary. Alex saves 40% of it. Sam spends $35\frac{3}{5}$ of it and saves the rest. Who saves more?

Work out Sam’s saving fraction. If Sam spends $35\frac{3}{5}$ , then the remaining fraction is $25\frac{2}{5}$ .
Convert Sam’s saving fraction to a percentage.
$25=40100=40%\frac{2}{5}=\frac{40}{100}=40\%$
Alex saves 40% and Sam saves 40%, so they save the same amount.

Example

Putting mixed forms in order

Put these in order from smallest to largest: 80%, $34\frac{3}{4}$ , 0.77, 0.9, $710\frac{7}{10}$ .

Convert everything to decimals.
$80%=0.880\%=0.8$ $34=0.75\frac{3}{4}=0.75$ $710=0.7\frac{7}{10}=0.7$
Now compare the decimal values: 0.8, 0.75, 0.77, 0.9, 0.7.
Put them in increasing order.
$0.90.7,\ 0.75,\ 0.77,\ 0.8,\ 0.9$
Change back to the original forms.
$0.9\frac{7}{10},\ \frac{3}{4},\ 0.77,\ 80\%,\ 0.9$

Common Mistake

Ordering without converting

Do not assume a percentage is bigger just because it has a larger-looking number. For example, 75% is 0.75, so it is smaller than 0.8.

Quick conversion rules

Here are the most useful rules to memorise:

Decimal to percentage: multiply by 100.
Percentage to decimal: divide by 100.
Percentage to fraction: put it over 100, then simplify if possible.
Decimal to fraction: use place value, then simplify.
Fraction to decimal: divide the numerator by the denominator.
Fraction to percentage: convert to a decimal or make the denominator 100.

Exam technique

In the exam

Convert mixed forms into the same type before comparing or ordering.
For one-mark conversion questions, write the answer clearly and check whether a fraction should be simplified.
In reasoning questions, show one clear conversion, such as changing 35% to 0.35 or changing $35\frac{3}{5}$ to 60%.

Self review

Check yourself

Can you explain why 0.03 is $3100\frac{3}{100}$ and not $310\frac{3}{10}$ ?
Can you convert 45% into both a decimal and a fraction?
If you had 0.6, $58\frac{5}{8}$ and 58%, which form would you convert them to before ordering?

Fractions, Decimals and Percentages

What you'll learn

How fractions, decimals and percentages describe the same value in different ways.
How to convert between decimals, percentages and fractions.
How to simplify fractions after converting.
How to compare and order mixed numbers like 60%, $35\frac{3}{5}$ and 0.58.

The big idea: three ways to say the same thing

A fraction, decimal and percentage can all represent the same part of a whole.

For example, half of something can be written as:

the fraction $12\frac{1}{2}$
the decimal 0.5
the percentage 50%

$Number line showing equivalent fractions, decimals and percentages from 0 to 1$

Definition

Key words

A fraction shows a part of a whole. In $35\frac{3}{5}$ , the top number 3 is the numerator and the bottom number 5 is the denominator.
A decimal is a number written using place value after a decimal point, such as 0.72.
A percentage means “out of 100”. So 23% means 23 out of 100.
Equivalent means “equal in value”, even if the numbers look different.

Key Idea

Same value, different form

When you are converting, you are not changing the amount. You are only changing how it is written.

Place value in decimals

Decimals are based on tenths, hundredths, thousandths, and so on.

0.3 means 3 tenths.
0.03 means 3 hundredths.
0.003 means 3 thousandths.

This matters because decimals often convert directly into fractions.

Example

Writing a decimal as a fraction

Write 0.06 as a fraction.

Read the place value. The 6 is in the hundredths column, so 0.06 means 6 hundredths.
Write this as a fraction.
$6100\frac{6}{100}$
Simplify by dividing the numerator and denominator by 2.
$6100=350\frac{6}{100}=\frac{3}{50}$

Common Mistake

Forgetting the zero

0.06 is not the same as 0.6. The zero after the decimal point changes the place value: 0.06 means 6 hundredths, but 0.6 means 6 tenths.

Percentages mean “out of 100”

The word percent comes from “per cent”, meaning “out of 100”.

