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Academic Skills Advice
Fractions
What do fractions mean?
Look at the bottom of the fraction first – this tells you how many pieces the shape (or
number) has been cut into. Then look at the top of the fraction – this tells you how
many pieces you are using.
𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 (𝑡𝑜𝑝)
𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 (𝑏𝑜𝑡𝑡𝑜𝑚)
means
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑝𝑖𝑒𝑐𝑒𝑠 𝑎𝑟𝑒 𝑦𝑜𝑢 𝑢𝑠𝑖𝑛𝑔?
𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝐸𝑄𝑈𝐴𝐿 𝑝𝑖𝑒𝑐𝑒𝑠 ℎ𝑎𝑣𝑒 𝑦𝑜𝑢 𝑠𝑝𝑙𝑖𝑡 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑖𝑛𝑡𝑜?
1
4
Examples:
1
4
3
5
“split the shape (or number) into 4
means
and use 1 piece”
3
5
“split the shape (or number) into 5
means
and use 3 pieces”
Remember:
If the top and bottom numbers of a fraction are the same then you have a whole one.
3
For example 3 means split the shape (or number) into 3 pieces and use 3 of them (in other
words the whole thing).
2 halves = 1
2
2
3 thirds = 1
3
=1
3
=1
4 quarters =1
4
4
=1
5 fifths =1
5
5
=1
6 sixths = 1 etc
6
6
=1
Remember:
A fraction is also a way of writing a division calculation.
2
For example 3 is a fraction and it also means 2÷3
(to do this calculation you would write 3 2.00 )
© H Jackson 2012 / ACADEMIC SKILLS
1
Finding Fractions of Numbers:
Remember our definition of a fraction: the bottom tells us how many pieces to split (divide)
the shape (or number) into and the top tells us how many pieces we are using. So it
follows, that we will divide by the bottom number and then multiply by the top number:
𝟏
To find:
𝟐
𝟏
𝟑
𝟏
𝟒
of a number
÷ by 2
of a number
÷ by 3
of a number
÷ by 4
etc.
Always divide by the bottom number:
1
3
1
5
of 12 = 12 ÷ 3 = 4. (to find a third divide by 3  the bottom number.)
of 35 = 35 ÷ 5 = 7. (to find a fifth divide by 5  the bottom number.)
Then multiply by the top number:
x
Answer
2
3
of 12
÷
12 ÷ 3 = 4
4x2=8
(to find 1 third ÷3)
(x2 to find 2 thirds)
bottom number
4
7
of 42
top number
42 ÷ 7 = 6
6 x 4 = 24
(to find 1 seventh ÷7)
(x4 to find 4 sevenths)
Examples:

