### Theory:

If a rational number $\frac{p}{q},\phantom{\rule{0.147em}{0ex}}q\ne 0$ can be expressed in the form of $\frac{p}{{2}^{m}×{5}^{n}}$, where $p\in \mathrm{ℤ}$ and $m,n\in W$, the rational number will have terminating decimal expansion. Otherwise, the rational number will have non-terminating and recurring decimal expansion.
Example:
Verify the following have terminating decimals or non-terminating and recurring decimal expansion.

1. $\frac{1}{20}$

$\frac{1}{20}=\frac{1}{4×5}=\frac{1}{{2}^{2}×{5}^{1}}$.

Thus, the given rational number follows the pattern $\frac{p}{{2}^{m}×{5}^{n}}$.

Therefore the number $\frac{1}{20}$ have terminating decimal expansion.

2. $\frac{-42}{625}$

$\frac{-42}{625}=\frac{-42}{{5}^{4}}=\frac{-42}{{2}^{0}×{5}^{4}}$.

Thus, the given rational number follows the pattern $\frac{p}{{2}^{m}×{5}^{n}}$.
Therefore the number$\frac{-42}{625}$ have terminating decimal expansion.

3. $\frac{19}{225}$

$\frac{19}{225}=\frac{19}{9×25}=\frac{19}{{3}^{2}×{5}^{2}}$.

Thus, the given rational number does not follow the pattern $\frac{p}{{2}^{m}×{5}^{n}}$.

Therefore the number $\frac{19}{225}$ does not have terminating decimal expansion. It has a non-terminating and recurring decimal expansion.

4. $\frac{-37}{350}$

$\frac{-37}{350}=\frac{-37}{2×25×7}$.

$\frac{-37}{350}=\frac{-37}{{2}^{1}×{5}^{2}×{7}^{1}}$.

Thus, the given rational number does not follow the pattern $\frac{p}{{2}^{m}×{5}^{n}}$.

Therefore the number $\frac{-37}{350}$ does not have terminating decimal expansion. But it has a non-terminating and recurring decimal expansion.