13 multiplication and division of fractions

Multiplication and Division of Fractions
Back to Algebra–Ready Review Content.

Rule for Multiplication of Fractions

c
d
=
a*c
b*d
a
b
*
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
*
Example A. 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
** =
a.
Each set of cancellation
produces a “1”, which
does not affect final the
product.

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.
Can't do this for addition and subtraction, i.e.
c
d
=
a c
b d
a
b
±
±
±

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

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

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

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

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

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

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

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

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

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

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

a
b
d
a
b
d
d
1
2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
against d = .
3
* 1082
3The statement translates into

a
b
d
a
b
d
d
1
2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
against d = .
3
* 1082
3
36
The statement translates into

a
b
d
a
b
d
d
1
2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
or* *
* *
* = 3 * 11 = 33
against d = .
3
* 108 = 2 * 362
3
36

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

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?

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 D: 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
*
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.
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.
6
1
= 30*
1
6
=5d. ÷ 5
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?

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
can we make?
We can make
3
4
÷ 1
16

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
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1

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
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1
4

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
can we make?
We can make
3
4
÷ 1
16
= 3
4
*
16
1
= 3 * 4 = 12 cookies.
4
HW: Do the web homework "Multiplication of Fractions"

Remember to cancel first!

Exercise. B.
12. In a class of 48 people, 1/3 of them are boys, how many girls are there?
13. In a class of 60 people, 3/4 of them are not boys, how many boys are there?
14. In a class of 72 people, 5/6 of them are not girls, how many boys are there?
15. In a class of 56 people, 3/7 of them are not boys, how many girls are there?
16. In a class of 60 people, 1/3 of them are girls, how many are not girls?
17. In a class of 60 people, 2/5 of them are not girls, how are not boys?
18. In a class of 108 people, 5/9 of them are girls, how many are not boys?
A mixed bag of candies has 72 pieces of colored candies, 1/8 of them are red, 1/3
of them are green, ½ of them are blue and the rest are yellow.
19. How many green ones are there?
20. How many are blue?
21. How many are not yellow?
20. How many are not blue and not green?
21. In a group of 108 people, 4/9 of them adults (aged 18 or over), 1/3 of them are
teens (aged from 12 to 17) and the rest are children. Of the adults 2/3 are females,
3/4 of the teens are males and 1/2 of the children are girls. Complete the following
table.
22. How many females are there and what is the fraction of the females to entire
group?
23. How many are not male–adults and what is the fraction of them to entire
group?

13 multiplication and division of fractions

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13 multiplication and division of fractions