3. (10 points) The simplest fraction $\frac{\mathrm{a}}{\mathrm{b}}$ satisfies $\frac{1}{5}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{1}{4}$, and $b$ does not exceed 19, then the product of the maximum possible value and the minimum possible value of $a+b$ is
253
【Analysis】Given the simplest fraction $\frac{\mathrm{a}}{\mathrm{b}}$ satisfies $\frac{1}{5}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{1}{4}$, we can first make the numerators the same to get $\frac{\mathrm{a}}{5 \mathrm{a}}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{\mathrm{a}}{4 \mathrm{a}}$. According to the comparison of fractions with the same numerator, we know that $4 a<b<5 a$. Since $b$ does not exceed 19, the maximum value of $a$ is 4, at which point $b$ can be as large as 19. When $a=2$, the minimum value of $b$ is 9. Therefore, the product of the maximum possible value of $a+b$ and the minimum possible value of $a+b$ is $(4+19) \times (2+9)=253$.
【Solution】From $\frac{1}{5}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{1}{4}$, we get: $\frac{\mathrm{a}}{5 \mathrm{a}}<\frac{\mathrm{a}}{\mathrm{b}}<\frac{\mathrm{a}}{4 \mathrm{a}}$,
Thus, $4 a<b<5 a$,
Since $b$ does not exceed 19, the maximum value of $a$ is 4, at which point $b$ can be as large as 19. When $a=2$, the minimum value of $b$ is 9,

Therefore, the product of the maximum possible value of $a+b$ and the minimum possible value of $a+b$ is: $(4+19) \times (2+9)=253$. Hence, the answer is: 253.