Question

Express as a fraction 2.1666666.....

Answer 1

$\large\begin{array}{l} \textsf{Express as a fraction}\\\\ \mathsf{2,\!1666\ldots} \end{array}$


$\large\begin{array}{l} \mathsf{x=2.16666\ldots\qquad(i)}\\\\\\ \textsf{Multiply it by 10:}\\\\ \mathsf{10x=10\cdot 2.16666\ldots}\\\\ \mathsf{10x=21.6666\ldots\qquad(ii)}\\\\\\ \textsf{Now, subtract }\mathsf{(ii)-(i):} \end{array}$

$\large\begin{array}{l} \mathsf{10x-x=21.6666\ldots-2,1666\ldots}\\\\ \mathsf{10x-x=(21.6+0.0666\ldots)-(2.1+0.0666\ldots)}\\\\ \mathsf{10x-x=21.6-2.1+0.6666\ldots-0.6666\ldots}\\\\ \mathsf{10x-x=21.6-2.1}\\\\ \mathsf{9x=19.5} \end{array}$


$\large\begin{array}{l} \textsf{Multiply both sides by 2:}\\\\ \mathsf{2\cdot 9x=2\cdot 19.5}\\\\ \mathsf{18x=39}\\\\ \mathsf{x=\dfrac{39}{18}}\\\\ \mathsf{x=\dfrac{\diagup\!\!\!\! 3\cdot 13}{\diagup\!\!\!\! 3\cdot 6}}\qquad\quad\textsf{(simplifying...)}\\\\ \mathsf{x=\dfrac{13}{6}}\\\\\\ \therefore~~\boxed{\begin{array}{l} \mathsf{2.16666\ldots=\dfrac{13}{6}} \end{array}}\qquad\checkmark \end{array}$


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Tags: repeating decimal fraction rational number