Prove algebrically that o. o15 can be written as 1/66

Call x = 0.0151515... (i)

Multiply both sides by 100:

100x = 100 * 0.0151515...

100x = 1.5151515...

That can be expressed as this sum:

100x = 1.5 + 0.0151515... (ii)

Now, subtract (i) from (ii):

100x - x = 1.5 + 0.0151515... - 0.0151515..

99x = 1.5

Multiply both sides by 10, so you get rid of decimals:

10 * 99x = 10 * 1.5

990x = 15

Therefore,

x = 15/990

Now, reduce that fraction by dividing both numerator and denominator by gcd(15, 990) = 15, and you finally get

x = (15 : 15)/(990 : 15)

x = 1/66 < there it is.

0.0151515... = 1/66

I hope this helps. =)