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The sum of age of a father and his son is \[45\] years. Five years ago, the product of their ages (in years) was \[124\]. Determine their present ages.

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Hint: Form linear equations in two variables based on the given data and then solve them to find the present ages of father and son.

We have some relations regarding the age of father and son. We have to find their exact present ages.
We will find their present ages by forming linear equations in two variables and then solving them.
Let’s assume that the present age of son is \[x\] years and the present age of father is \[y\] years.
We know that the sum of age of father and son is \[45\] years. Thus, we have \[x+y=45\]. \[...\left( 1 \right)\]
Five years ago, the son’s age would be \[x-5\] years and father’s age would be \[y-5\] years.
We know that five years ago, the product of son’s and father’s age was \[124\].
Thus, we have \[\left( x-5 \right)\left( y-5 \right)=124\]. \[...\left( 2 \right)\]
We now have two linear equations in two variables. We will solve them to get the present ages of father and son using elimination methods. In this method, we solve the system of linear equations by using the additional property of equality.
Rearranging the equation \[\left( 1 \right)\] to write variable \[x\] in terms of \[y\], we get \[x=45-y\].
Substituting the above equation in equation \[\left( 2 \right)\], we get \[\left( 45-y-5 \right)\left( y-5 \right)=124\].
Solving this equation, we have \[\left( 40-y \right)\left( y-5 \right)=124\].
\[\begin{align}
  & \Rightarrow 40y-200-{{y}^{2}}+5y=124 \\
 & \Rightarrow {{y}^{2}}-45y+324=0 \\
 & \Rightarrow {{y}^{2}}-9y-36y+324=0 \\
 & \Rightarrow y\left( y-9 \right)-36\left( y-9 \right)=0 \\
 & \Rightarrow \left( y-9 \right)\left( y-36 \right)=0 \\
 & \Rightarrow y=9,36 \\
\end{align}\]
If we substitute \[y=9\] in equation \[\left( 1 \right)\], we get \[x=36\]. But as \[x\] represents the age of the son, it can’t be greater than the age of the father. Thus, we can’t have \[y=9,x=36\].
If we substitute \[y=36\] in equation \[\left( 1 \right)\], we get \[x=9\].
Hence, we have the age of son as \[x=9\] years and the age of father as \[y=36\] years.
When we substitute these values in the given conditions, we observe that they satisfy the conditions.
Thus, we have the age of son as \[x=9\] years and the age of father as \[y=36\] years.

Note: We can also solve this question by using linear equations in one variable. We can assume the age of son as a variable\[x\]and then write the age of father in terms of age of son under the given conditions and solve the equations to get the present age of father and son.
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