4

–

7

2

–

3

a

–

1

ba

–

b

a

–

b

xa

–

b

a

–

b

da + bc

––

bd

c

–

d

a

–

b

31

–

40

15

–

40

16

–

40

5 ×3

–––

5 ×8

8 ×2

–––

8 ×5

3

–

8

2

–

5

3

–

8

2

–

5

a1+ a2

–––

b

a2

–

b

a1

–

b

120

–

200

30

–

50

9

–

15

6

–

10

3

–

5

a

–

b

2a

–

2b

fraction 205

ac

–

bd

c

–

d

a

–

b

2 ×4

–––

3 ×7

of boys does not alter the amount of pie each boy

receives: = . We have, in effect, “cancelled” a

common factor from the numerator and denominator

of the fraction. This principle also shows that it is pos-

sible to express the same fraction in an infinite number

of equivalent forms. For instance, ,,,, and

all represent the same fraction.

In continuing the model of sharing pies, we see that

if a1pies are shared among bboys, then another a2pies

are shared among the same boys, then in effect a1+ a2

pies were shared among those boys. This yields the rule:

Addition of Like Fractions: + =

thereby providing a method for adding fractions shar-

ing the same denominator. To add fractions with differ-

ent denominators, convert each fraction to forms that

share a

COMMON DENOMINATOR

and then add. For

instance, to sum and write + = +

= + to obtain the answer . This yields

the rule:

Addition of Unlike Fractions: + =

Subtraction is performed in a similar manner.

If, in sharing pies, one wished to double the

amount of pie each boy receives, one could simply dou-

ble the number of pies available. This suggests the rule:

Product Rule: x×=

This shows, along with the cancellation law, multiply-

ing a fraction by its denominator produces its numera-

tor as a result:

Denominator Product Rule: b×= = = a

The product rule also shows how to multiply for

fractions. For instance, to compute ×observe

that by the product rule. Multiplying

the numerator and denominator each by three gives

, which is . This process is summarized:

Multiplication of Fractions: ×=

The following division rule for fractions is proved in a

similar manner:

Division of Fractions:

(Multiply the numerator and denominator each by b

to obtain . Now multiply the numerator and

denominator each by dto yield .)

Types of Fractions

In writing a generic fraction a/b, it is often assumed

that aand bare each a

WHOLE NUMBER

. If this is

indeed the case, then the fraction is called a “common”

fraction (or sometimes a “simple” or a “vulgar” frac-

tion). Each common fraction represents a

RATIONAL

NUMBER

. Although there are infinitely many rational

numbers, surprisingly, they occupy absolutely no space

on the

NUMBER LINE

.

A common fraction with positive numerator and

denominator is called “proper” if its numerator is less

than its denominator, and “improper” otherwise. (Thus

a proper fraction represents a quantity less than one,

and an improper fraction a quantity greater than or

equal to one.) A “mixed number” is a number consist-

ing of an integer and a proper fraction. For example,

31/2is a mixed number.

A zero fraction is a fraction with numerator equal

to zero. If no pies are shared among bboys, then each

boy receives zero pie. Thus every zero fraction is equal

to zero: 0/b= 0.

An “undefined” fraction is a fraction with denomi-

nator equal to zero. Such a fraction is invalid, for it

cannot have any meaningful value. (If, for instance, 2/0

had value x, then multiplying through by the denomi-

nator yields the absurdity 2 = 0 ×x= 0. Also, dividing

ad

–

bc