Find Time In Compound Interest Word Problems Worksheet

6 problems

Finding time in compound interest word problems determines how long it takes for an investment to grow to a specified amount using compound interest. It uses t = \frac{\log\left(\frac{A}{P}\right)}{n \cdot \log\left(1 + \frac{r}{n}\right)}, where $$t$$ is time, $$A$$ is the final amount, $$P$$ is the principal, $$r$$ is the annual interest rate, and $$n$$ is the compounding frequency. Practice with find time in compound interest word problems worksheet for clarity.

Algebra 2
Exponential Functions

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.

How to Find Time in Compound Interest Word Problems?

• 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$$?$ $\newline$Use the formula $A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $A$ is the balance (final amount), $P$ is the principal (starting amount), $r$ is the interest rate expressed as a decimal, $n$ is the number of times per year that the interest is compounded, and $t$ is the time in years. $\newline$Round your answer to the nearest hundredth.
Solution:
1. Identify values: Identify the values of $P$, $r$, $n$, and $A$.$\newline$ $P = 2000$$\newline$ $r = 0.05$$\newline$ $n = 1$$\newline$ $A = 3000$
2. Use formula and solve: Use the formula $A = P\left(1 + \frac{r}{n}\right)^{nt}$ and solve for $t$.$\newline$$3000 = 2000\left(1 + 0.05\right)^t$
3. Divide and simplify: Divide both sides by $2000$.$\newline$ $\frac{3000}{2000} = (1.05)^t$ $1.5 = (1.05)^t$
4. Take natural logarithm: Take the natural logarithm (\ln) of both sides to solve for $t$. $\newline$$\ln(1.5) = \ln((1.05)^t)$ $\ln(1.5) = t \cdot \ln(1.05)$
5. Isolate and solve: Divide both sides by $\ln(1.05)$ to isolate $t$.$\newline$ $t = \frac{\ln(1.5)}{\ln(1.05)}$
6. 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.

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