Calculation:
To compare the numbers \(\displaystyle{0.58}\overline{{3}}{\quad\text{and}\quad}\frac{6}{{11}},\)

first we right the fraction \(\displaystyle\frac{6}{{11}}\) as decimal

and compare with the decimal \(\displaystyle{0.58}\overline{{3}}.\)

\(\displaystyle{11}\frac{0.5454}{{6.00}}\)

\(\displaystyle-\frac{55}{{50}}\)

\(\displaystyle-\frac{45}{{50}}\)

\(\displaystyle-\frac{45}{{50}}\)

\(\displaystyle-\frac{45}{{5}}\)

Therefore, \(\displaystyle\frac{6}{{11}}={0.545454}{\quad\text{or}\quad}{0}.\overline{{54}}\)

Now, we compare this with given decimal \(\displaystyle{0.58}\overline{{3}}.\)

Original numbers \(\displaystyle{0.58}\overline{{3}}\frac{6}{{11}}\)

Decimals \(\displaystyle{0.58}\overline{{3}}{0}.\overline{{54}}\)

Compare \(\displaystyle{0.58}\overline{{3}}>{0}.\overline{{54}}\)

Thus,\(\displaystyle{0.58}\overline{{3}}>\frac{6}{{11}}.\)

Answer: \(\displaystyle{0.58}\overline{{3}}>\frac{6}{{11}}\)

first we right the fraction \(\displaystyle\frac{6}{{11}}\) as decimal

and compare with the decimal \(\displaystyle{0.58}\overline{{3}}.\)

\(\displaystyle{11}\frac{0.5454}{{6.00}}\)

\(\displaystyle-\frac{55}{{50}}\)

\(\displaystyle-\frac{45}{{50}}\)

\(\displaystyle-\frac{45}{{50}}\)

\(\displaystyle-\frac{45}{{5}}\)

Therefore, \(\displaystyle\frac{6}{{11}}={0.545454}{\quad\text{or}\quad}{0}.\overline{{54}}\)

Now, we compare this with given decimal \(\displaystyle{0.58}\overline{{3}}.\)

Original numbers \(\displaystyle{0.58}\overline{{3}}\frac{6}{{11}}\)

Decimals \(\displaystyle{0.58}\overline{{3}}{0}.\overline{{54}}\)

Compare \(\displaystyle{0.58}\overline{{3}}>{0}.\overline{{54}}\)

Thus,\(\displaystyle{0.58}\overline{{3}}>\frac{6}{{11}}.\)

Answer: \(\displaystyle{0.58}\overline{{3}}>\frac{6}{{11}}\)