Question

okay so I don't really understand how to change a decimal to a fraction and the teacher never really explained it to us and she went really fast and like I hate showing my work because I'll always mess up

Answer 1

$\bf 0.2121\overline{21}\qquad \qquad \qquad x=0.2121\overline{21}\\\\ -------------------------------\\\\ \stackrel{\textit{moving the 21 to the left}}{\stackrel{100\cdot x}{100\cdot 0.2121\overline{21}}}\implies 21.2121\overline{21}\implies \begin{array}{llll} 21&+&0.2121\overline{21}\\ 21&+&x \end{array} \\\\\\ therefore\qquad \qquad 100x = 21+x\implies 99x=21 \\\\\\ x=\cfrac{21}{99}\implies x=\stackrel{simplified}{\cfrac{7}{33}}$