Find Rate Of Interest In Continuous Compound Interest Word Problems Worksheets [PDF]: Algebra 2 Math

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.

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 $15.7\%$ .