Question

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

Answer 1

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:-
$$\begin{gathered} \Rightarrow 47y-y^2=550 \\ \Rightarrow y^2-47y+550=0 \end{gathered}$$
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
$\Rightarrow (y-22)(y-25)=0$
$\Rightarrow y=22ory=25$
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,
$x=47-22$
$\text{x}=\text{25}$

$\therefore$ 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:-
$$x={\displaystyle \frac{550}{y}}$$ 
Putting this value in equation 1 we get:-
${\displaystyle \frac{550}{y}}+y=47$
On simplifying the above equation,
${\displaystyle \frac{550+y^2}{y}}=47$
$\Rightarrow 550+y^2=47y$
$\Rightarrow y^2-47y+550=0$

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:-
$$\Rightarrow x={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$$
Now applying this formula for the above equation we get:-
$$\Rightarrow y={\displaystyle \frac{-\left(-47\right)\pm \sqrt{{\left(47\right)}^2-4\left(1\right)\left(550\right)}}{2\left(1\right)}}$$

$y={\displaystyle \frac{47\pm \sqrt{2209-2200}}{2}}$
On further simplifications, we get
$\Rightarrow y={\displaystyle \frac{47\pm \sqrt{9}}{2}}$
$y={\displaystyle \frac{47\pm 3}{2}}$

Solving it further we get:-
$$\Rightarrow y={\displaystyle \frac{47+3}{2}}\text{or }y={\displaystyle \frac{47-3}{2}}$$
On simplifying the above equation,
$\Rightarrow y={\displaystyle \frac{50}{2}}\text{or }y={\displaystyle \frac{44}{2}}$
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.