Find Time In Compound Interest Word Problems Worksheets [PDF]: Algebra 2 Math

How Will This Worksheet on "Find Time in Compound Interest Word Problems" Benefit Your Student's Learning?

Helps students figure out how long it will take to reach their money goals, like saving for a car or a house.
Shows how math can solve everyday money situations, such as saving money, taking out loans, or investing.
Encourages thinking about different ways interest can change to decide how best to save or invest money.
Makes it easier to see how numbers work in money situations, especially with how interest rates can make money grow over time.
Gives skills to handle personal money well, including knowing how interest rates affect money over the years.

Determine the principal amount $ P $, the desired final amount $ A $, the annual interest rate $ r $, and the number of times interest compounds per year $ n $.
Use the formula `t = \frac{\log\left(\frac{A}{P}\right)}{n \cdot \log\left(1 + \frac{r}{n}\right)}` to calculate the time $ t $ required to reach $ A $ from $ P $ with compound interest.
Plug in the known values into the formula step-by-step, ensuring to use of natural logarithms (logarithm base $ e $).
Understand the calculated time $ t $ in the context of the problem, indicating how long it will take for the investment or loan to grow to the desired amount $ A $.

Solved Example

Q. You invest

\$2,000

in a savings account that offers an annual interest rate of

5\%

, compounded annually. How long will it take for your investment to grow to

\$3,000

?

\newline

Use the formula

A = P\left(1 + \frac{r}{n}\right)^{nt}

, where

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest hundredth.

Solution:

Identify values: Identify the values of $P$ , $r$ , $n$ , and $A$ . $\newline$ $P = 2000$ $\newline$ $r = 0.05$ $\newline$ $n = 1$ $\newline$ $A = 3000$
Use formula and solve: Use the formula $A = P\left(1 + \frac{r}{n}\right)^{nt}$ and solve for $t$ . $\newline$ $3000 = 2000\left(1 + 0.05\right)^t$
Divide and simplify: Divide both sides by $2000$ . $\newline$ $\frac{3000}{2000} = (1.05)^t$ $1.5 = (1.05)^t$
Take natural logarithm: Take the natural logarithm (\ln) of both sides to solve for $t$ . $\newline$ $\ln(1.5) = \ln((1.05)^t)$ $\ln(1.5) = t \cdot \ln(1.05)$
Isolate and solve: Divide both sides by $\ln(1.05)$ to isolate $t$ . $\newline$ $t = \frac{\ln(1.5)}{\ln(1.05)}$
Calculate value: Calculate the value of $t$ . $\newline$ $t = \frac{\ln(1.5)}{\ln(1.05)}$ $t \approx \frac{0.405465}{0.04879}$ $t \approx 8.31$ It will take approximately 8.31 years for your investment to grow.