Question

convert the following repeating decimal to a fraction in simplest form.
.\overline{78}

Explanation:

Step1: Let \( x = 0.\overline{78} \)

Let \( x \) represent the repeating decimal. So \( x = 0.787878\ldots \)

Step2: Multiply by 100

Since the repeating cycle has 2 digits, multiply \( x \) by 100. So \( 100x = 78.787878\ldots \)

Step3: Subtract the two equations

Subtract \( x = 0.787878\ldots \) from \( 100x = 78.787878\ldots \). We get \( 100x - x = 78.7878\ldots - 0.7878\ldots \) which simplifies to \( 99x = 78 \)

Step4: Solve for \( x \)

Divide both sides by 99: \( x=\frac{78}{99} \)

Step5: Simplify the fraction

Find the greatest common divisor (GCD) of 78 and 99, which is 3. Divide numerator and denominator by 3: \( \frac{78\div3}{99\div3}=\frac{26}{33} \)

Answer:

\(\frac{26}{33}\)