# Find Rate Of Interest In Continuous Compound Interest Word Problems Worksheet

## 6 problems

Understanding the formula A = Pe^{rt} is crucial for solving continuous compound interest word problems. To find the rate of interest, rearrange the formula to solve for $$r$$. A find rate of interest in continuous compound interest word problems worksheet offers practice examples. These resources help students master determining the interest rate in continuous compound interest scenarios.

Algebra 2
Exponential Functions

## 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:
1. Identify Values: Identify the values for $A$, $P$, and $t$.
A = $\15,000$
P = $\5,000$
t = $7$ years
2. Use Formula: Use the formula $A = Pe^{rt}$.$\newline$ Substitute $A = 15,000$, $P = 5,000$, and $t = 7$.$\newline$ $15,000 = 5,000 \times e^{7r}$
3. Isolate $e^{7r}$: Divide both sides by $5{,}000$ to isolate $e^{7r}$.$\newline$ $\frac{15{,}000} {5{,}000} = e^{7r}$$\newline$ $3 = e^{7r}$
4. Take Natural Logarithm: Take the natural logarithm ($\ln$) of both sides to solve for $r$.$\newline$ $\ln(3) = \ln(e^{7r})$$\newline$ $\ln(3) = 7r$
5. Divide by $7$: Divide both sides by $7$ to solve for $r$. $\newline$$r = \ln(\frac{3} {7})$ $r \approx 0.15694$
6. 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 $15.7\%$.

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