Final answer:
The repeating decimal $6.0\overline{89}$ is converted to a mixed number by setting the decimal to a variable, manipulating the equation to remove the repeating section, simplifying, and resulting in the mixed number 6 86/99.
Step-by-step explanation:
To write the repeating decimal $6.0\overline{89}$ as a mixed number, we need to first identify the non-repeating and repeating parts of the decimal. The digit '6' is the whole number part, and the repeating decimals are '89'. Let's set this decimal equal to a variable, say 'x':
x = 6.0\overline{89}
To isolate the repeating portion, we perform a small trick by multiplying 'x' by a power of 10 that matches the length of the repeating sequence. Since there are two repeating digits, we multiply by 10^2 (or 100) to shift the decimal point two places to the right:
100x = 608.\overline{89}
Subtracting the original 'x' from this equation gets rid of the repeating part:
100x - x = 608.\overline{89} - 6.0\overline{89}
99x = 602 and then divide by 99 to isolate x:
x = 602/99
Finally, we simplify the fraction, which remains a mixed number:
x = 6 + 86/99
Therefore, the mixed number equivalent of the repeating decimal $6.0\overline{89}$ is 6 \frac{86}{99}.