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GCSE: Fractions
Skipton Girls’ High School
Objectives: Be able to add, subtract, multiply and divide fractions,
whether improper fractions or mixed numbers. Find fractions of an
amount and solve problems involving successive fractions of an amount.
Teacher Guidance
Possible lesson structure:
Lesson 1: Equivalent fractions, adding/subtracting fractions.
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Lesson 2: Mixed numbers, adding/subtracting mixed numbers.
Simple multiplication/cross-cancelling.
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Lesson 3: Multiplying mixed numbers. Dividing fractions.
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Lesson 4: Fractions of amounts, reverse fractions of amounts.
Problem solving involving fractions of amounts.
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STARTER :: Equivalent Fractions
Give an equivalent fraction which is as simple as possible.
2 1
= ?
4 2
6 2
= ?
9 3
4
2
= ?
10 5
45 5
= ?
54 6
Adding Fractions
Adding fractions is very simple if the
denominators are the same.
2 3 5
+ = ?
7 7 7
“I ate 2 sevenths of the
pizza followed by 3
sevenths. How much
have I eaten?”
Therefore what should be our strategy if the
denominators are different?
?
2 1
8?
3?
11
+ =
+
=
3 4 12 12
12
?
Use equivalent fractions! Find a number both of the
denominators go into (preferably the smallest).
More Examples
1 1 2 1 𝟏
+ = + =?
3 6 6 6 𝟐
3 4 21 20 𝟒𝟏
+ =
+ ?=
5 7 35 35 𝟑𝟓
5 8 15 16 𝟑𝟏
+ =
+ ?=
6 9 18 18 𝟏𝟖
Check Your Understanding
1
3 1 3 2 𝟓
+ = + =?
8 4 8 8 𝟖
2
7 6 49 36 𝟏𝟑
?
− =
−
=
6 7 42 42 𝟒𝟐
3
1 1 1 1 30 20 15 12 𝟕𝟕
+ + + =
+
+
+? =
2 3 4 5 60 60 60 60 𝟔𝟎
N
1 1
𝒂
𝟐
𝒂+𝟐
+ =
+
=?
2 𝑎 𝟐𝒂 𝟐𝒂
𝟐𝒂
A Quicker (Mental) Method
We can multiply the numerators diagonally.
5 1
40 + 6
+ =
48
6 8
Step 1: Multiply the denominators (note: this
guarantees you get a number both 6 and 8
go into, but it may not be the smallest!)
Step 2: Since the 6 got multiplied by 8, so
does the 3. i.e. We are multiplying diagonally.
46 23
=
=
48 24
Disadvantages of this method:
• Because we’re not finding
smallest denominator, further
simplification may be
required.
• Doesn’t extend to more than
two fractions.
Step 3: And repeat with the other numerator.
?
A Quicker (Mental) Method
Another example:
2 3 10 + 27 37
? =
+ =
9 5
45
45
Quickfire Questions (in your head!)
3 2 𝟏𝟗
+ = ?
5 3 𝟏𝟓
3 3 𝟐𝟕
+ = ?
4 5 𝟐𝟎
5 1 𝟐𝟑
+ = ?
8 3 𝟐𝟒
1 1 𝟏
− = ?
2 3 𝟔
4 2 𝟑𝟒
+ = ?
7 5 𝟑𝟓
7 1 𝟑𝟑 𝟏𝟏
− =
=?
9 6 𝟓𝟒 𝟏𝟖
Exercise 1
No calculators!
1 Calculate the following, simplifying your fractions
6 Add the following fractions, giving your result
where possible:
a
c
e
2
1
3
3
8
6
5
1
3
1
4
5
4
3
2
+ =
+ =
+ =
7
12
13
8
27
10
?
?
?
1
5
5
6
7
6
b
d
f
1
5
𝟕
𝟏𝟓
−
−
−
in terms of any variables given.
1
1
=
10
10
3
7
= 30
5
2
17
=
9
18
?
?
?
a
c
?
?
16
1
2
3
16
8
1
4
1
8
?
?
?167
16
0
a
b
c
𝟏𝟔
𝟏𝟔𝟓
𝟏𝟗
𝟏𝟓
?
?
