Try to find the decimal expansions of $\dfrac{10}{3}$ and $\dfrac{11}{12}$. What do you observe about the repetition of the digits after the decimal point?

Decimal expansion of $\dfrac{10}{3}$ :

By long division :

$\begin{array}{l} \phantom{3)\quad1}{3.333\ldots} \ 3\overline{\smash{\big)}\quad 10.000\ldots} \ \phantom{3)}\phantom{)1}\underline{-9} \ \phantom{3)\quad))}10 \ \phantom{3)\quad,}\underline{-9} \ \phantom{3)\quad1))}10 \ \phantom{3)\quad10}\underline{-9} \ \phantom{3)\quad))10}10 \ \phantom{3)\quad100}\underline{-9} \ \phantom{3)\quad)1000}1\ldots \end{array}$

⇒ $\dfrac{10}{3} = 3.3333\ldots = 3.\overline{3}$

The digit 3 repeats infinitely after the decimal point.

Decimal expansion of $\dfrac{11}{12}$ :

By long division:

$\begin{array}{l} \phantom{12\overline{)}\,1}0.9166\ldots \ 12\overline{\smash{\big)}\phantom{)}11.0000\ldots} \ \phantom{12\overline{}}\underline{-108} \ \phantom{12\overline{)}111}20\phantom{00\ldots} \ \phantom{12\overline{)}\,0}\underline{-12}\phantom{00\ldots} \ \phantom{12\overline{)}1111}80\phantom{0\ldots} \ \phantom{12\overline{)}\,00}\underline{-72}\phantom{0\ldots} \ \phantom{12\overline{)}11111}80\phantom{\ldots} \ \phantom{12\overline{)}\,000}\underline{-72}\phantom{\ldots} \ \phantom{12\overline{)}111111}8\ldots \end{array}$

⇒ $\dfrac{11}{12} = 0.91666\ldots = 0.91\overline{6}$

After the decimal point, "91" appears once and then the digit 6 repeats infinitely.

Observations :

1. Both decimal expansions are non-terminating but repeating.

2. In $\dfrac{10}{3}$, the repetition starts immediately after the decimal point (pure repeating decimal).

3. In $\dfrac{11}{12}$, there are some non-repeating digits before the repeating block begins (general/mixed repeating decimal).

This happens because the denominator 3 (in $\dfrac{10}{3}$) has only the prime factor 3, and the denominator 12 = 2² × 3 (in $\dfrac{11}{12}$) has both 2 and 3 as factors. The presence of a prime factor other than 2 and 5 in the denominator makes the decimal repeating.

Hence, $\dfrac{10}{3} = 3.\overline{3}$ and $\dfrac{11}{12} = 0.91\overline{6}$. Both have repeating digits after the decimal point.