CASE III

Convert 2.8333.. to a fraction in the simplest form.

When there are non recurring digit(s) after the decimal point,

the decimal point is moved between the last non recurring

digit and the first recurring digit(s).

Let r stand for the recurring decimal

r=2.833333333...... I

Let this equation be equation I

Multiply by 10 to move the decimal point just before 3

10r=28.333333...

Let this be equation II

10r=28.333333....... II

Multiply equation II by ten since one digit is recurring.

100r=283.333.......

Let this be equation III

100r=283.333......... III

Subtract equation II from III

100r=283.333........ 

10r=28.333333.......... 

90r=255.0000...........

90r=255

r=255/90

r=2 75/90

r=2 5/6

But r=2.83

Therefore 2.83=2 5/6

b) Convert t3.45676767....... to a fraction in it simplest form.

Let r stand for 3.4567.......

r=3.456767.......

let this be equation I

r=3.456767……………I

Multiply equation I by 100 to move the decimal point to be

between the last non recurring digit and the first recurring digit.

100r=345.676767.......

Let this be equation II

100r=345.676767......... II

Multiply equation II by 100 since digits are recurring.

10000r=34567.676767.......

Let this be equation III

10000r=34567.676767......... III

Subtract equation II from equation III

10000r=34567.676767...... III

-100r=345.676767........... II

9900r=34222

r=34222/9900

r=3 4522/9900

r=3 2261/4950

But r=3.45676767...........

Therefore 3.45676767.........=3 2261/4950