Express 7.765765765 as a rational number, in the form p/q where p and q have no common factors.

Question

Express 7.765765765... as a rational number, in the form p/q where p and q have no common factors.

p = ?

q = ?

Answer 1

n=7.765765765...

3 digits repeat

multiply both sides by 10³ or 1000

1000n=7765.765765765...

subtract the first equation from the second equation

1000n=7765.765765765...

-

       n=      7.765765765...

999n=7758   (the decimal parts are eliminated)

     n=7758/999

     now simplify by dividing both terms by 9

     n=862/111

if the repetend has no lag, then put the repetend over 9, or 99, or 999, etc depending on how many digits repeat

7.765765765...=7 765/999=7758/999=862/111  (3 digits repeat thus the 999)

Answer 2

7.765765765=7+0.765+0.000 000765+...=

=7+765/1000+765/(1000²)+...

=7+765/1000(1+1/1000+1/1000^2+...) =

=|This infininte  geometric progression with q=1/1000|=

=7+765/1000(1/(1-1/1000)=

7+765/1000(1000/999)=7+765/999=7+85/111=

(777+85)/111=862/111.

 p=862, q=111 and GCD[862, 111]=1.