Turn 0.454545… into a fraction — without guessing

The question

Write the repeating decimal 0.454545… as a fraction in lowest terms, and justify each step.

What to notice first

The repeating block is two digits long, so multiplying by $10^2$ shifts the number by exactly one whole block. That is the entire idea: line the tails up so that subtracting cancels the infinite part. A block of length k wants $10^k$, which is why 10 would not work here.

Working it through

Name the number, so it can be manipulated like any other unknown:

The block '45' has length two, so multiply by 100. The digits after the point are now identical to the original's:

Subtract. Every digit after the decimal point cancels — this is the step that turns an infinite expression into a finite one:

So $99x=45$. Divide, then reduce by the common factor 9:

The answer

Divide 5 by 11 to check: it returns 0.454545…, which is the only verification this problem needs.

The mistake this one catches

Prefer to watch it? The lesson above walks through every line.