The problem involves compound interest growth. We are given the time it takes for an investment to double and asked to find the time it takes to become a larger multiple.

Understanding Compound Interest Growth

Compound interest causes investments to grow exponentially. This means the growth rate remains constant relative to the current amount.

Calculating Time for 8 Times Growth

We are told the sum doubles (becomes 2 times) in 4 years.

• To become 4 times itself, the amount needs to double twice (2 x 2 = 4). This will take another 4 years. Total time = 4 years + 4 years = 8 years.
• To become 8 times itself, the amount needs to double three times (2 x 2 x 2 = 8). This requires doubling the 4 times amount, which adds another 4 years. Total time = 8 years + 4 years = 12 years.

Mathematical Approach

Let the principal amount be $P$. Let the time taken for the amount to become $k$ times be $T_k$. We are given that $T_2 = 4$ years.

The formula for compound interest implies that the time taken to multiply the principal by a factor $M$ is proportional to the logarithm of $M$. In simpler terms, if the amount doubles in $T$ years, it will become $M$ times in $T \times \log_2(M)$ years.

We want the amount to become 8 times itself. Since $8 = 2^3$, we can write:

Time for 8 times = Time for 2 times $\times$ 3

Time for 8 times = $4 \text{ years} \times 3 = 12 \text{ years}$

Therefore, the sum will become 8 times itself in 12 years.