Compound Interest Growth Calculation

This question involves calculating the time required for an investment to grow to a specific multiple of its initial value under compound interest.

Understanding Compound Interest Growth

Compound interest means that the interest earned also earns interest. This leads to exponential growth. The key insight here is the relationship between the doubling time and the time it takes to reach higher multiples.

Problem Breakdown

• Let the principal amount be P.
• The sum doubles itself in 15 years. This means the amount becomes 2P in 15 years.
• We need to find the time (let's call it T) in which the amount becomes 8P.

Mathematical Formulation

The formula for compound interest is related to the growth factor over time. If the amount becomes 2P in 15 years, we can express this as:

Amount = Principal \times (1 + \text{rate})^{\text{time}}

So, $2P = P(1+r)^{15}$, which simplifies to $(1+r)^{15} = 2$.

We want to find the time T when the amount becomes $8P$:

$8P = P(1+r)^T$, which simplifies to $(1+r)^T = 8$.

Calculating the Time

We know that $8$ can be written as $2^3$. Using the information from the doubling time:

1. We have $(1+r)^{15} = 2$.
2. We need to find T such that $(1+r)^T = 8$.
3. Substitute $8 = 2^3$: $(1+r)^T = (2)^3$.
4. Substitute the value of $2$ from step 1: $(1+r)^T = \left((1+r)^{15}\right)^3$.
5. Using the exponent rule $(a^m)^n = a^{m \times n}$: $(1+r)^T = (1+r)^{15 \times 3}$.
6. Equating the exponents: $T = 15 \times 3$.
7. Therefore, $T = 45$ years.

Conclusion

The sum of money will become 8 times the original amount in 45 years.