Download Guide to Fraction Arithmetic

Pupils should be taught to:
•use common factors to simplify fractions; use common multiples to
express fractions in the same denomination.
•add and subtract fractions with different denominators and mixed
numbers, using the concept of equivalent fractions.
•multiply simple pairs of proper fractions, writing the answer in its
simplest form.
•divide proper fractions by whole numbers.
A fraction shows a whole thing that has been broken into equal
parts . On their own, they show something which is less than one
whole e.g. ¼, ¾, ½ etc…
3
4
Numerator- this number shows
how many parts of the whole is
being represented by the fraction.
Denominator- This number
shows how many equal parts the
whole thing has been split into.
Two important concepts for calculating with fractions are factors and
multiples. We will see why later on. Using 12 as our example number, here
are the definitions for these concepts and 12’s factors and multiples.
FACTORS
MULTIPLES
A whole number that divides exactly
into another whole number.
The product of multiplying a number by
an integer (whole number). Groups of a
given number.
12
1, 2, 3, 4, 6, 12
12, 24, 36, 48, 60, 72,
84, 96, 108, 120, 132,
144, 156, etc …
When we calculate with fractions, we can sometimes end up with
large numerators and denominators. If we can, we should simplify
these to express them with an equivalent fraction that has the
smallest numerator and denominator possible.
9
12
÷3
÷3
3
4
There are two main methods of simplifying fractions. See
the next page for how to do this…
When we calculate with fractions, we can sometimes end up with large
numerators and denominators. If we can, we should simplify these to express
them with an equivalent fraction that has the smallest numerator and
denominator possible. There are two ways to do this. The first is quickest!
1) Finding Highest Common Factors
9
To simplify 12 , write down all the factors of
9 and 12 and then choose the highest one
that they both share to divide by.
Factors
9
12
1
3
9
1
2
3
4
6
12
9
3
=
12 ÷ 3 4
÷3
2) Repeatedly divide by the smallest
prime number possible.
For harder numbers, or if you find it
difficult to find factors, divide by 2 as many
times as possible, then try 3, then 5 etc…
÷2
÷2
÷3
24
12
6
2
=
=
=
108 54 27 9
÷2
÷2
÷3
There are two ways that we can use fractions to show a quantity that is greater than
a whole. We need to be able to convert from improper fractions to mixed numbers
(and sometimes vice versa) when calculating using fractions.
What fraction of
these whole shapes
are white?
Improper Fractions
In improper fractions, the numerator is
greater than the denominator. This shows
that you have more than enough parts to
make at least one whole.
15
4
Mixed Numbers
Mixed numbers are used to show where there
are whole numbers and fractions. The whole
ones are shown in numbers before any
additional parts shown as fractions.
The quantity shown
in the diagram above
can be written in
these two ways…
3
3
4
Reciprocals are useful things when dividing fractions.
The reciprocal of a number is 1 divided by that number.
The reciprocal of a fraction is the fraction turned upside
down. Simple!
So the reciprocal of
3
4
is
4
3
For whole numbers, to work out the reciprocal of 4:
4
We can think of 4 as
1
1
So 4 turned upside down is
4
Roll
over!
Adding Fractions with the Same Denominator
Adding Fractions Where One has a Denominator that is a Factor of the Other
Adding Fractions Where One Denominator is Not a Factor of the Other
1
5
2
5
+ =?
Remember! Don’t
add the
denominators
together!
Adding fractions is as
easy as adding
whole numbers!
Step 1- The only things that are added together
are the numerators. The denominator will stay
the same. (Unless we simplify the answer later).
1
5
1
5
Step 2- Add the numerators together.
1
5
Step 3- Show as the answer to the question.
1
5
2
5
+ =?
2
5
+ =
+
2
5
=
+ =
2
5
5
+ =
5
3
5
1
5
+
4
10
If one of the denominators is a
multiple of the other, you can
complete the calculation by
converting one to the other.
(Usually the smaller one converts
to the larger.)
=?
Adding fractions is as
easy as adding
whole numbers!
