**Hint:** For solving such a type of numerical, let the unknown values as variables such that the number of unknown is equal to the number of given conditions. Now make the mathematical equations according to the conditions and solve them.

**Complete step by step solution:**Let the first part of $4500$ be $a$ and second part be $b$

Then, $a + b = 4500$-$(1)$

Given, $5\% $ of $a = 10\% $ of $b$.

$ \dfrac{{5a}}{{100}} = \dfrac{{10b}}{{100}} $

$ a = 2b $

Substituting the value of a in equation number $(1)$

$ 2b + b = 4500 $

$ 3b = 4500 $

$ b = 1500 $

Again, substituting the value of b in equation $(1)$

$ a + 1500 = 4500 $

$ a = 4500 - 1500 $

$ a = 3000 $

**Hence the two parts of $4500$ is $3000$ and $1500$ such that $5\% $ of the first part is equal to the $10\% $ of the second part.**

**Note:** Keep in mind that for solving such types of questions , first try to formulate algebraic equations using given conditions and then solve the equations by either elimination or substitution method. We know that in order to solve for n unknown variables, we need n equations. So first try to find as many numbers of equations as there are variables.