Introduction to fractions and concepts

Fractions
1 2/10
1/12
1/8
1 ½
11/12
55/60

What is a fraction?
A fraction is a quantity that cannot be represented by a
whole number.
Why do we need fractions?
Consider the following scenario.
Can you finish the whole cake?
If not, how many cakes did you
eat?
1 is not the answer,
neither is 0.
This suggest that we need a
new kind of number.

Definition:
A fraction is an ordered pair of whole numbers, the 1st one
is usually written on top of the other, such as ½ or ¾ .
numerator
a
b denominator
The denominator tells us how many equal pieces the
whole is divided into, and this number cannot be 0.
The numerator tells us how many pieces are being
considered.

Examples:
How much of a pizza do we have below?
• we first need to know the size of the original pizza.
The blue circle is our whole.
- if we divide the whole into 8
equal pieces,
- the denominator would be 8.
We can see that we have 7 of
these pieces.
Therefore the numerator is 7,
and we have 7
of a pizza.
8

Equivalent fractions
a fraction can have many different appearances, these
are called equivalent fractions
In the following picture we have ½ of a cake
because the whole cake is divided into two equal parts
and we have only one of those parts.
But if we cut the cake into smaller
congruent (equal) pieces, we can
see that
1 = 2
2
4
Or we can cut the original cake
into 6 congruent pieces,

Now we have 3 pieces out of 6 equal pieces, but
the total amount we have is still the same.
Therefore,
1 2
= 4
2 = 6
3
If you don’t like this, we can cut
the original cake into 8 congruent
pieces,

Then we have 4 pieces out of 8 equal pieces, but
the total amount we have is still the same.
1 =
2
2 =
4
3 =
6
4
8
whenever n is not 0
Therefore, by
amplification we get
a bunch of equivalent
fractions…

How do we know that two fractions are the same?
we cannot tell whether two fractions are the same until
we reduce them to their simplest form.
A fraction is in its simplest form (or is reduced) if we
cannot find a whole number (other than 1) that can divide
into both its numerator and denominator.
Examples:
It´s not in the simplest form because both
numbers NUMERATOR and
DENOMINATOR are divisible by 2.
It´s not in the simplest form because both
6
10
35
40

More examples:
Is not in the simplest form because both
Is already in the simplest form.
Is already in the simplest form.
110
260
8
15
11
23
To find out whether two fractions are equal, we need to
reduce them to their simplest form.

Examples:
Are
14 and
21
30 equal?
45
14 reduce
21
2
3
¸
14 7 =
¸
21 7
30 reduce
45
¸ 6
reduce
9
30 5 =
¸
45 5
2
3
¸
6 3 =
¸
9 3
Now we know that these two fractions are actually
the same!

Another example:
Are 24
and equal?
40
reduce
reduce
30
42
24
40
30
42
¸ 12
reduce
20
24 2 =
¸
40 2
3
5
¸
12 4 =
¸
20 4
5
7
¸
30 6 =
¸
42 6
This shows that these two fractions are not the same!

Improper Fractions and Mixed Numbers
An improper fraction is a fraction 5
with the numerator larger than or
equal to the denominator.
3
4 = 4
2 3
Any whole number can be
transformed into an improper
fraction.
A mixed number is a whole
number and a fraction together 7
,
1
An improper fraction can be converted to a mixed
number and vice versa.
1 = 7
7

Improper Fractions and Mixed Numbers
1 2
3
5 =
3
Converting improper fractions into
mixed numbers:
- divide the numerator by the denominator
- the quotient is the big (whole) number,
- the remainder as the new numerator.
-Keep the same denominator
2 1
17
7
1 3
More examples: 7 =
11 =
Converting mixed numbers
into improper fractions.
2 3 = 2 ´ 7 + 3
= 7
7
,
4
4
5
5

ONE WAY TO KNOW WHICH OF THESE FRACTIONS IS
BIGGER…
3
5
One way to know which one is bigger is by changing the
appearance of the fractions so that the denominators are the
same.
In that case, the pieces are all of the same size, and the one
with the larger numerator makes a bigger fraction.
The straight forward way to find a common denominator is to
multiply the two denominators together:
36
= ´ and
96
3 3 12
=
8 12
8
´
40
96
5 = 5 ´
8
=
12 8
12
´
8
12
Now it is easy to tell that 5/12 is actually a bit bigger than 3/8.

A more efficient and easy way to compare fractions
7
11
5
8
7 × 8 = 56 11 × 5 = 55
Which one is larger,
or 5
?
8
7
11
SSiinnccee 5566 >> 5555,, wwee sseeee tthhaatt
5
8
7 >
11
This method is called cross-multiplication, and make sure that you
remember to make the arrows go upward.

