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.

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