1 f3 multiplication and division of fractions

Multiplication and Division of Fractions
Frank Ma © 2011

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
a*c
a
=
*
d
b*d
b

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

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

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

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

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

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

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

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

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

c
a*c
a
=
*
d
b*d
b
3
3
15 * 12
12
15
9
3*3
a.
=
*
=
=
8 * 25
25
8
10
2*5
5
2
7*8*9*10
8
7
10
9
7
b.
*
*
*
=
=
9
8
11
10
11
8*9*10*11
c
a c
a
±
=
±
Can't do this for addition and subtraction, i.e.
d
b d
b
±

a
a
The fractional multiplications are important.
d
d
or
b
b
*
*
Often in these problems the denominator b can be cancelled
against d = .
d
1

a
a
d
d
or
b
b
*
*
against d = .
d
1
Example B: Multiply by cancelling first.
2
a.
18
*
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18
*
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6
*
*
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
11
b.
48
*
16

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
*
16

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11
*
16

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
The often used phrases " (fraction) of .." are translated to multiplications correspond to this kind of problems.

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
2
Example C: a. What is of $108?
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
2
3
2
* 108
The statement translates into
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
2
3
36
2
* 108
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
2
3
36
2
* 108 = 2 * 36
3

a
a
d
d
or
b
b
*
*
against d = .
d
1
6
2
a.
18 = 2 6 = 12
*
*
3
3
11
b.
48
= 3 * 11 = 33
*
16
2
3
36
2
* 108 = 2 * 36 = 72 $.
3

b. 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 lemon drops?

1
For chocolate, ¼ of 48 is
* 48
4

12
1
* 48 = 12,
4

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

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

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

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

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

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

12
1
* 48 = 12,
4
16
1
* 48
= 16,
3
the lemon drops is
20
20/4
=
48
48/4

12
1
* 48 = 12,
4
16
1
* 48
= 16,
3
the lemon drops is
20
20/4
5
=
=
48
48/4
12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

a
b
b
a
2
3
1
3
2
5
1
1
x
3
2
3
1
1
= 1,
5
= 1,
x
= 1,
x
*
*
*
3
2
5
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
1
3
2
5
1
1
x
3
2
3
1
1
= 1,
5
= 1,
x
= 1,
x
*
*
*
3
2
5
1
x
1
*
2
d
a
c
a
=
÷
*
c
b
d
b
reciprocate

a
b
b
a
2
3
1
3
2
5
1
1
x
3
2
3
1
1
= 1,
5
= 1,
x
= 1,
x
*
*
*
3
2
5
1
x
1
*
2
d
a*d
a
c
a
=
=
÷
*
c
b*c
b
d
b
reciprocate

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

15
12
8
12
*
=
a.
÷
8
25
15
25

3
15
12
8
12
*
=
a.
÷
8
25
15
25
2

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

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

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

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

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

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

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

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

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

3
3
15
12
8
12
9
=
*
=
a.
÷
8
25
15
25
10
5
2
3
1
9
3
9
6
÷
= * =
b.
8
8
6
16
2
6
1
= 30
*
=
5
d.
÷
5
1
6
Example E: We have ¾ cups of sugar. A cookie recipe calls for 1/16 cup of sugar for each cookie. How many cookies
can we make?

3
3
15
12
8
12
9
=
*
=
a.
÷
8
25
15
25
10
5
2
3
1
9
3
9
6
÷
= * =
b.
8
8
6
16
2
6
1
= 30
*
=
5
d.
÷
5
1
6
can we make?
3
1
÷
We can make
4
16

3
3
15
12
8
12
9
=
*
=
a.
÷
8
25
15
25
10
5
2
3
1
9
3
9
6
÷
= * =
b.
8
8
6
16
2
6
1
= 30
*
=
5
d.
÷
5
1
6
can we make?
3
1
3
16
÷
=
We can make
*
4
16
4
1

3
3
15
12
8
12
9
=
*
=
a.
÷
8
25
15
25
10
5
2
3
1
9
3
9
6
÷
= * =
b.
8
8
6
16
2
6
1
= 30
*
=
5
d.
÷
5
1
6
can we make?
4
3
1
3
16
÷
=
We can make
*
4
16
4
1

3
3
15
12
8
12
9
=
*
=
a.
÷
8
25
15
25
10
5
2
3
1
9
3
9
6
÷
= * =
b.
8
8
6
16
2
6
1
= 30
*
=
5
d.
÷
5
1
6
can we make?
4
3
1
3
16
÷
=
= 3 * 4 = 12 cookies.
We can make
*
4
16
4
1

1 f3 multiplication and division of fractions

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