So:

18% means $18100\frac{18}{100}$
4% means $4100\frac{4}{100}$
100% means one whole

This makes percentage-to-fraction questions very direct.

Example

Writing a percentage as a fraction

Write 17% as a fraction.

Percent means out of 100, so write 17 over 100.
$17%=1710017\%=\frac{17}{100}$
Check whether the fraction simplifies. 17 and 100 have no common factor other than 1.
The final answer is $17100\frac{17}{100}$ .

Example

Writing a small percentage as a decimal

Write 4% as a decimal.

Write 4% as a fraction out of 100.
$4%=41004\%=\frac{4}{100}$
Divide by 100 to convert to a decimal.
$4100=0.04\frac{4}{100}=0.04$

Converting decimals to percentages

To change a decimal to a percentage, multiply by 100.

This works because a percentage is the number of hundredths.

Tip

Decimal to percentage shortcut

To multiply a decimal by 100, move the decimal point two places to the right. For example, 0.29 becomes 29%.

Example

Writing a decimal as a percentage

Write 0.79 as a percentage.

Multiply the decimal by 100.
$0.79 \times 100 = 79$
Add the percentage sign.
$79\%$

Example

Another decimal to percentage example

Write 0.06 as a percentage.

Multiply by 100.
$0.06 \times 100 = 6$
Add the percentage sign.
$6\%$

Converting percentages to decimals

To change a percentage to a decimal, divide by 100.

This is the reverse of changing a decimal to a percentage.

Example

Writing a percentage as a decimal

Write 18% as a decimal.

Divide 18 by 100.
$18 \div 100 = 0.18$
So 18% as a decimal is 0.18.

Common Mistake

One-place percentages

30% is 0.3, not 0.30%. Once you have converted to a decimal, remove the percentage sign.

Converting fractions to decimals

A fraction bar means divide. So to change a fraction to a decimal, divide the numerator by the denominator.

For common fractions, you can also use equivalent fractions with denominator 10 or 100.

Example

Writing a fraction as a decimal

Write $710\frac{7}{10}$ as a decimal.

The denominator is 10, so the fraction is 7 tenths.
Write 7 tenths as a decimal.
$710=0.7\frac{7}{10}=0.7$

Example

Converting fifths to decimals

Write $25\frac{2}{5}$ as a decimal.

Make an equivalent fraction with denominator 10. Multiply the numerator and denominator by 2.
$25=410\frac{2}{5}=\frac{4}{10}$
Write 4 tenths as a decimal.
$410=0.4\frac{4}{10}=0.4$

Converting fractions to percentages

To change a fraction to a percentage, you can either:

change the fraction to a decimal, then multiply by 100
make an equivalent fraction out of 100, if possible

Example

Writing a fraction as a percentage

Write $350\frac{3}{50}$ as a percentage.

Percent means out of 100, so make the denominator 100.
Multiply the numerator and denominator by 2.
$350=6100\frac{3}{50}=\frac{6}{100}$
Since $6100\frac{6}{100}$ means 6 out of 100, the percentage is 6%.

Example

Using a decimal first

Write $920\frac{9}{20}$ as a percentage.

Make the denominator 100. Since 20 multiplied by 5 gives 100, multiply the numerator by 5 as well.
$920=45100\frac{9}{20}=\frac{45}{100}$
So the percentage is 45%.

Simplifying fractions

A fraction is in its simplest form when the numerator and denominator have no common factor except 1.

For example, $25100\frac{25}{100}$ can be simplified because 25 divides into both 25 and 100.

Example

Writing a decimal as a simplified fraction

Write 0.25 as a fraction.

0.25 means 25 hundredths.
$0.25=251000.25=\frac{25}{100}$
Divide the numerator and denominator by 25.
$25100=14\frac{25}{100}=\frac{1}{4}$

Tip

Useful fraction facts

Try to remember these common equivalents: $12=0.5=50%\frac{1}{2}=0.5=50\%$ , $14=0.25=25%\frac{1}{4}=0.25=25\%$ , and $34=0.75=75%\frac{3}{4}=0.75=75\%$ .

Comparing fractions, decimals and percentages

When numbers are written in different forms, convert them all into the same form before comparing.

Decimals are often the easiest form for ordering.

Example

Checking a statement

A student says 16% is greater than 0.2. Is the student correct?

