Theorem
Let 𝑥∈ℚ be a rational number.

Let the decimal expansion of 𝑥 be:

0⋅𝑎1𝑎2…𝑎𝑚𝑏˙1𝑏2…𝑏˙𝑛
where 𝑎𝑖:𝑖∈{1,2,…,𝑚} and 𝑏𝑗:𝑗∈{1,2,…,𝑛} be the digits in the base 10 expansion of 𝑥.

Then 𝑥 can be expressed as the following fraction:

𝑥=𝑎1𝑎2…𝑎𝑚𝑏1𝑏2…𝑏𝑛−𝑎1𝑎2…𝑎𝑚10𝑚×(10𝑛−1)


Proof
First we note that:

0⋅𝑏˙1𝑏2…𝑏˙𝑛×10𝑛 = 𝑏1𝑏2…𝑏𝑛⋅𝑏˙1𝑏2…𝑏˙𝑛
⇝ 0⋅𝑏˙1𝑏2…𝑏˙𝑛×10𝑛−0⋅𝑏˙1𝑏2…𝑏˙𝑛 = 𝑏1𝑏2…𝑏𝑛⋅𝑏˙1𝑏2…𝑏˙𝑛−0⋅𝑏˙1𝑏2…𝑏˙𝑛 subtracting 0⋅𝑏˙1𝑏2…𝑏˙𝑛 from both sides
⇝ 0⋅𝑏˙1𝑏2…𝑏˙𝑛×(10𝑛−1) = 𝑏1𝑏2…𝑏𝑛
⇝ 0⋅𝑏˙1𝑏2…𝑏˙𝑛 = 𝑏1𝑏2…𝑏𝑛(10𝑛−1)
⇝ 0⋅𝑎1𝑎2…𝑎𝑚𝑏˙1𝑏2…𝑏˙𝑛 = 𝑎1𝑎2…𝑎𝑚10𝑚+110𝑚×𝑏1𝑏2…𝑏𝑛(10𝑛−1) Definition of Decimal Expansion
= 𝑎1𝑎2…𝑎𝑚10𝑚×(10𝑛−1)(10𝑛−1)+110𝑚×𝑏1𝑏2…𝑏𝑛(10𝑛−1) multiplying top and bottom by 10𝑛−1
= 𝑎1𝑎2…𝑎𝑚000…000𝑛−12…𝑚10(10−1)+12…𝑛10(10−1)
= 𝑎1𝑎2…𝑎𝑚𝑏1𝑏2…𝑏𝑛−𝑎1𝑎2…𝑎𝑚10𝑚×(10𝑛−1)