Download Fraction Booklet - Maxwelltown High School

Fraction
Booklet
for
Pupils, Parents and Staff
©Mr J Scott Maxwelltown High School 2012
page 1
Simplifying Fractions
To simplify a fraction you must divide the top and bottom numbers by the highest number that can divide
both numbers exactly. The highest number that will divide both the top and bottom number exactly is
called the highest common factor (hcf).
Before we start to simplify fractions we must be able to find factors. Factors are whole numbers you
multiply together to get another number so the factors of 15 would be 1, 3, 5, 15
! 1! ×!15! =!15
! 3! ×!15! =!15
the factors can be found by reading the numbers from each side of the multiplication sign.
Practise finding factors by finding the factors of 6, 9, 12, 20 and 30 by filling in the blanks.
! 1! ×! 6! = ! 6
! 1 !× !9 != !9
! 1! ×!12! = !12
! 1! ×! _! = !20
! _! ×!30! = !30
! _! ×! 3! = ! 6
! _ !× !_ != !9
! 2! ×! _! = !12
! 2! ×!10! = !20
! 2! ×! _! = !30
! _! ×! _! = !12
! _! ×! _! = !20
! _! ×! _! = !30
! 5! ×! _! = !30
You should have noticed each number has one and itself as a factor, a number that only has itself and one
as a factor is called a prime number.
The factors of 12 are 1, 2, 3, 4, 6 and 12
The factors of 30 are 1, 2, 3, 5, 6, 10 and 30
1, 2, 3 and 6 are common to both lists so these are common factors to 12 and 30.
The largest common number is 6 so 6 would be the highest common factor of 12 and 30.
We are now ready to start simplifying fractions.
Simplify
12
30
The highest common factor of 12 and 30 is 6 so
12 ÷6 2
=
30 ÷6 5
Now try and simplify each of the following by filling in the blanks
6
8
10
15
9
21
15
36
6÷ _ _
=
8÷ _ _
10 ÷ _ _
=
15 ÷ _ _
9÷ _ _
=
21 ÷ _ _
15 ÷ _ _
=
36 ÷ _ _
Points to note
Simplifying or reducing a fraction means to make the fraction as simple as possible.
Factors are the numbers you multiply together to get another number.
A factor is a common factor when it is a factor of two or more numbers.
The highest common factor (hcf) is the largest of the common factors of two or more numbers.
A number that only has two factors, one and itself is called a prime number.
©Mr J Scott Maxwelltown High School 2012
page 2
Equivalent Fractions
A fraction has two numbers, the top number is called the numerator, the bottom number is called the
denominator.
The numerator is the number of parts you have and the denominator is the number of parts the whole
has been divided into.
If you cut a pizza into four equal parts and you had one part you would have a
1
4
Although some fractions look different they have the same
value. By looking at the table you can see that
1
4 2 1
= =
8 4 2
1
2
1 2
4
,
and
the fractions are equivalent.
2 4
8
We would say
1
2
1
4
1
8
1
4
1
8
1
8
1
4
1
8
1
8
1
4
1
8
1
8
1
8
To find equivalent fractions you must multiply or divide both the numerator and denominator by the same
number, it is important that after dividing the answer is still a whole number.
Remember you can only multiply or divide you cannot add or subtract.
Examples
3 3×5
=
4 4×5
15
=
20
10 10 ÷2
=
12 12 ÷2
5
=
6
Complete these to find equivalent fractions
3 _ ×2
=
5 _ ×2
_
=
10
2 _ ×3
=
7 _ ×3
_
=
_
3 _ ×2
=
8 _×_
_
=
_
6 _ ÷2
=
8 _ ÷2
_
=
4
12 _ ÷3
=
15 _ ÷3
_
=
_
6 _ ÷4
=
24 _ ÷ _
_
=
_
©Mr J Scott Maxwelltown High School 2012
page 3
Adding Fractions
Adding fractions is easy as long as the bottom number (the denominator) in the fractions being added is
the same. You should always show your answer in the simplest form by simplifying the fraction.
Examples
1 1 2
+ =
4 4 4
1
=
2
2 5 7
+ =
7 7 7
=1
1 3 4
+ =
8 8 8
1
=
2
1 1 2
+ =
3 3 3
Now try these questions by filling in the blanks
2 1 _
+ =
6 6 _
_
=
_
3 1 _
+ =
10 10 _
_
=
_
3 2 _
+ =
5 5 _
=_
When adding fractions that have different denominators there is an extra step you have to carry out
before you can complete the addition. This extra step involves making the denominators the same.
2 1
+
3 6
In this addition the two bottom numbers (denominators) are different, we could change the
3 to 6 by multiplying by 2 but remember what you do to the bottom number you must do
to the top number to keep the fractions equivalent.
