Fractions

What are They?

• Simply, a fraction is one number divided by another number.
1 77 123
1 ÷ 2 = , 77 ÷ 1291 = = 123 ÷ 6 =
2 1291 6

NB: Try to write your fractions with a horizontal line, rather than a
slant, this will help you keep everything neat and clear.
Why Are They Useful?
• Dividing is hard. Dividing a lot at once is even harder:
Imagine (3 x 2 x 3 + 5 ) ÷ 3 x 5 . This looks weird and messy.
3 𝑥 2 𝑥 3+5
But in fraction form we get
3𝑥5

11 1
• Cancelling and simplifying gives us or 2
5 5

• Notice how the fraction line here has made it so we don’t need to use
brackets.
 Different Kinds of Fractions:
• Fractions have a numerator (top) and denominator (bottom)
in a normal fraction, the numerator is smaller than the denominator
1 16 172 1234234
e.g.
8 80 83298 82384385389452987

• Because the number on the bottom is bigger, these fractions will be less than one. Any
number divided by a bigger number is less than one.

• Top Heavy or improper Fractions are those fractions where the numerator is bigger than the
denominator

10 152 13295730958309
e.g. , 38
8 2

• These will be bigger than one.

Different Kinds of Fractions:
• Improper fractions can be simplified into mixed numbers
These are a mix of numbers and fractions
4 1 3 1
E.G. goes to 1 and because 1 = . We write this as 1
3 3 3 3

To get from a improper fraction to a mixed number we divide the

numerator by the denominator to get our whole number. The
remainder will be our fraction.
5 1
E.g. . 5 ÷ 4 = 1 remainder 1, so the answer is 1
4 4
Different Kinds of Fractions:
• Lastly equivalent fractions are ones that are exactly the same.
• On this table, this is anywhere where
the lines are the same
3 6 1 3
e.g. = , = etc
4 8 2 6
 Checking For Equivalence.
• There’s a handy way to check if two fractions are equivalent.
6 15
• Take and 20 , for example.
8

• First we divide the larger numerator by the smaller one. 15/6 = 2.5 .

• Then we divide the larger denominator by the smaller one: 20/8 = 2.5.

• Because our answers are the same. We know that both sides have been enlarged by the
6 15
same amount from 8 to 20 , namely, they have both been x by 2.5. They are equivalent.

5 75
• Are and equivalent?
8 120
A Note about Equivalence.
• Fractions care about multiplication and dividing. If you try and find
equivalent fractions using adding or taking away, you’ll mess up.
1
• Imagine I want to find a fraction equivalent to 2 , but instead of multiplying
3
by 2 I add 2 to each side. This would give me 4 .

1 3
• = 0.5 ,
2 4
= 0.75 . I’ve gone wrong here! These clearly aren’t equivalent!

• It’s important that I think about equivalent fractions in terms of

What You Need to be Able to do With Fractions
• Convert improper fractions into mixed numbers and vice versa.

• Order fractions, either by finding common denominators or converting into decimals.

• Add and Subtract fractions by finding common denominators.

• Multiply and divide fractions.

15
• Find fractional amounts of numbers (e.g find. 20 of 200 )

• Recognise when to apply these skills with other questions. (higher level skill, 5+)
Converting Improper fractions to Mixed Numbers
• Like we said before: Take the numerator and divide by the
denominator. The remainder is the fraction.

• The proper fraction will always be less than one, so this is like finding
the amount of whole numbers and then expressing the bit after the
decimal point in fraction form.

7 9 16 129 12358329587324
• Let’s try: , , , ,
4 6 10 2 1
Converting Improper fractions to Mixed
Numbers
• Sometimes we want to go the other way.

• To Convert a mixed number to an improper fraction. Take the whole

number, multiply it by the denominator, then add it to the numerator.

3 2𝑥4 +3 8+3 13
• E.g. 2 = = =
4 4 4 4

2
• Can you turn 3 into an improper fraction?
3
Ordering Fractions.
• Sometimes we’ll be asked to order fractions. There are two ways to
do this.

1. Finding a common denominator.

2. Converting into decimals

Ordering Fractions Via Common Denominator
• E.g.

