Fractions, Decimals & Percentages
Fractions, decimals and percentages are different ways to write the same value. We will start with a shaded whole, then learn how to change from one form to another.
The main lesson builds Foundation skills used at both tiers. A short Higher extension at the end introduces recurring decimals.
Three ways to show the same amount
Imagine one whole bar split into equal parts. If one of two parts is shaded, one half is shaded.
We write one half as . The top number, the numerator, counts the parts we have. The bottom number, the denominator, says how many equal parts make one whole.
The second bar has the same amount shaded. It shows , or five tenths. These are equivalent fractions: fractions with the same value.
A decimal uses place value. The first place after the decimal point is tenths; the second is hundredths. Five tenths is written 0.5.
A percentage means “out of 100”. The symbol % means “per cent”. Half of 100 equal parts is 50 parts, so one half is 50%.
= 0.5 = 50%
The equals signs tell us these values are the same. Changing the way we write a number does not change its size.
Example Question:
Write as a decimal and a percentage.
Think about how many hundredths make one quarter. Example 1 shows this with a grid.
Exam Tip
Check which form the question asks for: a fraction, a decimal or a percentage.
Keep the % sign on a percentage. The numbers 25 and 25% have very different values.
If you are comparing values, write them all in the same form first.
Method — Changing Between Forms
Fractions and decimals
A fraction also means division: divide the numerator by the denominator. For one half, 1 ÷ 2 = 0.5.
It can be easier to use an equivalent fraction with 10, 100 or 1000 on the bottom. For example, = = 0.6.
To make an equivalent fraction, multiply or divide both the top and bottom by the same non-zero number. The value stays the same.
To change a decimal into a fraction, use its place value. For example, 0.7 is seven tenths: .
Then simplify the fraction if possible: divide the top and bottom by a shared factor until they have no shared factor greater than 1. A factor is a whole number that divides exactly.
Decimals and percentages
One whole is 100%. To write a decimal as a percentage, multiply the decimal by 100 and attach the % sign.
For 0.5: 0.5 × 100 = 50, so 0.5 = 50%.
To write a percentage as a decimal, divide its number by 100 and remove the % sign.
For 50%: 50 ÷ 100 = 0.5, so 50% = 0.5.
Percentages and fractions
A percentage is already “out of 100”. Write 20% as , then simplify to .
To change a fraction to a percentage, first find its decimal, then multiply that decimal by 100 and attach the % sign.
Worked Examples
Example 1 — One quarter in three forms
Write as a decimal and a percentage.
Each small square below is one hundredth of the whole. Exactly 25 are shaded.
Multiply the numerator and denominator by 25:
=
Twenty-five hundredths is 0.25. Twenty-five out of 100 is 25%.
= 0.25 = 25%
Example 2 — Start with a decimal
Write 0.35 as a fraction in its simplest form and as a percentage.
| Ones | Tenths | Hundredths |
|---|---|---|
| 0 | 3 | 5 |
There are 3 tenths and 5 hundredths. Three tenths is 30 hundredths, so altogether there are 35 hundredths.
0.35 =
Both 35 and 100 can be divided by 5:
35 ÷ 5 = 7
100 ÷ 5 = 20
0.35 =
To get the percentage number:
0.35 × 100 = 35
0.35 = 35%
Be careful with zeroes. The number 0.06 is six hundredths, so it is 6%. It is not the same as 0.6, which is 60%.
Example 3 — Start with a percentage
Write 18% as a decimal and a fraction in its simplest form.
Divide the percentage number by 100:
18 ÷ 100 = 0.18
18% = 0.18
For the fraction, 18% = .
Divide the top and bottom by 2:
18 ÷ 2 = 9
100 ÷ 2 = 50
18% =
Example 4 — A fraction that needs thousandths
Write as a decimal and a percentage.
There is no whole number we can multiply 8 by to get 100. We can use 1000 instead:
8 × 125 = 1000
Multiply the numerator by 125 too:
3 × 125 = 375
=
The third place after the decimal point is thousandths. So 375 thousandths is 0.375.
= 0.375
You can also calculate 3 ÷ 8 to get the same decimal.
