The sum of the ages of Vivek and his youger brother Amit is 47 years. The product of their ages in years is 550. Find their ages.

Let the age of Vivek be x years.

Since sum of ages of vivek and amit is 47, so amit's age = (47 - x) years

Product of ages = 550

∴ x(47 - x) = 550

$\Rightarrow 47x - x^2 = 550 \\[1em] \Rightarrow x^2 - 47x + 550 = 0 \\[1em] \Rightarrow x^2 - 25x - 22x + 550 = 0 \\[1em] \Rightarrow x(x - 25) - 22(x - 25) = 0 \\[1em] \Rightarrow (x - 22)(x - 25) = 0 \\[1em] \Rightarrow x - 22 = 0 \text{ or } x - 25 = 0 \\[1em] x = 22 \text{ or } x = 25.$

Vivek's age = 25 years and Amit's age = 22 years.