2
1
1
𝟏𝟑
−
+
=
3
4
5
𝟔𝟎
8
3
𝟐𝟖𝟏
− 49 = 𝟏𝟖𝟏𝟑
37
4
2
8
2
−
−
−
9
7
7
9
Complete the magic square (where
the total of each row, column and
long diagonal is the same).
?
?
d
1
3
1
8
+ +
1
27
+ =
𝟑
𝒂
𝒃+𝟐𝒂
𝒂𝒃
?
?
𝑎
2
b
𝑎
3
+ =
𝟓𝒂
𝟔
?
1
?
3 Identify the missing fraction.
1
1
𝟏
7
11
+
□
=
b
a 4+□=3
𝟏𝟐 ?
11
15
1
5
𝟏
3
7
1
d
+ −□=
c 12 + □ = 36 𝟏𝟖 ?
5
6
2
1
3
5
5 Calculate:
4
2
𝑎
2
𝑏
+ =
N Unit fractions are fractions where the
of my bananas are yellow and green. The rest are
pink. What fraction are pink?
1
𝑎
1
𝑎
𝟏𝟔
= −?
𝟐𝟏
=
𝟏𝟎𝟕
𝟐𝟏𝟔
?
numerator is 1, e.g. . Egyptian fractions are a
5
sum of unit fractions where all denominators
are different.
Can you express each of these unit fractions
as Egyptian fractions? There may be multiple
ways.
1 𝟏 𝟏
1 possibility
= +
2 𝟑 𝟔
1 𝟏 𝟏
= +
1 possibility
3 𝟒 𝟏𝟐
1
𝟏 𝟏 𝟏 𝟏
=
+ 2
=possibilities
+
4 𝟏𝟐 𝟔 𝟓 𝟐𝟎
1 𝟏 𝟏
= +
1 possibility
5 𝟔 𝟑𝟎
1
𝟏
𝟏
𝟏
𝟏 𝟏 𝟏
𝟏 𝟏
=
+
= 4 possibilities
+ = +
= +
6 𝟏𝟓 𝟏𝟎 𝟏𝟖 𝟗 𝟖 𝟐𝟒 𝟕 𝟒𝟐
Do the same for larger denominators. Can you
identify when you’ll have only 1 possibility?
(Note to teachers: I proved it in footnotes of this slide)
Mixed Numbers ⇔ Improper Fractions
Improper fractions are
where the numerator is
greater than the
denominator.
Mixed numbers have an
integer and fractional
part.
1?
?3
4
13
4
We could make up 3 wholes using quarters (because 4
goes into 13 three times).
The remainder was 1, so we have 1 quarter left over.
Mixed Numbers ⇔ Improper Fractions
14
4
⇒ 2?
5
5
17
2
⇒ 5?
3
3
13
1
⇒ 6?
2
2
24
⇒
7
3
3?
7
41
1
⇒ 4?
10
10
How many thirds does the 4
wholes give you?
?
1
13
4 ⇒
3
3
3
13
2
⇒ ?
5
5
1
15
7 ⇒ ?
2
2
2
26
8 ⇒ ?
3
3
6
83
11 ⇒ ?
7
7
Adding Mixed Numbers
To add mixed numbers:
1
1
3
?5 2
3 +5 =3 +
2
3
6
6
𝟓
=𝟓 ?
𝟔
5
1
𝟓
𝟐
2 + 5 = 𝟐 +?𝟓
6
3
𝟔
𝟔
𝟕
?
=𝟕
𝟔
𝟏
=𝟖
?
𝟔
Make fractional parts the same.
Add whole parts and fractional
parts separately.
If we’ve ‘overflown’ into the next
whole, we need to carry.
Subtracting Mixed Numbers
To subtract mixed numbers…
1
1
2
3
9 − 4 = 9 −?4
3
2
6
6
8
3
= 8 −?4
6
6
5
=4
?
6
We’d end up with a
negative number for
the fractions. Can we
borrow a whole? (In
the same way we borrow in
column subtraction)
(Alternatively you can just convert both to improper fractions and subtract as normal)
1
2
𝟑
𝟖
7 −3 =𝟕
−?𝟑
4
3
𝟏𝟐
𝟏𝟐
𝟏𝟓
𝟖
=𝟔
−?𝟑
𝟏𝟐
𝟏𝟐
𝟕
=𝟑
?
