If money is worth 12% per annum compounded monthly, how long will it take the principal P to become four times the original value?

To determine how long it will take for the principal \( P \) to quadruple in value with an interest rate of 12% per annum compounded monthly, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (the initial amount).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.

In this case:
- \( A = 4P \) (since we want the principal to become four times its original value).
- \( r = 0.12 \) (12% per annum).
- \( n = 12 \) (compounded monthly).
- \( t \) is what we need to find.

Substituting the values into the formula:

\[
4P = P \left(1 + \frac{0.12}{12}\right)^{12t}
\]

We can simplify the equation by dividing both sides by \( P \):

\[
4 = \left(1 + \frac{0.12}{12}\right)^{12t}
\]

Simplify the expression inside the parentheses:

\[
4 = \left(1 + 0.01\right)^{12t}
\]

\[
4 = \left(1.01\right)^{12t}
\]

To solve for \( t \), take the natural logarithm (ln) of both sides:

\[
\ln(4) = \ln\left((1.01)^{12t}\right)
\]

Using the logarithm power rule:

\[
\ln(4) = 12t \cdot \ln(1.01)
\]

Now, solve for \( t \):

\[
t = \frac{\ln(4)}{12 \cdot \ln(1.01)}
\]

Let's calculate the value:

\[
t = \frac{\ln(4)}{12 \cdot \ln(1.01)}
\]

Calculating:

\[
\ln(4) \approx 1.3863
\]
\[
\ln(1.01) \approx 0.00995
\]
\[
t = \frac{1.3863}{12 \cdot 0.00995} \approx \frac{1.3863}{0.1194} \approx 11.61
\]

So, it will take approximately **11.61 years** for the principal to become four times its original value.

Would you like more details on this calculation or have any other questions?

### Related Questions:
1. How would the time change if the interest were compounded quarterly instead of monthly?
2. What would be the effect of a different annual interest rate on the time required?
3. How does the compound interest formula differ from simple interest?
4. Can you calculate how long it would take for the principal to triple?
5. How does continuous compounding affect the time required for the principal to quadruple?
6. What is the effective annual rate for a nominal rate of 12% compounded monthly?
7. How does compounding frequency impact the growth of an investment?
8. How would inflation affect the real value of the investment over time?

**Tip:** When dealing with exponential growth, logarithms are very useful for solving equations involving time.

Learn how to calculate the time required for a principal amount to quadruple with compound interest at a rate of 12% per annum compounded monthly. Step-by-step solution using the compound interest formula and logarithmic functions. Suitable for high school students studying financial mathematics.

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