Fractional form of 0.94 bar where bar is on 4

Question

What is the fractional form of 0.94 bar  where bar is on 4

 

Answer 1

0.94444...

let x=0.9444...

multiply both sides by 10 because only 1 digit is in the repetend

10x=9.4444...

   x=0.9444...

subtract the equations

9x=8.5

x=8.5/9

x=85/90

simplify

x=17/18

another approach...

multiply 0.9444... by 10 and divide by 10

(10*0.9444...)/10

(9.444...)/10

0.444...=4/9

( 9 4/9)/10

(85/9)/10

85/90

17/18

 

 

Answer 2

0.944444444... = 0.9 + 0.04 + 0.004 + 0.0004 + ...

 

0.04 + 0.004 + 0.0004 + ... is a geometric series with common ratio r = 0.1

 

Since -1<r<1, the series converges and has sum 0.04/(1-0.1) = 2/45

 

So, 0.944444... = 0.9 + 2/45 = 9/10+ 2/45 = 85/90 = 17/18