𝟏𝟐
2
3
56 − 40
5
7
𝟏𝟒
𝟏𝟓
= 𝟓𝟔
−?𝟒𝟎
𝟑𝟓
𝟑𝟓
𝟒𝟗
𝟏𝟓
= 𝟓𝟓
−?𝟒𝟎
𝟑𝟓
𝟑𝟓
𝟑𝟒
?
= 𝟏𝟓
𝟑𝟓
Check Your Understanding
1
2
𝟏
2 + 142 = 𝟏𝟒𝟓?
2
3
𝟔
1
1
𝟏𝟏
76 − 30 = 𝟒𝟓 ?
4
3
𝟏𝟐
Multiplying Fractions
Skills to do with multiplying fractions:
• Multiply simple fractions.
• Multiply a mixture of whole numbers, proper fractions and mixed numbers.
• Understand ‘cross cancelling’ with fractions, including with several fractions where
there may be a pattern of cancelling.
• Find a fraction of an amount (including a fraction of a fraction).
• Understand how to deal with fractions nested inside fractions.
• Solve puzzles involving fractions, particularly involving successive fractions of an
amount and what’s left.
Multiplying fractions is fairly easy. Simply multiply numerators and denominators.
3 5 15
× = ?
4 7 28
Cross-Cancelling
Recall that we can simplify fractions by dividing top and bottom by a common factor.
Since numerators are being combined and denominators likewise, there’s nothing
preventing us cancelling any numerator with any denominator.
1
1
5
6 55 𝟓
×
= ?
11 18 𝟑
3
10 14 𝟒
×
= ?
21 15 𝟗
1 2 3
𝑛−1 𝟏 ?
× × × ⋯×
=
2 3 4
𝑛
𝒏
We can see the numbers between 2 and 𝑛 − 1 are common to top and
bottom. This leaves just 1 in the numerator and 𝑛 in the denominator.
Test Your Understanding
Ensure you simplify
your fraction.
A
3 5
𝟏𝟓
?
×
=
7 11 𝟕𝟕
B
13 20
𝟏
×
= ?
80 39 𝟏𝟐
C
5 21 𝟏
×
= ?
14 15 𝟐
N
1 2 3 4 5
998
𝟏×𝟐
𝟏
× × × × × ⋯×
=
?=
3 4 5 6 7
1000 𝟗𝟗𝟗 × 𝟏𝟎𝟎𝟎 𝟒𝟗𝟗𝟓𝟎𝟎
All numbers between 3 and 998 will cancel top and
bottom. This leaves just 1 and 2 in the numerator
and 999 and 1000 in the denominator.
Exercise 2
1 Convert the following to mixed numbers:
8
3
a 8 ⇒ 1 1?
⇒
1
b
7
7
5
5?
c
11
3
59
6
2
⇒ 3 3?
5 Bob and Dave go to PizzaScoff. Starting with 35
4
5
pizzas, Bob eats 6 pizzas and Dave eats 8
19
3
⇒
4
4
4
111
1
⇒ 22
5
5
?
?
d
5
pizzas. How many are left? 𝟏𝟗
f
⇒9 ?
e
6
2 Convert the following to improper fractions:
1
5
5
32
3
⇒
a 22 ⇒ 2 ?
b
9
9 ?
2
c 73 ⇒
23
3
4
?
d 35 ⇒
19
5
?𝟏𝟑𝟏𝟓
1
3
6 The perimeter of a rectangle is 20. If its width is
3
7
3 , what is its height? 𝟔
?
𝟒
𝟕
?
7 Calculate and simplify:
3 16 15
a 4× 5 × 9 =𝟒
3 Calculate the following (leave answer as mixed number)
1 2 3
98
𝟏
1
3
1
3
2
3
×
×
×
⋯
×
=
b
a 2 + =3
b 5 4 + 2 5 = 8 20
3 4 5
100 𝟒𝟗𝟓𝟎
2
4
4
5 7 9 11
101
1
2
8
1
5
2
N × × × ×⋯×
c 4 − =3
d 8 −4 =3
= 𝟑𝟑𝟑𝟑
5
3
15
2
6
3
1 3 5 7
97
e 7 3 + 10 1 = 17?16
7
3
21
1
7
f
𝟏𝟔
𝟑𝟓
g 68 − 17 = 𝟓𝟏
?
