4. (10 points) The sum of all simplest proper fractions with 100 as the denominator equals $\qquad$

Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
【Analysis】Let the simplest proper fraction with 100 as the denominator be $\frac{\mathrm{p}}{100}$, and $1 \leqslant p \leqslant 99$. Since $\frac{\mathrm{p}}{100}$ is the simplest fraction, $p$ and 100 cannot have a common factor greater than 1, meaning $p$ cannot have factors of 2 and 5. Then, we categorize and discuss: the sum of numbers less than 100 that have 2 as a factor (even numbers); the sum of numbers less than 100 that have 5 as a factor; the sum of numbers less than 100 that have 10 as a factor. This leads to the sum of numbers less than 100 that do not have 2 or 5 as factors, and further solves the problem.

【Solution】Solution: Let the simplest proper fraction with 100 as the denominator be $\frac{\mathrm{p}}{100}$, and $1 \leqslant p \leqslant 99$.
Since $\frac{\mathrm{p}}{100}$ is the simplest fraction, $p$ and 100 cannot have a common factor greater than 1, meaning $p$ cannot have factors of 2 and 5.
The sum of numbers less than 100 that have 2 as a factor (even numbers) is:
$$
2+4+6+\cdots+96+98=49 \times 50=2450 \text {. }
$$

The sum of numbers less than 100 that have 5 as a factor is:
$$
5+10+15+\cdots+90+95=\frac{5}{2} \times 19 \times 20=950 \text {. }
$$

The sum of numbers less than 100 that have 10 as a factor is:
$$
10+20+30+\cdots+90=\frac{10}{2} \times 9 \times 10=450 \text {. }
$$

The sum of numbers less than 100 that have 2 or 5 as a factor is:
$$
2450+950-450=2950 \text {. }
$$

Therefore, the sum of all simplest proper fractions with 100 as the denominator is:
$$
\begin{aligned}
& \left(\frac{1}{100}+\frac{2}{100}+\cdots+\frac{99}{100}\right)-\frac{2950}{100} \\
= & \frac{1}{100} \times \frac{99 \times 100}{2}-29.5 \\
= & 20 .
\end{aligned}
$$

The answer is: 20.