convert the following repeating decimal to a fraction in simplest form..6\\overline{5}

Answer:

$\frac{59}{90}$

Explanation:

Step1: Assign variable to decimal

Let $x = 0.6\overline{5}$, meaning $x = 0.6555...$

Step2: Multiply to shift non-repeating part

Multiply $x$ by 10: $10x = 6.555...$

Step3: Multiply to align repeating digits

Multiply $10x$ by 10: $100x = 65.555...$

Step4: Subtract to eliminate repeating decimals

Subtract $10x$ from $100x$: $$100x - 10x = 65.555... - 6.555...$$ $$90x = 59$$

Step5: Solve for x

$$x = \frac{59}{90}$$