4 Calculate (and simplify):
4
3
12
𝟔
a 5 × 2 = 10 = 𝟓?
b
c
e
10
11
54
55
×
×
?
?
33
𝟑
=
100
𝟏𝟎
77
𝟏𝟒
=
27
𝟓
?
?
?
?
?
3
5
?
d
f
?
2
3
13
10 5 − 7 4 = 2 20
?
1
5
1
4
𝟏𝟗
𝟐𝟎
h 39 − 29 = 𝟗?
?
?
?
7
24
𝟒
× =
6
21
𝟑
18
39
𝟗
× =
13
12
𝟐
81
17
𝟐𝟕
× =
51
56
𝟓𝟔
N Simplify:
3 5 4 6 5 7
𝑛−1 𝑛+1
× × × × × …×
×
4 4 5 5 6 6
𝑛
𝑛
After the first fraction, each pair of fractions
cancel. This leaves just the first and last
fraction, giving
𝟑 𝒏+𝟏
𝟒𝒏
?
Multiplying Mixed Numbers
If the numbers are not improper fractions, convert them into improper fractions.
Examples:
1
1 𝟕 𝟗 𝟔𝟑
3 × 2 = × ?=
2
4 𝟐 𝟒
𝟖
2 𝟒 𝟏𝟕 𝟔𝟖
4 × 3 = × ?=
5 𝟏 𝟓
𝟓
2
𝟐𝟑 𝟐 𝟒𝟔
7 ×2=
× ?=
3
𝟑 𝟏
𝟑
Quickfire Questions:
Schoolboy Error:
thinking the
8
answer is .
4 𝟖
2× = ?
3 𝟑
5
𝟏𝟎
×2=
7
𝟕
6
?
1 4 𝟕 𝟒 𝟐𝟖
2 × = × ?
=
3 5 𝟑 𝟓 𝟏𝟓
1
3 𝟒 𝟏𝟑 𝟓𝟐
1 × 2 = × ?=
3
5 𝟑 𝟓
𝟏𝟓
1
3 𝟕 𝟏𝟗 𝟏𝟑𝟑
3 × 4 = × ?=
2
4 𝟐 𝟒
𝟖
Dividing Fractions
Mental division
1 1
2 ÷ = 𝟏𝟎 ?
2 4
1
5 ÷ = 𝟐𝟎 ?
4
1
100 ÷ = 𝟑𝟎𝟎
?
3
What if this was phrased as “how many
quarter pizzas go into two and a half
pizzas”?
What appears to be the effect of
1
dividing by 4?
1
Or by 3?
Dividing Fractions
More generally…
Reciprocate (i.e. ‘flip’) the second fraction, and use multiplication instead.
4 2
÷
5 3
4 3 𝟔
= × = ?
5 2 𝟓
1
1
2 ÷1
2
3
5 4
= ÷
2 3?
5 3 𝟏𝟓
= × =
2 4
𝟖
3 4
÷
7 9
3 9 𝟐𝟕
= × = ?
7 4 𝟐𝟖
1
2÷3
2
2 7
= ÷
1 2
2 2? 𝟒
= × =
1 7 𝟕
4
8÷1
5
8 9
= ÷
1 5
8 5 ? 𝟒𝟎
= × =
1 9
𝟗
Test Your Understanding
A
D
2
3 ×4
5
17 4
=
×
5 1
𝟔𝟖?
=
𝟓
B
1
8÷
5
= 𝟒𝟎
?
C
[JMO 1996 A4] Evaluate
1
2
𝟏
?
𝟐𝟎
+
1
3
+
1
4
3
6÷1
4
6 7
= ÷
1?4
𝟔 𝟒
= ×
𝟏 𝟕
1
5
× −
1
6
Exercise 3
1 Calculate the following, simplifying/crosscancelling where possible.
a 4×1=𝟒
b 1 1 × 4 = 𝟏𝟔
c
e
g
?
?
?
?
