Question

When the son will be as old as his father today, the sum of their ages then will be 126. When the father was as old as his son is today, the sum of their ages then was 38. Find their present ages.

Answer 1

Let son 's current age is $x$ and father 's current age is $y$.

Now,where does the $(y-x)$ term come from?

It says 'when the son will be as old as the father is today, which means that the son has to age a certain amount of years. How many years does he have to age?

The difference between his current age and his father 's current age.

For example, if the son was $10$ and the father was $40$, it would take the son $30$ years (or $y-x=40-10$) to reach his current age. In the meantime, the father would age the same amount ($y-x=40-10=30$ years).

[Father 's new age] + [Son 's new age] $=126$

$[y+(y-x)]+\left[(x+(y-x))\right]=126$

$\Rightarrow 2y-x+y=126$

$\therefore 3y-x=126$ ..........$\left(1\right)$

[Father 's previous age] + [Son 's previous age]$=38$

$[y-(y-x)]+[x-(y-x)]=38$

$\Rightarrow x+x-y+x=38$

$\Rightarrow 3x-y=38$

$\therefore y=3x-38$ .........$\left(2\right)$

Now sub eqn$\left(2\right)$ into equation$\left(1\right)$

$\Rightarrow 3(3x-38)-x=126$

$\Rightarrow 9x-114-x=126$

$\Rightarrow 8x=240$

$\therefore x=30$

Now $y=3x-38$

$\Rightarrow y=3\left(30\right)-38$

$\therefore y=52$

The son is $30$ years old, and the father is $52$ years old.