CASE II

Convert 0.45.. into a fraction in its simplest form.

Let r stand for the recurring decimal given i.e.

r=0.45..

Let the above equation be equation I

r=0.45.. I

Multiply both sides of equation by 100

100r=45.4545..

Let this be equation II

100r=45.4545;. II

Now subtract equation I from equation II

110r=45.4545.. II

-r= 0.4545.. I

99r=45

99r=45

99 99

r=5/11

but r=0.4545.

Therefore 0.4545. =5/11

NB: In both case I and case II the decimal point is

just before the recurring number or numbers. 

 In the case I only one number is recurring hence

 we multiply by 10.

 In case II two numbers re recurring and hence

& we multiply by 100.

 If three numbers just after the decimal pointare

 recurring we need to multiply by 1000.

 In general if n numbers just after the decimal point a

 re recurring we multiply by 10n.

Example: convert 0.123123. to a fraction in its

simplest form.

Let r stand for the recurring decimal given that

r=0.123123.

Let the above equation be equation I

r=0.123123. I

Multiply both sides of equation I by 1000 since three numbers

are recurring after the decimal point.

1000r=123.123123.

Let this equation be equation II.

1000r=123.123123. II

Now subtract equation I from equation II

1000r=123.123123..

- r= 0.123123..

999r=123

999r/999=123/999

r=123/999

But r=0.1231223.

Therefore 0.123123. = 123/999