The addition of a rational number and its reciprocal is $\frac{74}{35}$. Calculate the number.

Let the rational number be $x$.According to the question:
$$x+\frac{1}{x}=\frac{74}{35}$$
$$\frac{x^2+1}{x}=\frac{74}{35}$$
$$35x^2+35=74x$$
$$35x^2-74x+35=0$$
Splitting the middle term:
$$35x^2-49x-25x+35=0$$
$$7x(5x-7)-5(5x-7)=0$$
$$(7x-5)(5x-7)=0$$
Thus, $x=\frac{5}{7}$ or $x=\frac{7}{5}$.Therefore, the numbers are $\frac{7}{5}$ and $\frac{5}{7}$.