Question

convert the following repeating decimal to a fraction in simplest form.
.\\(\overline{05}\\)

Explanation:

Step1: Let \( x = 0.0\overline{5} \)

Let \( x \) represent the repeating decimal. So \( x = 0.05555\ldots \)

Step2: Multiply \( x \) by 10 to shift the non - repeating part

Multiply \( x \) by 10: \( 10x=0.5555\ldots \)

Step3: Multiply \( x \) by 100 to shift more

Multiply \( x \) by 100: \( 100x = 5.5555\ldots \)

Step4: Subtract the two equations

Subtract \( 10x \) from \( 100x \):
\( 100x-10x=(5.555\ldots)-(0.555\ldots) \)
\( 90x = 5 \)

Step5: Solve for \( x \)

Divide both sides by 90: \( x=\frac{5}{90} \)
Simplify the fraction by dividing numerator and denominator by 5: \( x = \frac{1}{18} \)

Answer:

\(\frac{1}{18}\)