2 fractions multiplication and division of fractions

$: 2 fractions multiplication and division of fractions$
Fractions

Fractions are numbers of the form (or p/q) where

p, q 0 are whole numbers.

pq

Fractions


p, q 0 are whole numbers.

pq

Fractions

36


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

pq

Fractions

36


p, q 0 are whole numbers. Fractions are numbers that measure parts of whole items.Suppose a pizza is cut into 6 equal slices and we have 3 of

them, the fraction that represents this quantity is .

pq

36

Fractions

36




pq

36

36

Fractions




pq

36

The bottom number is the number of equal parts in the division and it is called the denominator.

36

Fractions




pq

36


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

36

Fractions




pq

36


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

36

Fractions

3/6 of a pizza

Whole numbers can be viewed as fractions with denominator 1.

Fractions


Thus 5 = and x = .

51

x1

Fractions


Thus 5 = and x = . The fraction = 0, where x 0.

51

x1

0x

Fractions



However, does not have any meaning, it is undefined.

51

x1

0x

x0

Fractions




51

x1

0x

x0

Fractions

The Ultimate No-No of Mathematics:




51

x1

0x

x0

Fractions

The Ultimate No-No of Mathematics:The denominator (bottom) of a fraction can't be 0.




51

x1

0x

x0

Fractions

The Ultimate No-No of Mathematics:The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)




51

x1

0x

x0

Fractions

The Ultimate No-No of Mathematics:The denominator (bottom) of a fraction can't be 0. (It's undefined if the denominator is 0.)Fractions that represents the same quantity are called equivalent fractions.




51

x1

0x

x0

Fractions


12

=24




51

x1

0x

x0

Fractions


12

=24

=36




51

x1

0x

x0

Fractions


… are equivalent fractions.12

=24

=36

=48




51

x1

0x

x0

Fractions


… are equivalent fractions.

The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction.

12

=24

=36

=48




51

x1

0x

x0

Fractions


… are equivalent fractions.

The fraction with the smallest denominator of all the equivalent fractions is called the reduced fraction.

12

=24

=36

=48

is the reduced one in the above list.12

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.

ab

ab = a / c

Fractions

b / c



that is, if the numerator and denominator are divided by the same quantity c, the result will be an equivalent fraction.In other words, a common factor of the numerator and the

denominator may be canceled as 1,

ab

ab = a / c

Fractions

b / c




denominator may be canceled as 1, i.e.

ab

ab = a / c

=a*cb*c

a*cb*c

1

Fractions

b / c





ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c





ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c

(Often we omit writing the 1’s after the cancellation.)





ab

ab = a / c

ab .=a*c

b*c =a*cb*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.






ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c

Example A: Reduce the fraction . 7854







ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

=







ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

= 78/254/2







ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

= 78/254/2


=39

27






ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

= 78/254/2


= 39/327/3

39

27






ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

= 78/254/2

= 139 .


= 39/327/3

39

27






ab

ab = a / c

ab .=a*c

b*c =a*cb*c

1

Fractions

b / c


7854

= 78/254/2

= 139


= 39/327/3

(or divide both by 6 in one step.)

39

27


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

FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.A participant in a sum or a difference is called a term.

FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression).

FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor.

FractionsOne common mistake in cancellation is to cancel a common number that is part of an addition (or subtraction) in the numerator or denominator.A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled.


2 + 12 + 3

35

=

A participant in a sum or a difference is called a term. The “2” in the expression “2 + 3” is a term (of the expression). The “2” is in the expression “2 * 3” is called a factor. Terms may not be cancelled. Only factors may be canceled.


2 + 12 + 3

35

=

This is addition. Can’t cancel!



2 + 12 + 3

= 2 + 1 2 + 3

35

=




2 + 12 + 3

= 2 + 1 2 + 3

= 13

35

=


!?



2 + 12 + 3

= 2 + 1 2 + 3

= 13

35

=


!? 2 * 12 * 3 = 1

3

Yes



2 + 12 + 3

= 2 + 1 2 + 3

= 13

35

=


!?

Improper Fractions and Mixed Numbers

2 * 12 * 3 = 1

3

Yes



2 + 12 + 3

= 2 + 1 2 + 3

= 13

35

=


!?

A fraction whose numerator is the same or more than its

denominator (e.g. ) is said to be improper .


