Question

How many years will it take for ​$ to grow to ​$ if​ it's invested at percent compounded​ annually?

a) n = 10.3 years
b) n = 8.6 years
c) n = 12.5 years
d) n = 15.2 years

Answer 1

To calculate the number of years needed for an investment to grow given an annual compound interest rate, use the formula n = ln(M) / ln(b).

Step-by-step explanation:

To determine how many years it will take for an investment to grow to a certain amount with annual compounding interest, the formula involving natural logarithms (ln) can be used: n = ln(M) / ln(b), where M is the multiplication factor of the original amount and b is 1 plus the annual growth rate (expressed as a decimal).

For instance, if we want to know when an investment would triple, M would be 3. If the annual growth rate is 5%, then b would be 1.05. Applying these values, n = ln(3) / ln(1.05), which is approximately 22.5 years. The power of compound interest is a key factor in the growth of investments over time.

Even a modest annual rate of return can significantly increase the value of an investment given enough time. For example, an investment compounded annually at a 7% rate will nearly fifteen fold after 40 years. This illustrates the importance of starting to save and invest early to maximize the benefits of compound interest over time.