5
𝟓
25
21
𝟏𝟓
× =
7
10
𝟐
1
1
1 ×1 =𝟐
2
3
1
8
4 × = 𝟏𝟐
2
3
d
f
h
?
?
?
?
3
𝟑
18
11
𝟑
× =
55
24
𝟐𝟎
12
39
𝟗
× =
13
8
𝟐
2
3
𝟗𝟏
8 ×1 =
3
4
𝟔
𝟑
1
𝟒
e 6 ÷ 42 = 𝟑?
g 10 ÷
9
9
10
=
𝟏𝟎𝟎
𝟗𝟗
?
1
1
𝟐𝟔
1
2
𝟕
2
5
3
1
4
4
𝟒
𝟑
h
÷
=
4 [JMC 1997 Q9] What is the value of
4
3
1−
4
2
−
𝟒𝟗
3
÷
1
4
?
4
× + =
𝟐𝟑
?
? 16
?
?
?
5
[IMC 2012 Q15] Which of the following has a
value that is closest to 0?
1
1
1
1
1
1
A + ×
B + ÷
C
2
1
×
2
1
−
2
3
4
1
1
÷
3
4
1
1
×
3
4
D
2
1
2
3
1
−
3
4
1
÷
4
Solution: E
?
[Kangaroo Pink 2003 Q15] What is the value of:
1
1
1
1 + × 1 + × ⋯× 1 +
?
2
3
2003
𝟑 𝟒 𝟓
𝟐𝟎𝟎𝟒 𝟐𝟎𝟎𝟒
× × ×⋯×
=
= 𝟏𝟎𝟎𝟐
𝟐 𝟑 𝟒
𝟐𝟎𝟎𝟑
𝟐
?
?
𝟏𝟏𝟑
𝟔𝟑𝟎
7
N1
f 39 ÷ 23 = 𝟔?
1
6
𝑎2 + 𝑐 =
𝑐 − 𝑏𝑐 =
𝟑𝟔
𝟏𝟐
[JMO 2010 A1] What is the value of
1
2
3
4
5
+ 1 + 1 + 1 + 1 ? Solution: 55
1
3
E
d 4 3 ÷ 2 2 = 𝟏𝟓?
1
7
If 𝑎 = , 𝑏 = −2, 𝑐 = 1 determine:
?
1
1
8
6
8
2 [JMC 2010 Q6] Which of the following has the
largest value?
1
1
1
1
1
A6÷
B 5÷
C4÷
D 3÷
E 2÷
2
3
4
5
6
Solution: C
c 1 2 ÷ 5 = 𝟏𝟎?
1
10
Calculate
1
3 Calculate the following:
3
𝟒
4
6
𝟐𝟐
a 1÷4=𝟑 ?
b 7 ÷ 11 = 𝟐𝟏?
1
9
5
N2 Calculate 1 −
1
1−
=
1
1−
1
1
1−3
𝟐
𝟑
?
Embedded Fractions
[JMO 2004 A1] Write
1
1
1+3
as a decimal.
There’s two ways you could deal with this:
1
3 fraction
3 by denominator
Multiply top and bottom
of outer
=inner one.=
of
1 3+1 4
1+
1
3
?
1
4
=
=1÷
4
1
3
Simplify denominator
then
treat
outer
fraction
as a division.
1+
3
3
1 3 ?3
= × = = 0.75
1 4 4
Check Your Understanding
A What is 1
3
?
1
+
4 3
𝟏 𝟏
𝟕
𝟑 𝟏𝟐 𝟑𝟔
𝟑÷
+
= 𝟑 ÷? = ×
=
𝟒 𝟑
𝟏𝟐 𝟏 𝟕
𝟕
N What is
1
1−
1
?
1
1−3
𝟏
𝟏
𝟏
=
=
=
= −𝟐
𝟏
𝟑
𝟏
?
𝟏− 𝟐
𝟏−
−
𝟐
𝟐
𝟑
Fractions of Amounts
3
4
Find of 80
Find
2
4
of
3
5
Find
1
2
of
2
3
3
4
of of
4
5
→
𝟑
𝟒
→
𝟐
𝟑
𝟒
𝟓
?
