1 f3 multiplication and division of fractions

p
q
Multiplication and Division of Fractions
Fractions (or p/q) are numbers that measure parts of
whole items.
3
6
Back to 123a-Home

p
q
3
6
3
6
whole items. For example, of a pizza represents “3 out of
6 slices” as shown here.

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

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

p
q
3
the denominator.
numerator.
3
6
Example A. Joe took 2/3 of a dozen eggs,
how many eggs did Joe take?

p
q
3
the denominator.
numerator.
3
6
The fraction 2/3 means to divide
the dozen eggs into 3 piles,
so each pile has 12/3 = 4 eggs,

p
q
3
the denominator.
numerator.
3
6
Joe took two piles so he took
2×4 = 8 eggs.

p
q
3
the denominator.
numerator.
3
6
2×4 = 8 eggs. The process is
recorded as multiplication:
2
3
12 =*
2
3
12
*
1

p
q
3
the denominator.
numerator.
3
6
2
3
12 =*
2
3
12
*
1
Step 1. 12÷ 3 = 4
so each pile has 4 eggs.
4
1

p
q
3
the denominator.
numerator.
3
6
2
3
12 =*
2
3
12
*
1
Step 1. 12÷ 3 = 4
so each pile has 4 eggs.
Step 2. 2x4 = 8 eggs
so 2 piles has 8 eggs.
= 2* 4 = 8
4
1

a
b
The fractional portion of a whole number x is expressed as
or
b
a x
*
1 b .
ax
*
1

a
b
To simplify these, always divide x by bor
b
a x
*
1 b .
ax
*
1
or cancel the common factor of x and b first.

Example B. Multiply by cancelling first.
2
3
18a.
*
a
b
b
a x
*
1 b .
ax
*
1

2
3
18a.
*
do 18÷ 3 = 6 first
a
b
b
a x
*
1 b .
ax
*
1

2
3
18a.
6
*
do 18÷ 3 = 6 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6a.
6
* * do 18÷ 3 = 6 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
* * do 18÷ 3 = 6 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
* *
*
do 18÷ 3 = 6 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
*
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11 = 33
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
b
b
a x
*
1 b .
ax
*
1
a

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

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11 = 33
Example C. a. What is of $108?2
3
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11 = 33
3
* 108
2
3
The statement translates into
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11 = 33
3
* 108
2
3
36
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
(do 108/3 = 36 first)
b
b
a x
*
1 b .
ax
*
1
a

2
3
18 = 2 6 = 12a.
6
11
16
48b.
3
* *
* = 3 * 11 = 33
3
* 108 = 2 * 36 = 72 $.
2
3
36
do 18÷ 3 = 6 first
do 48÷ 16 = 3 first
(do 108/3 = 36 first)
So 2/3 of $108 is $72.
b
b
a x
*
1 b .
ax
*
1
a

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
So the fraction of the lemon drops is 20
48

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

1
3
* 48
16
Example C. What is ¼ of 10 lb flour?
1
4
* 48 = 12,
12
48
=
20/4
48/4
=
5
12.

1
3
* 48
16
1
4
* 10
1
4
* 48 = 12,
12
“ ¼ of 10" is
48
=
20/4
48/4
=
5
12.

1
3
* 48
16
1
4
* 10
1
4
* 48 = 12,
12
“ ¼ of 10" is
(Reduce 10 and 4 by dividing by 2)
48
=
20/4
48/4
=
5
12.
5
2

1
3
* 48
16
1
4
* 10
1
4
* 48 = 12,
12
“ ¼ of 10" is
(Reduce 10 and 4 by dividing by 2)
48
=
20/4
48/4
=
5
12.
1
2
5
*
5
2
= or=
1
2
1
2
lb of flour.
5
2

Multiplication of Two Fractions

c
d
=
a*c
b*d
a
b
*
In general, we multiply two fractions as shown,

c
d
=
a*c
b*d
a
b
*
but as in the above examples, the keys is to reduce the product
by canceling any common “top–and–bottom factor”,
as much as possible first, then multiply.

Example B. 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
*
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
*

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
±
±
±

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
xthe reciprocal of is 3,1
3

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

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

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

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

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

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

a
b
b
a
2
3
3
2
1
5
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
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
65d. ÷

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 F. 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"
Hence, all division problems may be treated as multiplication.

Remember to cancel first!

HW (More) Factor completely and write the answer using
the exponential notation
a. 360 b. 756
In room of 120 people, 1/4 are male children, 3/8 are
female adults. There are 83 adults. Complete the table.
Male Female
Adults
Children
1
3
8
ft 22ft
How many are there?

1 f3 multiplication and division of fractions

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