Question 1199556
<br>
let x = Vivek's current age
let y = Arvind's current age<br>
The sum of their ages is x+y; the difference of their ages is x-y.<br>
The ratio of the sum of their ages and the difference of their ages is 2:1<br>
{{{(x+y)/(x-y)=2/1}}}
{{{2x-2y=x+y}}}
{{{x=3y}}}  [1]<br>
Vivek's age 4 years ago was x-4; Arvind's age 4 years ago was y-4.
The sum of their ages 4 years ago was x+y-8; the difference of their ages 4 years ago was (still!) x-y.
Four years ago, the ratio of the sum of their ages and the difference of their ages was 3:2<br>
{{{(x+y-8)/(x-y)=3/2}}}
{{{2x+2y-16=3x-3y}}}
{{{5y-16=x}}}  [2]<br>
[1] and [2] show two different expressions in terms of y for variable x.  Set those two expressions equal to each other and solve for y.<br>
{{{3y=5y-16}}}
{{{2y=16}}}
{{{y=8}}}  [3]<br>
Use [1] and [3] to find x.
{{{x=3(8)=24}}}<br>
Their current ages are Vivek = x = 24 and Arvind = y = 8.<br>
Their ages 12 years from now will be Vivek = 24+12 = 36 and Arvind = 8+12 = 20.<br>
The ratio of their ages 12 years from now will be 36:20 = 9:5<br>
ANSWER: 9:5<br>