Write the number 0.21212121.. as a fraction let x=

There are 2 repeating decimals, so you will need a 1 followed by 2 zeros which is 100. You'll use 100 soon to multiply an equation.

Let's call the decimal number x.

Write   

        x =     0.212121...

Then multiply the equation above by 100, and write it above that equation.
You end up with

    100x = 21.212121...
          x =   0.212121...

Now set it up as a subtraction, and subtract the second equation from the first equation:

    100x = 21.212121...
          x =   0.212121...
- ----------------------------
      99x = 21

Now solve for x. Divide both sides by 3:

33x = 7

Divide both sides by 33:

x = 7/33

Since x = 0.212121..., an d x = 7/33, that means 0.212121... must equal 7/33.

Answer: the fraction is 7/33