Work out
𝟐
𝟑
of 42
42 ÷ 3 = 14 then

Work out
𝟓
𝟗
of 63
63 ÷ 9 = 7 then

Work out
𝟑
𝟒
14 x 2 = 28
7 x 5 = 35
of 408
408 ÷ 4 = 102 then 102 x 3 = 306
At this stage you could try question 1-3 on the Fraction Practice Sheet.
© H Jackson 2012 / ACADEMIC SKILLS
2
Equivalent Fractions:
Some fractions are equivalent to (the same size as) others.
For example
1
2
is the same as
2
4
.
To make an equivalent fraction you can multiply the top number of a fraction by anything
as long as you do the same to the bottom number, and vice versa.
Examples:
2
3
X4
=
X4
8
3
12
5
X2
=
X2
6
5
10
7
X10
=
X10
50
70
The above pairs of fractions are equivalent because the top and bottom have been
multiplied by the same number every time. NB you should always work in PAIRS – think
to yourself “have I done the same to the top number as the bottom?”
Simplifying/Cancelling Fractions:
This is the same principle as above but you divide the top and bottom by the same
number instead of multiplying.
Examples:
15
20
÷5
=
÷5
3
14
4
21
÷7
=
÷7
2
12
3
15
÷3
=
÷3
4
5
To simplify a fraction you need to look for the number that will go into both the top and the
bottom number. When you can’t simplify any more then the fraction is in its simplest form.
You should write fractions in their simplest form wherever possible.
Creating Fractions from Real Scenarios:
If you are asked to write one number as a fraction of another just write the fraction then
simplify if possible. The number that you are writing the fraction of goes on the bottom
(normally this is the biggest number), e.g. If I have £10 and spend £7 I have spent 7 out of
7
10 which is 10.
Example:
Alex scored 20 out of 25 in a test. Write his score as a fraction in it’s lowest term.
Fraction:
20
25
simplifies to
© H Jackson 2012 / ACADEMIC SKILLS
4
5
(top and bottom both divided by 5)
3
Improper Fractions & Mixed Numbers:
An improper fraction is top heavy (𝑒𝑔
12
5
).
(read as twelve fifths)
2
A mixed number has both a whole number and a fraction (𝑒𝑔 5 3 ). (read as five and two thirds)
You may be asked to convert from one to the other.
Converting from Improper to Mixed:
Divide the bottom number into the top then write the remainder as the fraction.
Examples:
13
5
Whole ones
How many 5’s in 13 (answer is 2) how many left over? (answer is 3) = 2
(2 x 5 = 10)
17
3
Fifths
(13 – 10 = 3)
How many 3’s in 17 (answer is 5) how many left over? (answer is 2) = 5
3
5
2
3
Converting from Mixed to Improper:
Multiply the whole number by the bottom number of the fraction then add on the top
number of the fraction.
Examples:
Whole number
Bottom of fraction
4
2
3
4
3
7
Top of fraction
4 x 3 = 12 then add the extra 2 on. You have 14 thirds =
14
3
3 x 7 = 21 then add the extra 4 on. You have 25 sevenths =
Same denominator
25
7
Try doing all the examples the other way round and see if you get back to where we
started.
At this stage you could try question 4-8 on the Fraction Practice Sheet.
© H Jackson 2012 / ACADEMIC SKILLS
4
Multiplying Fractions:
To multiply fractions you need to multiply the top numbers (numerators) together and then
multiply the bottom numbers (denominators) together.
Example:
3
5
×
2
=
7
6
(3×2)
35
(5×7)
If possible simplify the numbers in the fractions before you multiply. You can simplify any
top number with any bottom number as long as you divide them by the same number.
Remember to work in PAIRS – you must always divide a top number and a bottom number.
(the 6 (top) and the 9 (bottom) will both divide by 3.)
Example:
𝟔
×
𝟕
2
𝟐
6
=
𝟗
7
×
2
9
=
3
4
(2×2)
21
(7×3)
Cancel as many pairs of numbers as you can so that the fractions are as simple as
possible before you multiply. However if you miss any it won’t matter – just simplify at the
end.
Example:
𝟔
𝟏𝟐
(the 6 (top) and the 9 (bottom) will both divide by 3.)
×
𝟒
𝟗
2
6
=
12
1
×
4
9
3
3
=
2
(2×1)
9
(3×3)
(the 4 (top) and the 12 (bottom) will both divide by 4.)
Dividing Fractions:
Dividing fractions is easy once you know how to multiply them.
To divide fractions change the divide sign into a multiplication sign and turn the second
fraction upside down. Then multiply as explained above.
Example:
1
3
÷
4
5
=
1
3
×
÷ becomes x
© H Jackson 2012 / ACADEMIC SKILLS
5
=
4
5
(1×5)
12
(3×4)
Second fraction is
turned upside down.
5
Adding Fractions:
Before you add fractions you need to make sure the denominators are the same. To do
this you need to be able to find equivalent fractions.
If the denominators are the same: it is a straight forward adding of the numerators only:
Example:
𝟐 𝟑
𝟓
+ =
𝟕 𝟕
𝟕
(2 + 3 = 5)
(7𝑡ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 − 𝒅𝒐 𝒏𝒐𝒕 𝑎𝑑𝑑 𝑡ℎ𝑒𝑠𝑒)
If the denominators are different: you need to use your knowledge of equivalent
fractions (see page 2) to make them the same. Look for a common denominator (a
number that both denominators will go into.)
Example:
𝟐
+
𝟑
These cannot just be added as the denominators are
different. So find the smallest number that 3 and 4 both
go into (12) then convert both fractions into 12th’s.
𝟑
𝟒
2
Top &
bottom x4
X4
3 X4
+
3 X3
4 X3
Top &
bottom x3
8
9
12
12
Now we have:
8
9
17
+
=
(𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑛 𝑛𝑜𝑤 𝑏𝑒 𝑎𝑑𝑑𝑒𝑑 𝑎𝑠 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒)
12 12
12
17
5
Remember to finish the question 12 should be written as 1 12 (see page 3).
Example:

𝟏
𝟒
+
𝟐
𝟓
1
Make the
denominators
the same.
4
2
+ =
5
20
5
+
8
20
=
© H Jackson 2012 / ACADEMIC SKILLS
𝟏𝟑
𝟐𝟎
6
Subtracting Fractions:
The method for subtracting fractions is exactly the same as that for adding.
Denominators the same: subtract the numerators only.
Example:
𝟖
𝟐
𝟔
−
=
𝟏𝟏 𝟏𝟏
𝟏𝟏
(8 − 2 = 6)
(11𝑡ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 − 𝒅𝒐 𝒏𝒐𝒕 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡 𝑡ℎ𝑒𝑠𝑒)
Denominators different: look for a common denominator.
Example:
𝟐
𝟑
−
These cannot just be subtracted as the denominators
are different. So find the smallest number that 3 and 2
both go into (6) then convert both fractions into 6th’s.
𝟏
𝟐
2
Top &
bottom x4
X2
3 X2
−
1
2
X3
X3
Top &
bottom x3
4
3
6
6
Now we have:
4 3
1
− =
(𝑡ℎ𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠 𝑐𝑎𝑛 𝑛𝑜𝑤 𝑏𝑒 𝑠𝑢𝑏𝑡𝑟𝑎𝑐𝑡𝑒𝑑 𝑎𝑠 𝑡ℎ𝑒 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟𝑠 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒)
6 6
6
Example:

𝟐
𝟑
−
𝟏
𝟓
=
2
Make the
denominators
the same.
3
−
10
15
1
5
−
=
3
15
=
𝟕
𝟏𝟓
Now try all the questions on the Fraction Practice Sheet.
© H Jackson 2012 / ACADEMIC SKILLS
7