Step 1: Understand the compound interest relation

If a sum becomes $n$ times in $t$ years under compound interest, the amount after $N$ years is given by the formula $A = P(1 + r)^N$.

$A = P \left(1 + \frac{r}{100}\right)^t$

Step 2: Express the given condition

Given that the sum doubles in $12$ years, we can write $2P = P(1 + r)^{12}$, which simplifies to: $(1 + r)^{12} = 2$

$(1 + r)^{12} = 2$

Step 3: Express the target condition

We need the sum to become $8$ times itself. Let the required time be $T$ years. Thus, $8P = P(1 + r)^T$, which simplifies to: $(1 + r)^T = 8$

$(1 + r)^T = 8$

Step 4: Solve for $T$

Since $8 = 2^3$, substitute $(1 + r)^{12} = 2$ into the equation: $(1 + r)^T = (2)^3 = ((1 + r)^{12})^3 = (1 + r)^{36}$ Therefore, $T = 36$ years.

$T = 36$