× =
𝟖
𝟏𝟓
→
𝟏
𝟐
𝟐
𝟑
𝟑
𝟒
?× 𝟓
𝟒
You can think of
the word ‘of’ as ×
× 𝟖𝟎? = 𝟔𝟎
× ×
=
𝟏
𝟓
[JMC 2004 Q15] Granny spends one third of her weekly pension on Thursday night,
and one quarter of what remains on Friday. What fraction of the original amount is
left for her big night out on Saturday?
3
2
She retains 4 of the 3 remaining.
𝟑
𝟒
𝟐
𝟑
→ × =
𝟏
𝟐
?
[JMO 2014 A10] My four pet monkeys and I harvested a large pile of peanuts.
Monkey A woke in the night and ate half of them; then Monkey B woke and ate one
third of what remained; then Monkey C woke and ate one quarter of the rest; finally
Monkey D ate one fifth of the much diminished remaining pile. What fraction of the
original harvest was left in the morning?
→
𝟒
𝟓
𝟑
𝟒
𝟐
𝟑
𝟏
𝟐
× × × =
?
𝟏
𝟓
Reverse Fractions of Amounts
3
of a number is 18. What
7
𝟏
is 6. Thus number is 42.
𝟕
was the original number?
?
[Kangaroo Grey 2014 Q4] A bucket was half full. A cleaner added two
litres of water to the bucket. The bucket was then three-quarters full.
How many litres can the bucket hold?
𝟏
is 2 litres. Thus whole
? bucket is 8 litres.
𝟒
2
Bob has just been on a diet and managed to lose of his body weight. He
5
is now 15 stone. How heavy was he pre-diet?
𝟑
𝟏
of his original weight is 15kg. is 5kg. Thus his original weight was
𝟓
𝟓
?
25kg.
Exercise 4
1 a
b
2
4
If of a number is 20, what is the number? 35
7
5
2
6
3
If of a number is 30, what is of it? 24
?
7
[IMC 2015 Q3] What is a half of a third, plus a third of
𝟑
a quarter, plus a quarter of a fifth? Solution:
?
3
?
𝟏𝟎
[JMC 2013 Q13] When painting the lounge, I used half
of a 3 litre can to complete the first coat of paint. I
then used two thirds of what was left to complete the
second coat. How much paint was left after both
coats were complete?
Solution: 500ml
?
8
?
4
[IMC 2007 Q4] Between them, Ginger and Victoria eat
two thirds of a cake. If Ginger eats one quarter of the
cake, what fraction of the cake does Victoria eat?
?
Solution:
5
𝟓
𝟏𝟐
[IMC 1997 Q15] On the first day after the flood, half
of Noah’s animals escaped. On the second day one
third of the remainder wandered off. On the third day
one quarter of the rest hopped it. What fraction of
𝟏
Noah’s original menagerie was then left?
𝟒
6
?
[JMO 1997 A6] One half of the class got As. One third
of the rest got Bs. One quarter of the remainder got
Cs. One fifth of the others got Ds. What fraction of the
𝟏
class got Es or worse?
Solution:
?
𝟏𝟎
[JMO 2007 A4] The hobbits Frodo, Sam, Pippin and
Merry have breakfast at different times. Each one takes
a quarter of the porridge in the pan, thinking that the
other three have not yet eaten. What fraction of the
porridge is left after all four hobbits had their breakfast?
𝟖𝟏
Solution:
𝟐𝟓𝟔
[TMC Regional 2013 Q10] Dean spent one fifth of the
amount of money in his wallet and then one fifth of
what remained. He spent a total of £72. Find the
amount of money in his wallet to start with. £200
?
9
[TMC Regional 2009 Q10] Three squirrels, Steve, Keith
and Benjamin, have spent all day collecting nuts. At the
end of the day they are very tired and go to bed. During
the night, Steve wakes up and eats a nut. He then
decides to take half of the remaining pile and hides it
before going back to sleep. Keith then wakes up. He too
is hungry, so eats a nut. He also takes half of the
remaining pile and hides it before going back to sleep.
Finally, Benjamin wakes up, eats a nut and hides half of
the remaining pile. In the morning, all of the squirrels
share the remaining nuts equally between them and
each goes off with four more nuts. Each squirrel then
eats all the nuts he now has. How many nuts has Steve
eaten in total?
Solution: 56
?