Question

convert the following repeating decimal to a fraction in simplest form..3\overline{1}

Explanation:

Step1: Let \( x = 0.3\overline{1} \)

Let \( x \) represent the repeating decimal. So, \( x = 0.3111\ldots \)

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

Multiply both sides of \( x = 0.3111\ldots \) by 10. We get \( 10x=3.111\ldots \)

Step3: Multiply \( x \) by 100 to move past the repeating part

Multiply both sides of \( x = 0.3111\ldots \) by 100. We get \( 100x = 31.111\ldots \)

Step4: Subtract the two equations

Subtract \( 10x = 3.111\ldots \) from \( 100x=31.111\ldots \). So, \( 100x - 10x=(31.111\ldots)-(3.111\ldots) \)
Simplify the left - hand side: \( 90x \)
Simplify the right - hand side: \( 28 \)
So we have the equation \( 90x = 28 \)

Step5: Solve for \( x \)

Divide both sides of the equation \( 90x = 28 \) by 90. \( x=\frac{28}{90} \)

Step6: Simplify the fraction

Simplify \( \frac{28}{90} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So, \( \frac{28\div2}{90\div2}=\frac{14}{45} \)

Answer:

\(\frac{14}{45}\)