The given numbers are \(\bar{0.6}\ and\ \frac{5}{6}.\)

To compare the numbers \(\displaystyle\overline{{0.6}}{\quad\text{and}\quad}\frac{5}{{6}},\ \text{first we write the fraction}\ \frac{5}{6}\ \text{as decimal and compare with the decimal}\ \bar{0.6}.\)

\(\displaystyle{6}\frac{0.833}{{5.00}}\)

\(\displaystyle\frac{{-{48}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{2}}\)

Thus, \(\displaystyle\frac{5}{{6}}={0.833}\ldots{\quad\text{or}\quad}\overline{{0.83}}.\)

Now, we compare this with given decimal \(\bar{0.6}.\)

Original number \(\displaystyle\overline{{0.6}}\frac{5}{{6}}\)

Decimals\(\displaystyle\overline{{0.6}}\overline{{0.83}}\)

Compare \(\displaystyle\overline{{0.6}}<\overline{{0.83}}\)</span>

Thus, \(\displaystyle\overline{{0.6}}<\frac{5}{{6}}.\)</span>

Final statement:

\(\displaystyle\overline{{0.6}}<\frac{5}{{6}}.\)</span>

To compare the numbers \(\displaystyle\overline{{0.6}}{\quad\text{and}\quad}\frac{5}{{6}},\ \text{first we write the fraction}\ \frac{5}{6}\ \text{as decimal and compare with the decimal}\ \bar{0.6}.\)

\(\displaystyle{6}\frac{0.833}{{5.00}}\)

\(\displaystyle\frac{{-{48}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{20}}\)

\(\displaystyle\frac{{-{18}}}{{2}}\)

Thus, \(\displaystyle\frac{5}{{6}}={0.833}\ldots{\quad\text{or}\quad}\overline{{0.83}}.\)

Now, we compare this with given decimal \(\bar{0.6}.\)

Original number \(\displaystyle\overline{{0.6}}\frac{5}{{6}}\)

Decimals\(\displaystyle\overline{{0.6}}\overline{{0.83}}\)

Compare \(\displaystyle\overline{{0.6}}<\overline{{0.83}}\)</span>

Thus, \(\displaystyle\overline{{0.6}}<\frac{5}{{6}}.\)</span>

Final statement:

\(\displaystyle\overline{{0.6}}<\frac{5}{{6}}.\)</span>