• To do this, we need to give all the fractions the same bottom number.
• This means we’re looking for the lowest common multiple of 2, 3, 12, 6, and 4.

𝑥
• This is 12 (2x6, 3x4, 12x1, 6x2, 4x3) so we need our fractions in the form
12

• We’ll do this by x the tops by the same number we had to x each of the bottoms
1 𝑥6 6
by to get to 12. (e.g. = )
2 𝑥6 12

• Can you complete this question?

Ordering Fractions by converting to Decimals.
• Sometimes the lowest common multiple is really big, so it’s better to convert to
decimals.
E.g

• Here the LCM is 40, so instead lets convert to decimals. We do this by doing 1
divided by the denominator, then multiplied by the numerator. Or by doing a bus-
stop division.
3 1
• E.g. . Either do = 0.25 , 0.25 x 3 = 0.75 | or |
4 4

• See if you can work out the answer from here.

Adding and Subtracting.
• To Add and subtract fractions we also need them to have a common
denominator.

• First we make the denominators the same by finding the LCM, then
we add and subtract the numerators as normal, whilst keeping the
denominator the same.
1 2 5 8 13
E.g. + = + =
4 5 20 20 20
Adding and Subtracting Fractions
3 1
• 1. Solve +
7 3

1 1
• 2. Solve +
4 2

5 1
• 3. Solve -
8 3

1 3
• 4. Solve 2 -
4 8
Multiplying Fractions.
• Multiplying fractions follows a simple rule:
Times the top by the top, and the bottom by the bottom.

• This means we x the numerators together to get the numerator for our
answer, and x the denominators together to get the denominator for our
answer.

7 1 7𝑥1 7
• E.g. x = =
4 2 4𝑥2 8

3 2
• Solve x 5 4
Dividing Fractions.
We also follow a rule to divide fractions.

Flip the fraction your dividing by, then multiply.

We need to find the reciprocal of the fraction we’re dividing by, this means flipping it: The
denominator becomes the numerator and the numerator becomes the denominator.

3 4
e.g. the reciprocal of 4
is 3 , the numbers have flipped.
From there we multiply by doing top x top and bottom x bottom.
3 1
Can you solve: ÷
4 2
Multiplying and Dividing Fractions:
2 2
• 1. x
7 3

1 6
• 2. 3 x
2 9

4 4
• 3. ÷
11 11

2 5
• 4. 4 ÷
3 6

3 1 1 4
• 5. ( x )+( ÷ )
4 2 2 5
Working Out Fractional amounts
• Sometimes we want to work out what a fractional amount of a value
is.

2
• E.g. Lets say you win a quiz and want to give of the £20 prize money
5
1
to your captain, and each to the other three players.
5

• There are two ways we can solve this.

Working Out Fractional amounts
• Our first method is to convert the amount we’re trying to find a
fraction of into fraction form, then multiply the two fractions.

𝑥 20
• Any whole number x has the fraction form , so £20 is £
1 1

2 20 40
• x£ = £ . £(40 ÷ 5) = £8 for the captain.
5 1 5
Working Out Fractional amounts
• The second method is to divide by the denominator and then multiply
by the numerator.
2 1 2
E.g. of £20 : = £20 ÷ 5 = £4 . = £4 x 2 = £8
5 5 5

• £8 for the quiz captain and £4 for everyone else.

Working Out Fractional amounts
2
• 1. What’s of £12
5

7
• 2. What’s of £25
5

3
• 3. What’s of £100
8

3
• 4. What’s of £144
12
Using Fractions in Other Questions.
• Sometimes we’ll see fractions in questions that require us to use
other knowledge. It’s important that we know our fraction rules so
that we don’t get confused with these questions.
E.g
This question is about powers, but we need to know that the fraction
line means divide to work it out.

• E.g. This question needs you to convert

words into fractions.
Checklist.
• How do we convert between improper fractions and mixed numbers?

• What are the two methods to order fractions?

• What is the key step required for adding and subtracting fractions?

• What is the method for multiplying fractions?

• What does reciprocal mean? Show with an example.