To express $3.1648\overline{648}$ as a ratio of integers, we need to represent the repeating decimal as a fraction.

Step-by-Step Solution

1. Let $x = 3.1648\overline{648}$.

   This means: $x = 3.1648648648648\ldots$

2. Separate the non-repeating and repeating parts.

   The number $x$ can be decomposed as: $x = 3.1 + 0.0648\overline{648}$

3. Set up an equation for the repeating part.

   Let $y = 0.0648\overline{648}$.

4. Express $y$ as a fraction.

   Since $y = 0.0648\overline{648}$, we can rewrite $y$ with repeating decimals:

   Let: $y = 0.0648648648648\ldots$

   The repeating part has 3 digits ("648"), so we will multiply by $10^3 = 1000$ to shift the repeating part:

   $1000y = 64.8648648648\ldots$

   Now we have: $1000y = 64.8648648648\ldots$ $y = 0.0648648648\ldots$

Subtract