Convert 16% to a decimal by dividing by 100.
$16%=0.1616\%=0.16$
Compare 0.16 and 0.2. You can write 0.2 as 0.20 to compare the digits.
Since 0.16 is less than 0.20, the student is not correct.

Example

Comparing two people’s savings

Two people earn the same monthly salary. Alex saves 40% of it. Sam spends $35\frac{3}{5}$ of it and saves the rest. Who saves more?

Work out Sam’s saving fraction. If Sam spends $35\frac{3}{5}$ , then the remaining fraction is $25\frac{2}{5}$ .
Convert Sam’s saving fraction to a percentage.
$25=40100=40%\frac{2}{5}=\frac{40}{100}=40\%$
Alex saves 40% and Sam saves 40%, so they save the same amount.

Example

Putting mixed forms in order

Put these in order from smallest to largest: 80%, $34\frac{3}{4}$ , 0.77, 0.9, $710\frac{7}{10}$ .

Convert everything to decimals.
$80%=0.880\%=0.8$ $34=0.75\frac{3}{4}=0.75$ $710=0.7\frac{7}{10}=0.7$
Now compare the decimal values: 0.8, 0.75, 0.77, 0.9, 0.7.
Put them in increasing order.
$0.90.7,\ 0.75,\ 0.77,\ 0.8,\ 0.9$
Change back to the original forms.
$0.9\frac{7}{10},\ \frac{3}{4},\ 0.77,\ 80\%,\ 0.9$

Common Mistake

Ordering without converting

Do not assume a percentage is bigger just because it has a larger-looking number. For example, 75% is 0.75, so it is smaller than 0.8.

Quick conversion rules

Here are the most useful rules to memorise:

Decimal to percentage: multiply by 100.
Percentage to decimal: divide by 100.
Percentage to fraction: put it over 100, then simplify if possible.
Decimal to fraction: use place value, then simplify.
Fraction to decimal: divide the numerator by the denominator.
Fraction to percentage: convert to a decimal or make the denominator 100.

Exam technique

In the exam

Convert mixed forms into the same type before comparing or ordering.
For one-mark conversion questions, write the answer clearly and check whether a fraction should be simplified.
In reasoning questions, show one clear conversion, such as changing 35% to 0.35 or changing $35\frac{3}{5}$ to 60%.

Self review

Check yourself

Can you explain why 0.03 is $3100\frac{3}{100}$ and not $310\frac{3}{10}$ ?
Can you convert 45% into both a decimal and a fraction?
If you had 0.6, $58\frac{5}{8}$ and 58%, which form would you convert them to before ordering?

Fractions, Decimals and Percentages

What you'll learn

The big idea: three ways to say the same thing

Place value in decimals

Percentages mean “out of 100”

Converting decimals to percentages

Converting percentages to decimals

Converting fractions to decimals

Converting fractions to percentages

Simplifying fractions

Comparing fractions, decimals and percentages

Quick conversion rules

Recap questions

You've reached the end

Fractions, Decimals and Percentages

What you'll learn

The big idea: three ways to say the same thing

Place value in decimals

Percentages mean “out of 100”

Converting decimals to percentages

Converting percentages to decimals

Converting fractions to decimals

Converting fractions to percentages

Simplifying fractions

Comparing fractions, decimals and percentages

Quick conversion rules

Recap questions

You've reached the end

Fractions, Decimals and Percentages

What you'll learn

The big idea: three ways to say the same thing

Place value in decimals

Percentages mean “out of 100”

Converting decimals to percentages

Converting percentages to decimals

Converting fractions to decimals

Converting fractions to percentages

Simplifying fractions

Comparing fractions, decimals and percentages

Quick conversion rules

Recap questions

You've reached the end

Fractions, Decimals and Percentages Revision Guide

Fractions, Decimals and Percentages

What you'll learn

The big idea: three ways to say the same thing

Place value in decimals

Percentages mean “out of 100”

Converting decimals to percentages

Converting percentages to decimals

Converting fractions to decimals

Converting fractions to percentages

Simplifying fractions

Comparing fractions, decimals and percentages

Quick conversion rules

Recap questions

You've reached the end

Fractions, Decimals and Percentages Revision Guide