COMPARING NUMBERS BY CROSS
MULTIPLICATION:
Which one is larger, ?
MULTIPLICATION:
2
or 2
7
5
or 8
8
WWhhiicchh oonnee iiss llaarrggeerr,, ?
25
23
or 41
41
267
135
Ans : 2
5
Ans : 8
23
Ans : 41
135

MULTIPLICATION:
or 5
7
Which one is
larger, ?
Which one is
larger,
12
12
or 13
8
20
20
or 63
45
Ans : 7
12
Ans : 13
100
100
Ans : 63
100
20
MULTIPLICATION:

Addition of Fractions
- Addition means combining objects in two or
more sets.
- The objects must be of the same type, i.e. we
combine chickens with chickens and cows
with cows.
- In fractions, we can only combine pieces of the
same size. In other words, the denominators
must be the same.

Addition of Fractions with equal denominators
+ = ?
3
8 1
+
Example: 8
Click to see animation

3
8 1
+
Example: 8
+ =

+ =
3
8 1
+
Example: 8
The answer is
(1+ 3) which can be simplified to 2
8
The denominator stays equal 8.
1

MMoorree eexxaammpplleess
1
+ =
5
2
5
3
5
6 7
=
+ =
10
10
13
10
1 3
10
8
+ =
15
6
15
14
15

Addition of Fractions with
different or unlikely denominators
In this case, we need to first convert them into equivalent
fraction with the same denominator.
Example:
= ´
2
2 2 3
=
1 +
= ´ 5
15
15
1 1 5
=
3 5
3
´
6
5 3
5
´
5
3
An easy choice for a common denominator is 3×5 = 15
Therefore,
11
15
1 + = + 6
=
15
5
15
2
5
3

Addition of Fractions with
different or unlike denominators
Remark: When we have more than 2 denominators , we
need to find the least common denominator (LCM) by
PRIME FACTORIZATION.
When there are only 2 denominators, then the easiest way
is multiply the denominators, to make them equal.
Also, you can use amplification to make the denominat

Exercises to practice:
1
8
´
3 2 +
´
4 2
1
8
3 +
4
2
7
3 +
5
4
9
5 +
6
=
=
=
+ ´
2 5
7 5
´
3 7
5 7
´
´
+ ´
4 6
9 6
´
5 9
6 9
´
´
=
6 +1 =
6 + = 8
=
=
1
8
8
7
8
10
35
21 +
35
31
35
21+10 =
35
=
24
54
45 +
54
115
54
45+ 24 = 69
=
54
54
=

Adding Mixed Numbers
Example:
2 3
5
3 1 +
5
= 3+ 1 + +
2 3
5
5
3
5
= 3+ 2 + 1 +
5
= 5 + 1+ 3
5
= 5 + 4
5
= 5 4
5

Adding Mixed Numbers
Another Example:
13
8
2 4 +
7
= 2 + 4 + +
1 3
8
7
3
8
= 2 +1+ 4 +
7
= 3+ 4´8 + 3´7
56
= 3+ 53
56
= 3 53
56

Subtraction of Fractions
- Subtraction means taking objects away.
-The objects must be of the same type, i.e. we
can only take away apples from a group of
apples.
- In fractions, we can only take away pieces of
the same size. In other words, the denominators
must be the same.

Subtraction of Fractions with equal or like
denominators
11 -
3
12
12
Example:
This means to take away
3 from
12
11
12
11 take away
12
(Click to see animation)

Subtraction of Fractions with equal denominators
Example:
3
12
11 -
12
11
12
3 from
12
11
12

Example:
3
12
11 -
12
3 from
12
11
12

Example:
3
12
11 -
12
3 from
11
Now you can see that there are only 8 pieces left,
therefore
11 - = -
11 3
12
3
12
12
2
3
= 8 =
12
12
12

Subtraction of Fractions
Examples to practice in class:
15 7
- =
- =
16
16
15 7
16
1
2
8 =
16
- ´
´
6 4
=
- =
9
7
54 - =
6 9 4 7
- =
´
´
9 7
7 9
28
63
63
54 28 26
63
63
- ´
´
7 11
=
- =
23
10
´ - ´
10 23
7 23 11 10
=
´
´
23 10
10 23
7 23 11 10
´
161-110
230
= 51
230
Did you get all the answers right?

Subtraction of mixed numbers
This is more difficult than before, so please take notes.
Example:
11
2
3 1 -
4
Since 1/4 is not enough to be subtracted by 1/2, we better
convert all mixed numbers into improper fractions first.
Since 1/4 is not enough to be subtracted by 1/2, we better
convert all mixed numbers into improper fractions first.
3 1 - = ´ + - ´ +
1 2 1
2
3 4 1
4
11
2
4
3
2
= 13 -
4
6
4
= 13 -
4
1 3
4
= 7 =
4

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