Step 1: Recall the projection idea.
Rather than computing each cosine separately, use the cosine rule in the compact form $\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}$ and similarly for B and C, since a common denominator will appear.
Step 2: Write each divided term.
$\dfrac{\cos A}{a} = \dfrac{b^2 + c^2 - a^2}{2abc}$, $\dfrac{\cos B}{b} = \dfrac{c^2 + a^2 - b^2}{2abc}$, $\dfrac{\cos C}{c} = \dfrac{a^2 + b^2 - c^2}{2abc}$.
Step 3: Add them over the common denominator.
The sum is $\dfrac{(b^2 + c^2 - a^2) + (c^2 + a^2 - b^2) + (a^2 + b^2 - c^2)}{2abc}$.
Step 4: Simplify the numerator.
Each squared term appears once with $+$ and once with $-$ except that the totals leave $a^2 + b^2 + c^2$. So the numerator is $a^2 + b^2 + c^2$.
Step 5: Plug in the values.
With $a = 2, b = 3, c = 5$: numerator $= 4 + 9 + 25 = 38$, denominator $= 2(2)(3)(5) = 60$.
Step 6: Reduce the fraction.
$\dfrac{38}{60} = \dfrac{19}{30}$.
\[ \boxed{\dfrac{19}{30}} \]