2 fractions multiplication and division of fractions

Fractions are numbers of the form (or p/q) where
p, q  0 are whole numbers.
p
q
Fractions

p, q  0 are whole numbers.
p
q
Fractions
3
6

p, q  0 are whole numbers. Fractions are numbers that
measure parts of whole items.
p
q
Fractions
3
6

Suppose a pizza is cut into 6 equal slices and we have 3 of
them, the fraction that represents this quantity is .
p
q
3
6
Fractions
3
6

p
q
3
6
3
6
Fractions

p
q
3
6
The bottom number is the
number of equal parts in the
division and it is called the
denominator.
3
6
Fractions

p
q
3
6
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions

p
q
3
6
denominator.
The top number “3” is the
number of parts that we
have and it is called the
numerator.
3
6
Fractions
3/6 of a pizza

Whole numbers can be viewed as fractions with denominator 1.
Fractions

Thus 5 = and x = .5
1
x
1
Fractions

Thus 5 = and x = . The fraction = 0, where x  0.5
1
x
1
0
x
Fractions

Thus 5 = and x = . The fraction = 0, where x  0.
However, does not have any meaning, it is undefined.
5
1
x
1
0
x
x
0
Fractions

5
1
x
1
0
x
x
0
Fractions
The Ultimate No-No of Mathematics:

5
1
x
1
0
x
x
0
Fractions
The denominator (bottom) of a fraction can't
be 0.

5
1
x
1
0
x
x
0
Fractions
be 0. (It's undefined if the denominator is 0.)

5
1
x
1
0
x
x
0
Fractions
Fractions that represents the same quantity are called
equivalent fractions.

5
1
x
1
0
x
x
0
Fractions
1
2
=
2
4

5
1
x
1
0
x
x
0
Fractions
1
2
=
2
4
=
3
6

5
1
x
1
0
x
x
0
Fractions
… are equivalent fractions.1
2
=
2
4
=
3
6
=
4
8

5
1
x
1
0
x
x
0
Fractions
… are equivalent fractions.
The fraction with the smallest denominator of all the
equivalent fractions is called the reduced fraction.
1
2
=
2
4
=
3
6
=
4
8

5
1
x
1
0
x
x
0
Fractions
… are equivalent fractions.
The fraction with the smallest denominator of all the
equivalent fractions is called the reduced fraction.
1
2
=
2
4
=
3
6
=
4
8
is the reduced one in the above list.
1
2

Factor-Cancellation Rule
Given a fraction , then
that is, if the numerator and denominator are divided by the
same quantity c, the result will be an equivalent fraction.
a
b
a
b =
a / c
Fractions
b / c

In other words, a common factor of the numerator and the
denominator may be canceled as 1,
a
b
a
b =
a / c
Fractions
b / c

denominator may be canceled as 1, i.e.
a
b
a
b =
a / c
=
a*c
b*c
a*c
b*c
1
Fractions
b / c

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
(Often we omit writing the 1’s after the cancellation.)

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
To reduce a fraction, we keep divide the top and bottom by
common numbers until no more division is possible.
What's left is the reduced version.

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
Example A: Reduce the fraction .78
54

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=
78/2
54/2

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=
78/2
54/2
=
39
27

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=
78/2
54/2
=
39/3
27/3
39
27

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=
78/2
54/2
= 13
9 .
=
39/3
27/3
39
27

a
b
a
b =
a / c
a
b .=
a*c
b*c =
a*c
b*c
1
Fractions
b / c
54
78
54
=
78/2
54/2
= 13
9
=
39/3
27/3
(or divide both by 6 in one step.)
39
27

Fractions
One common mistake in cancellation is to cancel a common
number that is part of an addition (or subtraction) in the
numerator or denominator.

Fractions
A participant in a sum or a difference is called a term.

Fractions
The “2” in the expression “2 + 3” is a term (of the expression).

Fractions
The “2” is in the expression “2 * 3” is called a factor.

Fractions
Terms may not be cancelled. Only factors may be canceled.

Fractions
2 + 1
2 + 3
3
5
=

Fractions
2 + 1
2 + 3
3
5
=
This is addition. Can’t cancel!

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
3
5
=

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
!?

