Age of elder brother

The elder brother's age $>\sqrt{550}$ ,so it is at least 24.Since it is a divisor of 550,the only option can choose is 25.

In this problem the elder brother is Vivek so you should take his age always grater than Amit

Let Vivek's age be $x$

So, Amit's age is $(47 - x)$

Now the product of ages is 550

$x \times (47 - x) = 550$

$x^2 - 47x - 550 = 0$

$x^2 - 25x - 22x + 550 = 0$

$x(x - 25) - 22(x - 25) = 0$

$(x - 25) (x - 22) = 0$

$\implies x = 25~(or)~x = 22$

Here both the values of $x$ are positive .

But we should take the value of x as 25 because we should remember that Vivek is elder than Amit .

Therefore , the age of Vivek is 25 years .

Since the product of the two positive integers is $550$ , it is obvious that the unit digits of the two numbers are $2$ and $5$ . This is confirmed when their sum $47$ ends with $7 (=2+5)$ . Then the two tenth digits must add up to $4$ . It is obvious that both the tenth digits are $2$ . So that $22+25=47$ and $22\times 25 = 550$ . So the eldest is $\boxed{25}$ .

If $x$ and $y$ are the ages of Amit and Vivek, then

$\begin{cases} x + y & = 47~~~~ ... (1) \\xy & = 550 ~ ~~...(2)\\ \end{cases}$

From $(1):$

$x = 47 - y ... (3)$

Substituting $x = 47 - y$ in $(2):$

$\begin{aligned} (47 - y)y & = 550 \\47y - y^2 & = 550 \\47y - y^2 - 550 & = 0 \\ \end{aligned}$

$\implies y = 22, y = 25$ $~~~~~~~$ Quadratic equation

Since Vivek is older than Amit, we will take $y = \boxed{25}$