Conversion Between Recurring Decimals and Fractions

Steps to Convert Recurring Decimals to Fractions

Step 1: Assign the recurring decimal to a variable (represented by letters) with unknown value, e.g., let \(x=0.\dot{3}\).

Step 2: If there are \(m\) nonrepeating decimal places followed by a repeating block of \(n\) digits, form \(10^{m+n}x\) and \(10^m x\). Here \(m=0\), \(n=1\).

\(10\):\(10x=3.\dot{3}\).

Step 3: Subtract \(10^m x\) from \(10^{m+n}x\) to cancel the identical recurring tails. The resulting equation has integer coefficients.

\(10x-x=3.\dot{3}-0.\dot{3}\)

Step 4: Solve for \(x\), the value of the recurring decimal expressed as \(\frac{p}{q}\).

\(\begin{align} & 9x=3 \\ & x=\frac{1}{3} \end{align}\)