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
!? 2 * 1
2 * 3
=
1
3
Yes

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
!?
Improper Fractions and Mixed Numbers
2 * 1
2 * 3
=
1
3
Yes

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
!?
A fraction whose numerator is the same or more than its
denominator (e.g. ) is said to be improper .
3
2
2 * 1
2 * 3
=
1
3
Yes

Fractions
2 + 1
2 + 3
= 2 + 1
2 + 3
= 1
3
3
5
=
!?
A fraction whose numerator is the same or more than its
denominator (e.g. ) is said to be improper .
We may put an improper fraction into mixed form by division.
3
2
2 * 1
2 * 3
=
1
3
Yes

23
4
Example B.
a. Put into mixed form.
b. Convert into improper form.5 3
4

23
4
23 4 = 5 with remainder 3.·
·
Example B.
4

23
4
·
23
4
= 5 + 5 3
4 .
Example B.
3
4
=
4
Hence,

23
4
·
23
4
= 5 + 5 3
4 .
Example B.
3
4
=
We may put a mixed number into improper fraction by doing
the reverse via a multiplication followed by an addition.
4
Hence,

23
4
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
Example B.
3
4
=
4
Hence,

23
4
·
23
4
= 5 + 5 3
4 .
5 3
4
= 4*5 + 3
4
23
4
=
Example B.
3
4
=
4
Hence,

Multiplication and Division of Fractions
The phrase “3/6 of a pizza” instructs us to
divide a pizza evenly into 6 slices
and take 3 of the slices.

3/6 of a pizza

Likewise the phrase “2/3 of $108”
instructs us to divide $108 into 3 piles
so each pile consists of 108/3 = $36
3/6 of a pizza

3/6 of a pizza
$108 $36$36$36
2
3
of
Divide into 3 piles

and we take 2 piles or 2 x 36 = $72.
3/6 of a pizza
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles

This procedure is recorded as multiplication
3/6 of a pizza
* 108
2
3
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles

3/6 of a pizza
* 108
2
3
36
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles

3/6 of a pizza
* 108
2
3
36
= 2 * 36 = $72.
$108 $36$36$36 = $72
2
3
of
Divide into 3 piles
Take 2 piles

a
b
d
as in the last example are important because they corresponds
to the commonly used phrases: “(fractional amount) of ..”.
Fractional multiplications of the form
* or
a
b
d *

a
b
d
To calculate these, always divide or cancel first, then multiply.
* or
a
b
d *

a
b
d
Example C. Multiply by cancelling first.
2
3 18a.
11
16
48b.
*
*
*
or
a
b
d *
c. How many minutes are there in 2 3
5
hours?
*

a
b
d
2
3 18a.
6
11
16
48b.
*
*
*
or
a
b
d *
5
hours?
*

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
*
* *
*
or
a
b
d *
5
hours?
*

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
*
or
a
b
d *
5
hours?
*

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
*

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
There are 60 minutes in one hour, 3
5
2
*
* 60 min.so there are

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
5
2
*
*
We split the mixed fraction to do the multiplication, i.e.
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
5
2
*
*
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120
Distribute

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
5
2
*
*
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120 + 36
Distribute
12

a
b
d
2
3 18 = 2 6 = 12a.
6
11
16
48b.
3
*
* *
* = 3 * 11 = 33
or
a
b
d *
5
hours?
5
2
*
*
3
5
2
60 min.so there are
* 60 = (2 + ) * 603
5 = 120 + 36 = 156,
Distribute
12
3
5
hours is 156 minutes.so 2

Rule for Multiplication of Fractions
To multiply fractions, multiply the numerators and multiply
the denominators, but always cancel as much as possible
first then multiply.

c
d
=
a*c
b*d
a
b
*

c
d
=
a*c
b*d
a
b
*
Example D. Multiply by reducing first.
12
25
15
8
*a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
=
3*3
2*5
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
b.
8
9
7
8
*
10
11
9
10
**
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** =
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.

c
d
=
a*c
b*d
a
b
*
=
15 * 12
8 * 25
12
25
15
8
*
2
3
5
3
= =
9
10
3*3
2*5
7*8*9*10
8*9*10*11
b.
8
9
7
8
*
10
11
9
10
** = =
7
11
a.
We can’t cancel like this for ± , i.e.
c
d = a c
b d
a
b ± ±
±
think of the case that: 1
2 = 2
4
1
2
+

Example E. A bag of mixed candy contains 48 pieces of
chocolate, caramel and lemon drops. 1/4 of them are
chocolate, 1/3 of them are caramel. How many pieces of each
are there? What fraction of the candies are the lemon drops?

For chocolate, ¼ of 48 is
1
4
* 48

1
4
* 48 = 12,
12

1
4
* 48 = 12,
12
so there are 12 pieces of chocolate candies.

