How do I express 0.6969... in the form of p/q where p and q are integers?

How do I express #0.dot6dot9# in the form of #p/q# where p and q are integers?

Your help is greatly appreciated!

3 Answers
Jul 22, 2018

I'm assuming it is 0.69 recurring

Explanation:

Let #x=0.6969...#

So #100x=69.6969...#

Subtract the first from the second and all the decimal numbers will disappear to leave

#99x=69#

Divide by 99

#x=69/99#

Divide by 3

#x=23/33#

Jul 22, 2018

#0.dot6dot9=23/33#

Explanation:

Let #0.dot6dot9=0.6969696969....................=x#

Then #100x=69.6969696969..............................#

Subtracting former from latter we get

#99x=69#

Hence, #x=69/99=23/33#

i.e. #0.dot6dot9=23/33#

#0.6969....=23/33#

Explanation:

Let

#x=0.69696969............\ \quad (1)#

#100x=69.69696969............\ \quad (2)#

subtracting (1) from (2) as follows

#100x-x=(69.696969...)-(0.696969....)#

#99x=69#

#x=69/99#

#x=23/33#