Question

Write the following infinite decimal expansion as a fraction. You do not need to simplify
0.31\overline

Answer 1

Let x = 0.31\overline{{}}.

Multiply both sides by 100 to move the decimal point to the right two places:
100x = 31.\overline{{}}.

Subtract the original equation from this new equation to eliminate the repeating decimal:
100x - x = 31.\overline{{}} - 0.31\overline{{}}.
99x = 31

Divide both sides by 99:
99x/99 = 31/99
x = 31/99.

Therefore, the infinite decimal expansion 0.31\overline{{}} can be written as the fraction 31/99.