The ages of two friends Ani and Biju differ by 3 years. Anis father Dharam is twice as old as Ani and Biju as twice as old as his sister Cathy. The ages of Cathy and Dharam differ by 30 years. Find the ages of Ani and Biju.

Explanation

For easiness, we let the ages of all persons according to their names.
According to the problem,
A - B = `pm` 3 …(1) (don't know, who is elder)
Also, D = 2A
and `{:("B = 2C " implies " "C = (B)/(2)),():}}{:("All will be converted in terms"),("of A and B as we want only"),("two variables A and B"):}`
Also, D - C = 30 (obviously `D gt C`)
But if you don't know, who is elder, then you can write
`D - C = pm 30`
implies `2A - (B)/(2) = pm 30`
implies `4A - B = pm 60`
Now, from equations (1) and (2), we have four possibilities
`{:(" A - B = 3"),("4A - B = 60 "),(ul"- + -"),(" 3A = 57"),(therefore " A = 19"),(therefore " B = 19 - 3"),(" = 16"):}:|{:(" A - B = 3"),(" 4A - B = - 60"),(" "ul " - + +"),(" 3A = - 63"),(therefore " A = - 21"),("which is not possible as "),("age cannot be negative"):}:|{:(" A - B = - 3"),("4A - B = 60"),(ul"- + -"),(" 3A = 63"),(therefore " A = 21"),(therefore " B = 21 + 3"),(" = 24"):}:|{:(" A - B = - 3"),("4A - B = - 60"),(ul" - + +"),(" 3A = - 57"),(therefore " A = - 19"),("which is not possible as"),("age cannot be negative"):}`
Hence, `{:("Ani's age = 19 years and Biju's age = 16 years"),("Ani's age = 21 years and Biju's age = 24 years "):}}`