To get the percentage number:
0.375 × 100 = 37.5
= 37.5%
A percentage does not have to be a whole number.
Example 5 — More than one whole
Write 1.25 as a fraction and a percentage.
The 1 represents one whole. That is 100 hundredths. Add the other 25 hundredths to get 125 hundredths:
1.25 =
Divide the numerator and denominator by 25:
1.25 =
To get the percentage number:
1.25 × 100 = 125
1.25 = 125%
Values above 1 give percentages above 100%. This means more than one whole.
Example 6 — Compare different forms
Put 0.62, and 65% in order, smallest first.
Convert all three to decimals:
= = 0.6
65% = 0.65
Write 0.6 as 0.60 to make the place values easier to compare. Adding a zero at the end of a decimal does not change its value.
Compare 0.60, 0.62 and 0.65. All have 6 tenths, so compare their hundredths: 0, 2 and 5.
Smallest to largest:
, 0.62, 65%
Example 7 — A GCSE-style test score
A student scores 18 out of 20 on a test. Write the score as a fraction of the total, a decimal and a percentage.
The fraction is . The whole test is 20 marks, not 100 marks.
Divide the numerator and denominator by 2:
=
Nine tenths is 0.9.
0.9 × 100 = 90
= 0.9 = 90%
The student earned 90% of the available marks.
Example 8 — Higher extension: a recurring decimal
Some decimals never end. In 0.444…, the dots mean the 4 repeats forever. This is a recurring decimal.
Write 0.444… as an exact fraction. We will use a little algebra to remove the repeating part.
Let x stand for the exact value 0.444…:
x = 0.444…
Multiply both sides by 10. The expression 10x means 10 × x:
10x = 4.444…
Subtract the first equation from the second. The identical repeating parts cancel:
10x − x = 4.444… − 0.444…
9x = 4
Divide both sides by 9:
x =
So 0.444… = . This is exact.
The decimal 0.444, with no repeating dots, stops after three places. It is a different value. Do not use a rounded decimal when the question asks for an exact fraction.
Useful equivalents
These common values are useful to recognise. Each row shows the same amount in three forms.
| Fraction | Decimal | Percentage |
|---|---|---|
| 0.5 | 50% | |
| 0.25 | 25% | |
| 0.75 | 75% | |
| 0.2 | 20% | |
| 0.1 | 10% |
Check the size of your answer
In Example 4, is less than , which is one half.
Its decimal should be below 0.5, and its percentage below 50%. The answers 0.375 and 37.5% fit.
This would help you spot an answer such as 375%. That is more than three wholes, so it cannot equal .
Comparing signed fractions and linking to ratio
Fractions can be negative too. As with negative decimals, the number further right on a number line is greater. Use an exact common denominator when that is easier than division.
Worked example — order −3/4, −2/3, 0 and 1/2
Write the negative fractions in twelfths: − = − and − = −.
−9 twelfths is less than −8 twelfths. Increasing order is −, −, 0, . Negative numbers come before zero and positive numbers.
Quick practice — put −5/8, −0.6 and 1/4 in increasing order.
= 0.625, so − = −0.625. Write −0.6 as −0.600.
−0.625 lies to the left of −0.600. The order is −, −0.6, .
A short bridge from ratio to fractions
A ratio compares parts. If red and blue counters are in the ratio 2 : 3, read this as “2 red parts to 3 blue parts”. There are 2 + 3 = 5 parts altogether.
The red fraction of the whole is ; the blue fraction is . The fraction would compare red with blue, not red with the whole. Understanding and Simplifying Ratios develops these comparisons further.
Quick practice — yellow and green counters are in the ratio 3 : 5. What fraction of all the counters is yellow?
There are 3 + 5 = 8 equal parts altogether. Yellow uses 3 of them, so the fraction is .
A rational number can be written as a fraction of two integers with a non-zero denominator. Integers, terminating decimals and recurring decimals are rational. For example, 4 = and 0.25 = .
Think Like a Mathematician
Why does equal ? Look back at the two shaded bars.
Split each half into five smaller equal pieces. The whole now has ten pieces, and the shaded half has five.
