The sum of two numbers is $25$ and their HCF and LCM are $3$ and $105$ respectively. The sum of the reciprocals of the two numbers is:

Let the two numbers be $a$ and $b$.
Given:

* Sum of the numbers: $a+b=25$
* Product of the numbers: $a\times b=\text{HCF}\times \text{LCM}=3\times 105=315$

We need to find the sum of their reciprocals:

$$\text{Sum of reciprocals}=\frac{1}{a}+\frac{1}{b}=\frac{a+b}{a\times b}$$

Substitute the values into the formula:

$$\frac{1}{a}+\frac{1}{b}=\frac{25}{315}$$

Simplifying the fraction by dividing both numerator and denominator by $5$:

$$\frac{25÷5}{315÷5}=\frac{5}{21}$$