Question

Is the following situation possible? If so, determine their present ages.

The sum of the ages of two friends is $20years$. Four years ago, the product of their ages in years was $48$.

Answer 1

Step 1: Write mathematical equations using given conditions.

Let us assume the age of one friend is $xyears$.

Given: The sum of the ages of two friends is $20years$.

Therefore, the age of second friend will be $(20-x)years$.

Given: Four years ago, the product of their ages in years was $48$.

So, we can write,

$$\begin{gathered} \left(x-4\right)\left(20-x-4\right)=48 \\ \Rightarrow 16x-x^2-64+4x=48 \\ \Rightarrow -x^2+20x-112=0 \\ \Rightarrow x^2-20x+112=0 \end{gathered}$$

Step 2: Check the solution of the quadratic equation is real or not.

$x^2-20x+112=0$

Comparing the equation with $a{x}^2+bx+c=0$, we get

$a=1,b=-20$ and $c=112$

The discriminant $\left(D\right)$ is given by $b^2-4ac$

$$\begin{gathered} D={\left(-20\right)}^2-4\left(1\right)\left(112\right) \\ \Rightarrow D=400-448 \\ \Rightarrow D=-48 \\ \Rightarrow D<0 \end{gathered}$$

The discriminant is less than zero.

Therefore, there will be no real solution possible for the equation.

Hence, the condition does not exist.