There are more pieces, but each piece is smaller. The shaded amount has not changed.
That is why multiplying both the numerator and denominator by 5 keeps the fraction’s value the same.
Common mistakes to avoid
Using unequal parts. A fraction’s denominator counts equal parts of a whole. A large slice and a tiny slice are not two halves.
Dividing in the wrong order. means 3 ÷ 8, not 8 ÷ 3.
Losing place value. 0.06 is 6%, while 0.6 is 60%. Check the tenths and hundredths.
Changing only the top or bottom. To keep an equivalent fraction, multiply or divide both by the same non-zero number.
Comparing the written digits alone. Convert fractions, decimals and percentages to one form before ordering them.
Higher: stopping a recurring decimal. 0.444… repeats forever. Writing 0.444 changes the value.
Try it yourself
Try these in order. Show your conversion steps before opening each worked answer. Question 9 is a Higher extension.
1. What fraction of this whole is shaded? Give the simplest fraction, decimal and percentage.
Two of four equal parts are shaded: .
Divide the numerator and denominator by 2:
=
One half is five tenths, or 0.5. Half of 100 is 50.
= 0.5 = 50%
2. Write 0.4 as a fraction in its simplest form and as a percentage.
Four tenths is .
Divide the numerator and denominator by 2:
=
0.4 × 100 = 40
0.4 = = 40%
3. Write 7% as a decimal and a fraction.
7% means 7 out of 100.
7% =
7 ÷ 100 = 0.07
7% = 0.07 =
The fraction is already in its simplest form.
4. Write as a decimal and a percentage.
Multiply the numerator and denominator by 2:
=
Six tenths is 0.6.
0.6 × 100 = 60
= 0.6 = 60%
5. Write 0.125 as a fraction in its simplest form and as a percentage.
There are three decimal places, so 0.125 = .
Divide both numbers by 125:
125 ÷ 125 = 1
1000 ÷ 125 = 8
0.125 =
0.125 × 100 = 12.5
0.125 = = 12.5%
6. Write 140% as a decimal and a fraction in its simplest form.
140 ÷ 100 = 1.4
140% =
Divide the top and bottom by 20:
140 ÷ 20 = 7
100 ÷ 20 = 5
140% = 1.4 =
All three forms represent more than one whole.
7. Put 0.45, and 42% in order, smallest first.
Convert to decimals:
= = 0.4
42% = 0.42
Compare 0.40, 0.42 and 0.45. Each has 4 tenths, so compare the hundredths: 0, 2 and 5.
Smallest to largest: , 42%, 0.45.
8. A student scores 16 out of 20 on a quiz. Write the score as a fraction in its simplest form, a decimal and a percentage.
The fraction of the total marks is .
Divide the top and bottom by 4:
=
Make tenths:
=
Eight tenths is 0.8.
0.8 × 100 = 80
= 0.8 = 80%
9. Higher extension: write 0.666… as an exact fraction in its simplest form. The 6 repeats forever.
Let x stand for the recurring decimal:
x = 0.666…
Multiply both sides by 10:
10x = 6.666…
Subtract the first equation from the second:
10x − x = 6.666… − 0.666…
9x = 6
Divide both sides by 9:
x =
Divide the numerator and denominator by 3:
0.666… =
Frequently Asked Questions
What does per cent mean?
It means “out of 100”. So 1% is one hundredth of a whole, and 100% is one whole.
Why is 0.5 the same as 0.50?
Five tenths is fifty hundredths. The extra zero changes how many decimal places are shown, but not the value.
Can a percentage be more than 100%?
Yes. It then represents more than one whole. For example, 125% equals 1.25.
Does every fraction give a decimal that stops?
No. For example, = 0.333… . The 3 repeats forever. Keep the exact fraction or show the repeating decimal unless a rounded answer is requested.
Do I always need a calculator?
No. Equivalent fractions and familiar values often make conversions straightforward. Dividing the numerator by the denominator is another method; use a calculator when the question allows it.
Is this the same as finding a percentage of an amount?
This lesson changes how a value is written. Finding, for example, 20% of £30 is a further calculation covered by Percentages of an Amount.