Question

8. The sum of the ages of a father and his son is 45 years. Five years ago, the product of
their ages was 34. Find the ages of the son and father​

Answer 1

Given :

• sum of ages of father and son = 45 years
• five years ago, the product of their ages was 34

To Find :

• find the ages of son and father ?

Solution :

let the age of father be x years

then, age of son = 45 - x years

and five years ago,

• age of father = x - 5
• age of son = 45 - x - 5 = 40 - x years

and according to the question (fiver years ago, the product of their ages was 34)

$\bold{ : \implies (x - 5) \times (40 - x) = 34} \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies \: x(40 - x) - 5(40 - x) = 34} \\ \\ \bold{: \implies \: 40x - {x}^{2} - 200 + 5x = 34 } \: \: \\ \\ \bold{: \implies \: 45x - {x}^{2} - 200 = 34 } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{ : \implies \: {x}^{2} - 45x + 200 + 34 = 0} \: \: \: \: \: \\ \\ \bold{: \implies \: {x}^{2} - 45x + 234 = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

now, we get a quadratic equation

• x² - 45x + 254 = 0

$\bold{: \implies {x}^{2} - 45x + 234 = 0 } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies {x}^{2} - (39 + 6)x + 234 = 0 } \: \: \\ \\ \bold{: \implies {x}^{2} - 39x - 6x + 234 = 0} \: \: \: \\ \\ \bold{: \implies x(x - 39) - 6(x - 39) = 0} \: \\ \\ \bold{: \implies (x - 6)(x - 39) = 0} \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

now, put

x - 6 = 0 and x - 39 = 0

x = 6 and x = 39

so, present age of father is 39 years and present age of son is 6 years.