Question

If X is 4 times as old as Y. 4 years ago, X was 10 times as old as Y. What will be the ratio of
their ages three 6 years hence?

Answer 1

10(x-4)=y-4
10x-y=36 this is our first eq.
after 6  years 
x+6=y+6
x-y=0 this is our second eq.
subtracting 1 and 2 eq we obtain
10x-y=36
-x+y=0
9x=36
x=1/4
putting the value of x in 2 eq.
1/4-y=0
taking the lcm 4
1+4y=0
-1/4=y
hence the ratio is 1:4

Answer 2

(1)=>  x:y=4:1  (difference=3)
4 years ago (2)=>  x:y=10:1  (difference=9)
ratio of differences= 3:9 =>1:3
so to equalise both ratios multiply them with the ratio of their differences
ratio(1)=> 3*[4:1]=12:3
ratio(2)=>1*[10:1]=10:1
now the differences=9 thus equalised
ratio(1)-ratio(2)=12-10=3-1=2
2 units=4 years
1 unit=2 years 

therefore present age=2*[ratio(1)]=2*[12:3]
i.e.,x=12*2=24 ,  y=3*2=6

6 years hence x:y=[24+6 : 6+6] =>30:12 => 5:2
answer=5:2