3 2

2 * 12 * 3 = 1

3

Yes



2 + 12 + 3

= 2 + 1 2 + 3

= 13

35

=


!?

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 * 12 * 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.



23 4


23 4

= 5 + 5 3 4 .


Example B.


3 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.


Hence,

23 4


23 4

= 5 + 5 3 4 .

5 3 4

= 4*5 + 3 4


Example B.


3 4

=



Hence,

23 4


23 4

= 5 + 5 3 4 .

5 3 4

= 4*5 + 3 4

23 4=


Example B.


3 4

=



Hence,

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

Multiplication and Division of Fractions

3/6 of a pizza

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


Likewise the phrase “2/3 of $108” instructs us to divide $108 into 3 pilesso each pile consists of 108/3 = $36

3/6 of a pizza



Likewise the phrase “2/3 of $108” instructs us to divide $108 into 3 pilesso each pile consists of 108/3 = $36

3/6 of a pizza


$108 $36$36$3623

of

Divide into 3 piles


Likewise the phrase “2/3 of $108” instructs us to divide $108 into 3 piles so each pile consists of 108/3 = $36 and we take 2 piles or 2 x 36 = $72.

3/6 of a pizza


$108 $36$36$36 = $7223

of

Divide into 3 piles

Take 2 piles



This procedure is recorded as multiplication

3/6 of a pizza


* 10823

$108 $36$36$36 = $7223

of

Divide into 3 piles

Take 2 piles




3/6 of a pizza


* 10823

36

$108 $36$36$36 = $7223

of

Divide into 3 piles

Take 2 piles




3/6 of a pizza


* 10823

36= 2 * 36 = $72.

$108 $36$36$36 = $7223

of

Divide into 3 piles

Take 2 piles

ab d

as in the last example are important because they corresponds to the commonly used phrases: “(fractional amount) of ..”.

Fractional multiplications of the form

Multiplication and Division of Fractions* or a

bd *

ab d

as in the last example are important because they corresponds to the commonly used phrases: “(fractional amount) of ..”. To calculate these, always divide or cancel first, then multiply.


Multiplication and Division of Fractions* or a

bd *

ab d


Example C. Multiply by cancelling first.23 18a.


1116

48

b.

Multiplication and Division of Fractions*

*

*

or abd *

c. How many minutes are there in 2 35

hours?

*

ab d


Example C. Multiply by cancelling first.23 18a.


6

1116

48

b.


*

*

or abd *


hours?

*

ab d


Example C. Multiply by cancelling first.23 18 = 2 6 = 12a.


6

1116

48

b.


* *

*

or abd *


hours?

*

ab d




6

1116

48

b.3


* *

*

or abd *


hours?

*

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?

*

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?

There are 60 minutes in one hour, 352

*

* 60 min. so there are

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?


*

*

We split the mixed fraction to do the multiplication, i.e.352

60 min. so there are

* 60 = (2 + ) * 6035

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?


*

*



* 60 = (2 + ) * 6035 = 120

Distribute

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?


*

*



* 60 = (2 + ) * 6035 = 120 + 36

Distribute

12

ab d




6

1116

48

b.3


* *

* = 3 * 11 = 33

or abd *


hours?


*

*



* 60 = (2 + ) * 6035 = 120 + 36 = 156,

Distribute

12

35 hours is 156 minutes.so 2

Rule for Multiplication of FractionsTo 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*

Multiplication and Division of FractionsRule for Multiplication of FractionsTo 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 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

3a.


c

d=

a*c

b*d

a

b*



=15 * 12

8 * 25

12

25

15

8*

2

3

5

3a.


c

d=

a*c

b*d

a

b*



=15 * 12

8 * 25

12

25

15

8*

2

3

5

3

=3*3

2*5a.


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*5a.


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*11b.

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*11b.

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*11b.

8

9

7

8*

10

11

9

10** = =

7

11

a.

We can’t cancel like this for ± , i.e. cd = a c

b dab ± ±

±


think of the case that: 12 = 2

4

12 +

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 14

* 48




* 48 = 12,12




* 48 = 12,12

so there are 12 pieces of chocolate candies.


