To determine how long it will take for an investment to triple with compound interest, we can use the compound interest formula and logarithms.

Step-by-Step Solution

1. Set Up the Compound Interest Formula:

   The compound interest formula for an investment is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

   • $A$ is the final amount after time $t$,
   • $P$ is the initial investment (in this case, $M$),
   • $r$ is the annual interest rate (5% or 0.05 in decimal form),
   • $n$ is the number of compounding periods per year (3, since it's compounded thrice a year),
   • $t$ is the time in years.

2. Apply the Conditions Given in the Problem:

   Since we want the investment to triple, $A = 3P$.

   Plugging in the values, we get: $3M = M \left(1 + \frac{0.05}{3}\right)^{3t}$

3. Simplify the Equation:

   Divide both sides by $M$ (since $M \neq 0$): $3 = \left(1 + \frac{0.05}{3}\right)^{3t}$

   Calculate the base inside the parentheses: $1 + \frac{0.05}{3} = 1 + 0.0167 = 1.0167$ So the equation becomes: $3 = (1.0167)^{3t}$

4. Use Logarithms to Solve for $t$:

   Taking the natural logarithm on both sides: $\ln(3) = \ln((1.0167)^{3t})$

   Using the power rule of logarithms, $\ln(a^b) = b \ln(a)$: $\ln(3) = 3t \cdot \ln(1.0167)$

   Now, solve for $t$: $t = \frac{\ln(3)}{3 \cdot \ln(1.0167)}$

5. Calculate the Result:

   • $\ln(3) \approx 1.0986$
   • $\ln(1.0167) \approx 0.0166$

   Substituting these values: $t \approx \frac{1.0986}{3 \cdot 0.0166} = \frac{1.0986}{0.0498} \approx 22.06 \text{ years}$

Answer:

It will take approximately 22 years for the investment to triple under these conditions.

Would you like more details on any step, or do you have further questions?

Here are some additional related questions you might consider:

1. How does the number of compounding periods affect the time required for an investment to triple?
2. What would the time be if the interest rate were different, say 8% annually?
3. How would continuous compounding (instead of thrice per year) change the result?
4. How does this formula compare with a simple interest calculation?
5. What happens if we want to quadruple the investment instead of tripling it?

Tip: Logarithms are particularly useful in compound interest problems, as they help isolate the variable for time, making it easier to solve exponent-based equations.