Why does $(10^4 - 10^2) \cdot 0.0012121212\dots = 12$?

Question

When you answer this question $(10^4 - 10^2) \cdot 0.0012121212\dots$ you get $12$. However, that seems to defy PEMDAS. Please explain. Doing PEMDAS wouldn't you get $(10^4 - 10^2)$ = $10^2$ and then multiply that by $0.0012121212\dots$?

Answer 1

First, let $n=0.0012121212\dots$ so that $100n=0.12121212\dots$. Subtracting, $100n-n=99n=0.12=\frac{12}{100}$, so $n=\frac{12}{9900}$ (I'm intentionally not simplifying those fractions).

Now, as pointed out in the comments, $10^4-10^2=10000-100=9900$, so $$(10^4-10^2)(0.0012121212\dots)=9900\cdot\frac{12}{9900}=12.$$

Answer 2

I figure it might be easier to see like this.

$10^4\times0.001212...=12.1212..$

$10^2\times0.001212...=0.1212...$

Now subtract.

$(10^4\times0.001212...)-(10^2\times0.001212...)=12$

Using the distributive property, we can rewrite this as

$(10^4-10^2)\times0.001212...=12$