13

* 48



* 48 = 12,12

For caramel, 1/3 of 48 is



13

* 48 16



* 48 = 12,12

For caramel, 1/3 of 48 is = 16,



13

* 48 16



* 48 = 12,12


so there are 16 pieces of caramel candies.



13

* 48 16



* 48 = 12,12


so there are 16 pieces of caramel candies.The rest 48 – 12 – 16 = 20 are lemon drops.



13

* 48 16



* 48 = 12,12


so there are 16 pieces of caramel candies.The rest 48 – 12 – 16 = 20 are lemon drops. The fraction of

the lemon drops is 2048



13

* 48 16



* 48 = 12,12



the lemon drops is 2048 = 20/4

48/4



13

* 48 16



* 48 = 12,12




48/4 = 512



13

* 48 16

c. A class has x students, ¾ of them are girls, how many girls are there?



* 48 = 12,12




48/4 = 512



13

* 48 16

c. A class has x students, ¾ of them are girls, how many girls are there?

34 * x.



* 48 = 12,12




48/4 = 512

It translates into multiplication as


The reciprocal (multiplicative inverse) of is . ab

ba

Reciprocal and Division of Fractions


ba


So the reciprocal of is , 23

32


ba



32 the reciprocal of 5 is , 1

5


ba




5

the reciprocal of is 3, 13


ba




5

and the reciprocal of x is . 1xthe reciprocal of is 3, 1

3


ba


Two Important Facts About Reciprocals



5


3


ba


Two Important Facts About ReciprocalsI. The product of x with its reciprocal is 1.



5


3


ba





5


3

23

32* = 1,


ba





5


3

23

32* = 1, 5 1

5* = 1,


ba





5


3

23

32* = 1, 5 1

5* = 1, x 1x* = 1,


ba





5


3

23

32*

II. Dividing by x is the same as multiplying by its reciprocal .

= 1, 5 15* = 1, x 1

x* = 1,1x


ba





5


3

23

32*


= 1, 5 15* = 1, x 1

x* = 1,1x

For example, 10 ÷ 2 is the same as 10 , *12


ba





5


3

23

32*


= 1, 5 15* = 1, x 1

x* = 1,1x

For example, 10 ÷ 2 is the same as 10 , both yield 5. *12


ba





5


3

23

32*


= 1, 5 15* = 1, x 1

x* = 1,1x


Rule for Division of FractionsTo divide by a fraction x, restate it as multiplying by the

reciprocal 1/x , that is,


ba





5


3

23

32*


= 1, 5 15* = 1, x 1

x* = 1,1x



reciprocal 1/x , that is, dc

ab *

cd = a

b ÷

reciprocate


ba





5


3

23

32*


= 1, 5 15* = 1, x 1

x* = 1,1x



reciprocal 1/x , that is, dc = a*d

b*cab *

cd = a

b ÷

reciprocate

Example F. Divide the following fractions.

8

15 =

12

25a. ÷



15

8

12

25*

8

15 =

12

25a. ÷



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

10a. ÷



15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

÷

÷ =b.



15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

÷

÷ = * b.



15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

÷

÷ = * b.



15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

3

16

÷

÷ = * = b.



15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

3

16

÷

÷ = * = b.


16 = 5d. ÷


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

3

16

÷

÷ = * = b.


61 = 30 *

16 = 5d. ÷ 5


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

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?

61 = 30 *

16 = 5d. ÷ 5


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

3

16

÷

÷ = * = b.



In general, we convert mixes fractions into improper ones to do multiplication or division.

61 = 30 *

16 = 5d. ÷ 5


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

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?Using division, the number of cakes can be made is

12 ÷ 11 =


7 4

61 = 30 *

16 = 5d. ÷ 5


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

19

8 6

3

2

3

16

÷

÷ = * = b.



12 ÷ 11 = 15

254


7 4÷

61 = 30 *

16 = 5d. ÷ 5

Using division, the number of cakes can be made is


15

8 =

12

25*

8

15 =

12

25 5

3

2

3 9

10a.

698

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?

12 ÷ 11 = 15

254

3= 6 cakes.


7 4÷ 15

2 54 2

*=

61 = 30 *

16 = 5d. ÷ 5

Using division, the number of cakes can be made is


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.46 ,

812 ,

159 ,

2418 ,

3042 ,

5436 ,

6048 ,

72108

2 fractions multiplication and division of fractions

Education