To determine how many years it will take for an investment to triple in value at an annual interest rate of 6.5% compounded yearly, we use the formula for compound interest:

$A = P(1 + r)^t$

Where:

• $A$ is the final amount (tripled value, $3P$),
• $P$ is the initial principal,
• $r$ is the annual interest rate (6.5% = 0.065),
• $t$ is the time in years.

Step 1: Substitute known values

$3P = P(1 + 0.065)^t$

Cancel out $P$ (since $P \neq 0$): $3 = (1.065)^t$

Step 2: Solve for $t$

Take the natural logarithm (ln) on both sides to handle the exponent: $\ln(3) = t \cdot \ln(1.065)$

Rearranging for $t$: $t = \frac{\ln(3)}{\ln(1.065)}$

Step 3: Calculate

$\ln(3) \approx 1.0986, \quad \ln(1.065) \approx 0.0631$

$t = \frac{1.0986}{0.0631} \approx 17.41 \, \text{years}$

Final Answer:

It will take approximately 17.41 years for the investment to triple in value at 6.5% annual interest compounded yearly.

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Would you like further clarification or a breakdown of the calculation?

Related Questions:

1. What is the formula for compound interest, and how is it applied in different cases?
2. How does the frequency of compounding affect the time required for an investment to triple?
3. How would the time change if the interest rate was 8% instead of 6.5%?
4. Can this calculation be performed using other bases for logarithms (e.g., base 10)?
5. What happens if the interest rate is compounded monthly instead of yearly?

Tip:

Always remember to convert percentages to decimals and ensure