Step 1- Convert one fraction to the same denominator
as the other (if it is a factor of it) by multiplying the
numerator by the same number as the denominator
must be multiplied by.
Step 2- Add only the numerators together.
Step 3- Simplify the answer where possible by
dividing the numerator and denominator by
their highest common factor.
x2
x2
1
5
+
2
10
2
10
4
10
+
+
+
=?
4
10
4
10
=?
=
=
6
3
=
10 5
6
10
1
3
+
4
10
To add these together, both fractions
must be converted. Look for the
lowest common multiple of each of
the denominators and convert to that
fraction by multiplying the
numerator by the same numbers as
their denominators.
=?
Adding fractions is as
easy as adding
whole numbers!
Step 1- Convert both fractions to the same
denominator by finding the lowest common multiple
of the two numbers. Multiply the numerators by the
same number as the denominator to convert them.
Step 2- Add only the numerators together.
1
3
x10
x10
10
30
10
30
+
4
10
+
12
30
+
12
30
÷2
Step 3- Simplify the answer where possible by
dividing the numerator and denominator by
their highest common factor.
=
x3
x3
22
30
22 11
=
30 ÷ 2 15
Subtracting Fractions with the Same Denominator
Subtracting Fractions Where One has a Denominator that is a Factor of the
Other
Subtracting Fractions Where One Denominator is Not a Factor of the Other
Subtracting Fractions from Mixed Numbers
7
8
3
8
− =?
Remember! Don’t
subtract the
denominators away
from each other!
If you can add
fractions, you can
also subtract them!
Step 1- You do not have to alter the fractions if
the denominators are the same.
7
8
7
8
Step 2- Simply subtract the numerator away
from the other.
7
8
Step 3- Show as the answer to the question.
7
8
3
8
− =?
3
8
− =
-
3
8
=
− =
3
8
− =
8
4
8
9
15
2
5
− =?
If one of the denominators is a
multiple of the other, you can
complete the calculation by
converting one to the other.
(Usually the smaller one converts
to the larger.)
This is the same process
as the addition with
similar denominators.
Step 1- Convert one fraction to the same denominator
as the other (if it is a factor of it) by multiplying the
numerator by the same number as the denominator
must be multiplied by.
Step 2- Subtract only the numerators.
9
15
9
15
9
15
−
2
5
−
6
15
−
6
15
÷3
Step 3- Simplify the answer where possible by
dividing the numerator and denominator by
their highest common factor.
x3
=
=
x3
3
15
3
1
=
15
5
÷3
6
7
1
4
− =?
If you can add numbers like
this, you can subtract
numbers just as easy by
converting both fractions
using the lowest common
multiple.
Adding fractions is as
easy as adding
whole numbers!
Step 1- Convert both fractions to the same
denominator by finding the lowest common multiple
of the two numbers. Multiply the numerators by the
same number as the denominator to convert them.
Step 2- Subtract one numerator away from the
other in the usual way.
Step 3- Simplify the answer where possible. Be
aware: it is not always possible. as it isn’t here!
6
7
x4
x4
24
28
24
28
−
1
4
−
7
28
−
7
28
17
28
x7
=
x7
17
28
1
2
3
2
3
− =?
The easiest way to compete
calculations like this is to
convert the mixed numbers
into improper fractions.
Converting from mixed
numbers to improper
fractions is easy!
1
23 =
3
3
3
1
1
2
3
7
+3+3=3
Step 1- Convert the mixed number to an improper
fraction. Think of the whole numbers in terms of the
3
fraction. 2 whole ones equal .
3
Step 2- Convert both fractions to the same
denominator by finding the lowest common
multiple.
Step 3- Convert the improper fraction back to
a mixed number:
25
12
12
1
1
=
+
+
=
2
12
12
12 12
12
Simplify the fraction, if this is possible.
7
3
28
12
1
4
− =?
1
4
− =?
