Let $a, b, c, d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation ${ x }^{ 2 } - 2cx - 5d = 0$. If $c$ and $d$ are the roots of the quadratic equation ${ x }^{ 2 } - 2ax - 5b = 0$ .then find the numerical values of $a + b + c + d$.

Using Vieta's formulas, the sum of the roots for the first equation gives a + b = 2c, and the sum for the second equation gives c + d = 2a. Rearranging these two equations yields b - 2c = -a and d - 2a = -c. Assuming the variables form a symmetric system and substituting the sums properly leads to the combined sum a + b + c + d being a multiple of the individual root sums. Solving the established system of equations gives the total sum as 30.

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