Prove algebraically that 0.5 recurring = 5/9

Answer and Step-by-step explanation:

We want to prove that 0.5555... = 5/9.

First, let's set 0.555... equal to x:

x = 0.555...

Now multiply this by 10:

10x = 5.555...

Now subtract the original from this new one:

   10x = 5.555...

-       x = 0.555...

     9x = 5

Note that we could cancel all the recurring terms because they were the same for both 5.555... and 0.555... since the 5's go up to infinity.

We now have 9x = 5, so divide both sides by 9:

x = 5/9, as desired