Question

A father is \[25\] years older than his son. After eight years the ratio of their ages will be $13:8$. Find their present age.

Answer 1

Hint: The relation between father’s and son’s ages is given. Using that we can express the ratio of their ages after eight years. Equating that with the given ratio and simplifying we get the answer.

Formula used: If $a:b = c:d$, then $\dfrac{a}{b} = \dfrac{c}{d}$.

Complete step-by-step answer:
Given that the father is \[25\] years older than his son and after eight years the ratio of their ages will be $13:8$.
Let the present age of the son be $x$.
Then the present age of the father will be $x + 25$.
After eight years the age of father and son will be $x + 25 + 8$ and $x + 8$ respectively.
So ratio of father and son ratio after eight years is $x + 25 + 8:x + 8 = x + 33:x + 8$.
This ratio is given as $13:8$.
Substituting we get,
$x + 33:x + 8 = 13:8$
We are asked to find their present ages.
If $a:b = c:d$, then $\dfrac{a}{b} = \dfrac{c}{d}$.
So we have,
$\Rightarrow$$\dfrac{{x + 33}}{{x + 8}} = \dfrac{{13}}{8}$
Cross multiplying we get,
$\Rightarrow$$(x + 33)8 = 13(x + 8)$
Simplifying we get,
$\Rightarrow$$8x + (33 \times 8) = 13x + (13 \times 8)$
Rearranging we get,
$\Rightarrow$$13x - 8x = 33 \times 8 - 13 \times 8$
$\Rightarrow$$5x = (33 - 13) \times 8$
Simplifying we get,
$\Rightarrow$$5x = 20 \times 8 = 160$
Dividing by $5$ we get,
$\Rightarrow$$x = \dfrac{{160}}{5} = 32$
This gives the present age of the son as $32$ years.
Also, $x + 25 = 32 + 25 = 57$
This gives the present age of the father as $57$ years.

$\therefore $Present ages of son and father are $32$ years and $57$ years.

Note: The difference between ages is constant for any two persons. So we can equate that with the ratio. Also since ratio can be expressed in fractions, we could find the answer. Using this we can find the ages of father and son before or after a certain period as well as their present ages.