2 ×2 4
so
=
3 ×2 6
1 3
+
5 4
2 1 4 1
+ = +
3 6 6 6
5
=
6
In this addition you will have to change both denominators, the easiest way to do this is
cross multiplication.
Multiply the first fraction by the second fractions denominator and multiply the second
fraction by the first fractions denominator.
1 3 1× 4 3×5
+ =
+
5 4 5× 4 4×5
4 15
=
+
20 20
19
=
20
Try this question by filling in the blanks
©Mr J Scott Maxwelltown High School 2012
1 3 1 × _ 3 ×2
+ =
+
2 7 2×7 7× _
_ 6
= +
14 _
_
=
_
page 4
Subtracting Fractions
When subtracting fractions you must sure the bottom number (the denominator) in the fractions being
subtracted is the same. You should always show your answer in the simplest form by simplifying the
fraction.
Examples
3 1 2
- =
4 4 4
1
=
2
5 1 4
− =
7 7 7
5 3 2
- =
8 8 8
1
=
4
2 1 1
- =
3 3 3
Now try these questions by filling in the blanks
5 1 _
− =
6 6 _
_
=
_
7 1 _
− =
10 10 _
_
=
_
3 2 _
− =
5 5 _
When subtracting fractions that have different denominators there is an extra step you have to carry out
before you can complete the subtraction. This extra step involves making the denominators the same.
2 1
3 6
In this subtraction the two bottom numbers (denominators) are different, we could change
the 3 to 6 by multiplying by 2 but remember what you do to the bottom number you must
do to the top number to keep the fractions equivalent.
2 ×2 4
so
=
3 ×2 6
3 1
4 5
2 1 4 1
- = 3 6 6 6
3
=
6
1
=
2
In this subtraction you will have to change both denominators, the easiest way to do this
is cross multiplication.
Multiply the first fraction by the second fractions denominator and multiply the second
fraction by the first fractions denominator.
3 1 3×5 1× 4
- =
4 5 4×5 5× 4
15 4
=
20 20
11
=
20
Try this question by filling in the blanks
©Mr J Scott Maxwelltown High School 2012
1 3 1 × _ 3 ×2
− =
−
2 7 2×7 7× _
_ 6
= +
14 _
_
=
_
page 5
Multiplying Fractions
When multiplying two fractions you do not need to worry about having the same bottom numbers, common
denominators, you simply multiply the two top numbers (numerators) together and then the two bottom
numbers (denominators) together. You should always give your answer in the simplest form by simplifying.
Examples
2 1 2×1
× =
3 4 3× 4
2
=
12
1
=
6
3 1 3×1
× =
8 3 8 ×3
3
=
24
1
=
8
5 1 5×1
× =
6 2 6 ×2
5
=
12
Now try these questions by filling in the blanks
3 1 3×1
× =
5 3 5×3
_
=
15
_
=
_
2 2 2× _
× =
7 3 7× _
4
=
_
Dividing Fractions
When dividing two fractions it does not matter if the bottom numbers (denominators) are different but
unlike multiplication there is a slight twist.
When dividing fractions
! 1!
invert (turn upside down) the second fraction
! 2! change the division sign to a multiplication sign
! 3!
multiply the fractions
As with all fraction work you should always give your answer in the simplest form by simplifying.
Examples
1 1 1 2
÷ = ×
8 2 8 1
2
=
8
1
=
4
3 2 3 3
÷ = ×
5 3 5 2
9
=
10
Now try these questions by filling in the blanks
2 1 2 _
÷ = ×
7 3 7 1
_
=
_
©Mr J Scott Maxwelltown High School 2012
3 1 _ 2
÷ = ×
10 2 _ 1
_
=
_
_
=
_
page 6
Fractions of Quantity
When asked to find a fraction of a quantity there are two types of questions.
! 1!
When you have to find a unitary fraction (when the top number is one) of a quantity
! !
Example
! !
!
Find
1
of £160
4
! 2! When the numerator (top number) is greater than one
! !
Example
! !
!
Find
3
of £160
4
If it is a unitary fraction you divide the quantity by the denominator (bottom number)
To find a
1
! divide by 2
2
! !
!
1
divide by 3
3
! !
!
1
divide by 4 and so on
4
Example
! Find!
1
£160
4
160 ÷ 4 = 40
! answer £40
When the numerator is greater than one you have two steps to follow in your calculation
! 1!
Divide by the bottom number
! 2! Multiply your answer from step 1 by the top number
To find!!
3
divide by 5 then multiply your answer by 3
5
! !
5
divide by 7 then multiply your answer by 5
7
!
Example
! Find!
3
£160
4
160 ÷ 4 = 40
40 × 3 = 120
! answer £120
©Mr J Scott Maxwelltown High School 2012
page 7