Compound Interest Growth Calculation

The problem involves calculating the time required for an investment to grow to 25 times its initial value, given its growth rate over 10 years.

Understanding Compound Interest Growth

Let the initial principal amount be P. The formula for the future value (A) with compound interest is $A = P(1+r)^t$, where r is the annual interest rate and t is the number of years.

Step 1: Determine the Growth Factor over 10 Years

We are given that the amount increases to five times its initial value after 10 years. This means:

$5P = P(1+r)^{10}$

Simplifying this equation by dividing both sides by P, we get:

$5 = (1+r)^{10}$

This equation tells us the growth factor over 10 years is 5.

Step 2: Calculate Time for 25 Times Growth

We need to find the time (let's call it T) when the amount becomes twenty-five times its original value, i.e., $25P$.

$25P = P(1+r)^T$

Simplifying this gives:

$25 = (1+r)^T$

We know that $25 = 5^2$. Substitute the value of 5 from Step 1 into this equation:

$5^2 = ((1+r)^{10})^2$

Using the rule of exponents $(a^m)^n = a^{m \times n}$, we get:

$25 = (1+r)^{10 \times 2}$ $25 = (1+r)^{20}$

Comparing this result ($25 = (1+r)^{20}$) with the equation we need to solve ($25 = (1+r)^T$), we can directly see that:

$T = 20$

Conclusion

It will take 20 years for the initial amount to grow to twenty-five times its original value.