**How Will This Worksheet on "Find Rate of Interest in Continuous Compound Interest Word Problems" Benefit Your Student's Learning?**

- Students improve their algebra and logarithm skills by working with exponential equations.
- This helps them understand how interest rates impact investments and loans, making them more financially savvy.
- They learn to break down complex problems into simpler ones, boosting their analytical skills.
- Encourages critical thinking to understand how different financial variables relate to each other.
- Prepares students for advanced math and finance courses in college.

**How to Find Rate of Interest in Continuous Compound Interest Word Problems?**

- Use the continuous compound interest formula \(A = Pe^{rt}\), where \(A\) is the final amount, \(P\) is the principal, \(r\) is the rate of interest, and \(t\) is the time.
- To find the rate \(r\), rearrange the formula to `r = \frac{\ln(\frac{A}{P})}{t}`. This step involves using logarithms to isolate \(r\).
- Substitute the known values for \(A\) (final amount), \(P\) (principal), and \(t\) (time) into the rearranged formula.
- Use a calculator to compute the natural logarithm and division, giving you the rate of interest \(r\).

## Solved Example

Q. Lucas invested $\$5,000$ in an account to save for a trip abroad. After $7$ years, his investment grew to $\$15,000$. What is the annual interest rate, compounded continuously, that Lucas's account earned$?$ $\newline$Use the formula $A = Pe^{rt}$, where $A$ is the balance (final amount), $P$ is the principal (starting amount), $e$ is the base of natural logarithms ($\approx 2.71828$), $r$ is the interest rate expressed as a decimal, and $t$ is the time in years.$\newline$Round your answer to the nearest two decimal places in percentage form.

**Solution:****Identify Values:** Identify the values for $A$, $P$, and $t$.

A = $\$15,000$

P = $\$5,000$

t = $7$ years**Use Formula:** Use the formula $A = Pe^{rt}$.$\newline$ Substitute $A = 15,000$, $P = 5,000$, and $t = 7$.$\newline$ $15,000 = 5,000 \times e^{7r}$**Isolate $e^{7r}$:** Divide both sides by $5{,}000$ to isolate $e^{7r}$.$\newline$ $\frac{15{,}000} {5{,}000} = e^{7r}$$\newline$ $3 = e^{7r}$**Take Natural Logarithm:** Take the natural logarithm ($\ln$) of both sides to solve for $r$.$\newline$ $\ln(3) = \ln(e^{7r})$$\newline$ $\ln(3) = 7r$**Divide by $7$:** Divide both sides by $7$ to solve for $r$. $\newline$$r = \ln(\frac{3} {7})$ $r \approx 0.15694$**Convert to Percentage:** Convert $r$ to a percentage by multiplying by $100$.$\newline$ $r \approx 0.15694\times 100$ $r \approx 15.7\%$So, the annual interest rate Lucas's account earned is approximately $15.7\%$.