At the two different rates of annual simple interest a certain quantity of Principal becomes double in `5` years and triple in `12` years. In what case the investments is profitable?

Explanation

Let the principal be `Rs x`
The rate of annual simple interest for the first case be `a%` and that for the second case be `b%`.
As per question , the internet for `1st` case `=Rs(2x-x)=Rs x`
and for all the `2nd` case `=Rs(3x-x)=Rs 2x`
`:.(x xxaxx5)/(100)=x` (for all `1st` case) `impliesa=(100xx x)/(x xx5)impliesa=20`
`:.` For the `1st` case, the rate of annual simple interest `=20%`
Again, `(x xxbxx12)/(100)=2x` (for the `2nd` case)
or, `b=(2x xx100)/(x xx12)impliesb=(50)/(3)=16(2)/(3)`
`:.` For the `2nd `case, the rate of annual simple interest `=16(2)/(3)%`
Now , since the rate of annual simple interest for the `1st ` case is better than that for the `2nd` case.
`:.` The investment is more profitable in the `1st` case.