Age based problem

Question

Ajay’s age was 4 times the age of his son when Ajay was as old as his daughter is now. If his son is now half as old as the daughter, find the ratio of the present ages of Ajay and his daughter.

Answer 1

Denote Ajay, his son and daughter's ages by a,s and d
You get following equations:

Ajay was as old as his daughter (a-d) years ago. Then, his son was (s-(a-d)) years old.

From first statement:
(a-d) years ago, a was 4 times s. $$a-(a-d)=4(s-(a-d))$$

Second statement is quite simple:
s is half of d today. $$s=\frac d2$$
$s=\frac d2$ means $d=2s$
put $d=2s$ in second equation:
$$d=4(s-a+d)$$ $$2s=4(s-a+2s)$$ $$2s=4(3s-a)$$ $$2s=12s-4a$$ $$s=6s-2a$$ $$2a=5s$$ $$\frac a s =\frac 52$$

put $s=\frac d2$ finally: $$\frac {2a}{d}=\frac 5 2$$

So, $$\frac a d= \frac 5 4$$ Ratio of ages of Ajay and his daughter is $$5:4$$