Express 5.165 (recurring) as a fraction in its simplest form.

Assumption: The notation $5.1\dot{6}\dot{5}$ is interpreted as $5.\dot{1}\dot{6}\dot{5}$, meaning the digits $165$ repeat.

Step 1: Set up the equation for the repeating decimal. Let $m$ be the repeating decimal: $m = 5.165165165... \quad (1)$

Step 2: Multiply the equation to shift the repeating part to the left of the decimal. Since there are three repeating digits ($165$), multiply equation $(1)$ by $1000$: $1000m = 5165.165165... \quad (2)$

Step 3: Subtract the original equation from the multiplied equation. Subtract equation $(1)$ from equation $(2)$: $1000m - m = 5165.165165... - 5.165165165...$ $999m = 5160$

Step 4: Solve for $m$. Divide both sides by $999$: $m = \frac{5160}{999}$

Step 5: Simplify the fraction. Both the numerator ($5160$) and the denominator ($999$) are divisible by $3$. $5160 \div 3 = 1720$ $999 \div 3 = 333$ So, the simplified improper fraction is: $m = \frac{1720}{333}$

Step 6: Convert the improper fraction to a mixed number. Divide $1720$ by $333$: $1720 = 5 \times 333 + 55$ So, the mixed number is: $m = 5 \frac{55}{333}$

The final answer is $\boxed{5 \frac