Converting recurring decimals to rational numbers

We have $x=0.\overline{72}=\mathrm{0.727272...}$ and $100x=\mathrm{72.727272...}$. Both have the same infinitely repeating decimal part $\mathrm{0.727272...}$

Since the decimal tails are identical, subtract the smaller from the larger: $100x-x=\mathrm{72.727272...}-\mathrm{0.727272...}$ $99x=72$

The correct next step is to subtract $x$ from $100x$, giving $99x=72$. Then $x=\frac{72}{99}=\frac{8}{11}$.

The goal is to eliminate the repeating decimal part by choosing the right operation after setting up $x=0.\overline{72}$ and $100x=72.\overline{72}$.

Subtracting $10x$ would require $10x=7.\overline{27}$, which doesn't have the same decimal tail as $100x=72.\overline{72}$. The recurring parts would not cancel. Eliminate.

Multiplying by 10 again gives $1000x=720.\overline{72}$, but this doesn't eliminate the recurring part — it just makes the numbers larger. Eliminate.

Adding $x$ to $100x$ gives $101x=72.\overline{72}+0.\overline{72}$, which still contains a recurring decimal. This doesn't simplify the problem. Eliminate.

Subtracting $x=0.\overline{72}$ from $100x=72.\overline{72}$ cancels the recurring decimal tails perfectly, giving $99x=72$. This is correct.