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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.

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Last updated date: 05th Mar 2024
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IVSAT 2024
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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