−
3
12
=
25
12
25
1
=2
12
12
Multiplying Fractions by Fractions
Multiplying Fractions by Whole Numbers
Multiplying Mixed Numbers by Whole Numbers
3 1
𝑥
9 3
Fractions can be multiplied
together easily. Just multiply
the numerators together
and then the denominators
together!
=?
If you know your
tables, you know
how to do this!
Step 1- Multiply the numerators together and
the denominators together.
Step 2- Write the answers to these calculations
as the new numerator and denominator.
Step 3- Simplify the fraction if possible by
dividing the numerator and the denominator
by their highest common factor.
3 1
𝑥 =
9 3
x
=
3 1
𝑥 =
9 x 3 =
3 1
𝑥
9 3
3
27
=
?
3
27
÷3
=
÷3
1
9
3
𝑥
5
6=?
To multiply a fraction by a
whole number (integer), just
turn the whole number into
a fraction and do the same
as on the previous page.
Multiplying fractions
is easy! Follow this
simple guide!
Step 1- Convert the whole number to an
improper fraction with a denominator of 1.
Step 2- Multiply the numerators together and
the denominators together and write the
answer as an improper fraction.
Step 3- Convert the improper fraction to a
mixed number.
3
2 𝑥
5
6=?
Complete this type of calculation
in stages. Partition the whole
number and the fraction and
then do the same as in the
previous question type.
Just one more step to
do to multiply mixed
numbers by whole
numbers.
Step 1- Partition the mixed number into a
whole number and a fraction. Multiply the
whole numbers first.
Step 2- Next, multiply the fraction with the
whole number by converting it to an improper
fraction with a denominator of 1.
Step 3- Convert the improper fraction to a
mixed number.
Step 4- Add the whole number answer to the
mixed number answer to recombine.
Dividing Fractions by Fractions
Dividing Fractions by Whole Numbers
Dividing Whole Numbers by Fractions
2
3
1
3
÷ =?
To divide by a fraction, turn
the divisor on its head to
generate the reciprocal.
Then, just multiply them
together!
If you can multiply
fractions, you can
divide them easily!
Step 1- Use the reciprocal of the divisor (the
number you divide by) only! The dividend
(the number you are dividing) is kept the
same.
2
3
1
3
÷ =?
2 3
𝑥
3 1
x
Step 2- Multiply the numerators together and
the denominators together and write the
answer as an improper fraction.
2 3
𝑥
3 x 1
Step 3- Convert the improper fraction to a
mixed number (or whole number in this case).
6
3
=?
=
=
=
=2
6
3
2
3
÷3=?
Dividing fractions by whole
numbers can be done easily
by converting the whole
number to a fraction and
using the reciprocal.
Just one additional
step to doing the
same job with whole
number divisors.
Step 1- Turn the whole number into a fraction.
Step 2- Turn the divisor upside down to use
the reciprocal. Multiplying by the reciprocal is
the same as dividing by the original number.
Step 3- Multiply the numerator by the
numerator and the denominator by the
denominator.
Step 4- If possible, simplify the fraction by
dividing the numerator and the denominator
by their highest common factor!
2
3
÷3=?
2
3
3
=
1
÷
2 1
𝑥
3 3
x
?
=?
=
2 1 2
𝑥 =
3x 3 = 9
2
9
1
3
9 ÷ =?
For this question type, turn the
whole number (integer) into a
fraction and then use the
reciprocal of the divisor to
multiply and find your answer.
You will notice
something odd about
the answers to these
questions. The number
gets bigger!
Step 1- Turn the whole number into a fraction.
Step 2- Turn the divisor upside down the use
the reciprocal. Multiplying by the reciprocal is
the same as dividing by the original number.
Step 3- Multiply the numerator by the
numerator and the denominator by the
denominator.
Step 4- Convert the improper fraction to a
mixed number (or an integer).
1
3
9 ÷ =?
9
1
1
3
÷ =?
9 3
𝑥
1 1
x
=?
=
9 3 27
𝑥 =
1 x1 = 1
27
= 27
1
Use this space to make notes about any other methods that you find useful
or any really difficult examples that you come across.