Answer

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**Hint:**Here we will assume the ages of Vivek and Amit to be x and y respectively and make the equations with the help of given information and solve for the values of x and y to get the ages of both the boys.

**Complete step by step solution:**

Let the age of Vivek be x

Let the age of Amit be y

Then since it is given that the sum of their ages is 47

Therefore,

\[x + y = 47\]………………………………….(1)

Also, it is given that the product of their ages is 550

Therefore,

\[xy = 550\]…………………………………(2)

Now we will solve equations (1) and (2) in order to get the values of x and y

Therefore,

From equation (1) we get:-

\[x = 47 - y\;\]

Putting this value in equation 2 we get:-

\[(47 - y)y = 550\]

Solving it further we get:-

\[

\Rightarrow 47y - {y^2} = 550 \\

\Rightarrow {y^2} - 47y + 550 = 0 \\

\]

Now applying middle term split to solve the quadratic equation we get:-

\[\Rightarrow {y^2} - 22y - 25y + 550 = 0 \]

On simplifying the above quadratic equation,

$\Rightarrow y\left( {y - 22} \right) - 25\left( {y - 22} \right) = 0 $

On further simplifications, we get

$\Rightarrow (y - 22)(y - 25) = 0 $

$\Rightarrow y = 22\;or\;y = 25 $

Now since y is the age of Amit and he is younger

Therefore, $y=22$

Now putting this value in equation 1 we get:-

\[ x + 22 = 47 \]

On simplifying the above equation,

$ x = 47 - 22{\text{ }}\;{\text{ }} $

$ {\text{x}} = {\text{25}}$

**$\therefore$ The age of Vivek is 25 years and the age of Amit is 22 years.**

**Note:**

Another approach to solve the equations can be:-

From equation (2) we get:-

\[x = \dfrac{{550}}{y}\]

Putting this value in equation 1 we get:-

$\dfrac{{550}}{y} + y = 47 $

On simplifying the above equation,

$\dfrac{{550 + {y^2}}}{y} = 47 $

$\Rightarrow 550 + {y^2} = 47y $

$\Rightarrow {y^2} - 47y + 550 = 0 $

We got the equation in form of a quadratic eqaution. Now we will apply the quadratic formula to solve the above equation

For any equation of the form \[a{x^2} + bx + c = 0\]

The roots of the equation using quadratic formula are given by:-

\[\Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Now applying this formula for the above equation we get:-

\[\Rightarrow y = \dfrac{{ - \left( { - 47} \right) \pm \sqrt {{{\left( {47} \right)}^2} - 4\left( 1 \right)\left( {550} \right)} }}{{2\left( 1 \right)}} \]

$y = \dfrac{{47 \pm \sqrt {2209 - 2200} }}{2}$

On further simplifications, we get

$\Rightarrow y = \dfrac{{47 \pm \sqrt 9 }}{2}$

$y = \dfrac{{47 \pm 3}}{2} $

Solving it further we get:-

\[\Rightarrow y = \dfrac{{47 + 3}}{2}{\text{or }}y = \dfrac{{47 - 3}}{2} \]

On simplifying the above equation,

$\Rightarrow y = \dfrac{{50}}{2}{\text{or }}y = \dfrac{{44}}{2} $

On further simplifications, we get

$\Rightarrow y = 25{\text{or }}y = 22 $

Now since $y$ is the age of Amit and he is younger

Therefore, $y=22$

Now putting this value in equation 1 we get:-

\[\Rightarrow x + 22 = 47 \]

On simplifying the above equation,

$x=25$

Hence the age of Vivek is 25 years and the age of Amit is 22 years.

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