1
3
* 48
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is

1
3
* 48
16
1
4
* 48 = 12,
12
For caramel, 1/3 of 48 is = 16,

1
3
* 48
16
1
4
* 48 = 12,
12
so there are 16 pieces of caramel candies.

1
3
* 48
16
1
4
* 48 = 12,
12
The rest 48 – 12 – 16 = 20 are lemon drops.

1
3
* 48
16
1
4
* 48 = 12,
12
The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of
the lemon drops is 20
48

1
3
* 48
16
1
4
* 48 = 12,
12
48 = 20/4
48/4

1
3
* 48
16
1
4
* 48 = 12,
12
48 = 20/4
48/4 = 5
12

1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
1
4
* 48 = 12,
12
48 = 20/4
48/4 = 5
12

1
3
* 48
16
c. A class has x students, ¾ of them are girls, how many girls
are there?
3
4 * x.
1
4
* 48 = 12,
12
48 = 20/4
48/4 = 5
12
It translates into multiplication as

The reciprocal (multiplicative inverse) of is .
a
b
b
a
Reciprocal and Division of Fractions

a
b
b
a
So the reciprocal of is ,
2
3
3
2

a
b
b
a
2
3
3
2
the reciprocal of 5 is ,
1
5

a
b
b
a
2
3
3
2
1
5
the reciprocal of is 3,1
3

a
b
b
a
2
3
3
2
1
5
and the reciprocal of x is .1
x
3

a
b
b
a
Two Important Facts About Reciprocals
2
3
3
2
1
5
x
3

a
b
b
a
I. The product of x with its reciprocal is 1.
2
3
3
2
1
5
x
3

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2* = 1,

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2* = 1, 5
1
5* = 1,

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2* = 1, 5
1
5* = 1, x
1
x* = 1,

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
II. Dividing by x is the same as multiplying by its reciprocal .
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 ,*
1
2

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
For example, 10 ÷ 2 is the same as 10 , both yield 5.*
1
2

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
1
2
Rule for Division of Fractions
To divide by a fraction x, restate it as multiplying by the
reciprocal 1/x , that is,

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
1
2
reciprocal 1/x , that is, d
c
a
b
*
c
d =
a
b ÷
reciprocate

a
b
b
a
2
3
3
2
1
5
x
3
2
3
3
2*
= 1, 5
1
5* = 1, x
1
x* = 1,
1
x
1
2
reciprocal 1/x , that is, d
c =
a*d
b*c
a
b
*
c
d =
a
b ÷
reciprocate

Example F. Divide the following fractions.
8
15
=
12
25
a. ÷

15
8
12
25
*
8
15
=
12
25
a. ÷

15
8
12
25
*
8
15
=
12
25 2
3
a. ÷

15
8
12
25
*
8
15
=
12
25 5
3
2
3
a. ÷

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a. ÷

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
÷
÷ =b.

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
÷
÷ = *b.

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
÷
÷ = *b.

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
1
6
=5d. ÷

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
6
1
= 30*
1
6
=5d. ÷ 5

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Example G.
A cake recipe calls for 1¼ cups of sugar for each cake.
How many cakes can we make with 7½ cups of sugar?
6
1
= 30*
1
6
=5d. ÷ 5

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Example G.
In general, we convert mixes fractions into improper ones to
do multiplication or division.
6
1
= 30*
1
6
=5d. ÷ 5

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Example G.
Using division, the number of cakes can be made is
1
2
÷
11 =
7 4
6
1
= 30*
1
6
=5d. ÷ 5

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Example G.
1
2
÷
11 =
15
2
5
4
7 4
÷
6
1
= 30*
1
6
=5d. ÷ 5

15
8
=
12
25
*
8
15
=
12
25 5
3
2
3
9
10
a.
6
9
8
19
8 6
3
2
3
16
÷
÷ = * =b.
Example G.
1
2
÷
11 =
15
2
5
4
3
= 6 cakes.
7 4
÷
15
2 5
4
2
*=
6
1
= 30*
1
6
=5d. ÷ 5

B. Convert the following improper fractions into mixed
numbers then convert the mixed numbers back to the
improper form.
9
2
11
3
9
4
13
5
37
12
86
11
121
17
1. 2. 3. 4. 5. 6. 7.
Exercise. A. Reduce the following fractions.
4
6 ,
8
12 ,
15
9 ,
24
18 ,
30
42 ,
54
36 ,
60
48 ,
72
108

2 fractions multiplication and division of fractions

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