Compound Interest Math Worksheet with Problems and a Festive Turtle Illustration

To solve these compound interest problems, we will use the compound interest formula:

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

Where:
- \( A \) is the amount of money accumulated after \( n \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.

The interest earned can be calculated as:

\[
\text{Interest} = A - P
\]

Let's solve each problem step by step.

---

Problem 1:

How much interest is earned on \$760 at 7% compounded semiannually for nine years?

#### Given:
- \( P = 760 \)
- \( r = 0.07 \)
- \( n = 2 \) (semiannually)
- \( t = 9 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 760 \left(1 + \frac{0.07}{2}\right)^{2 \cdot 9}
\]
\[
A = 760 \left(1 + 0.035\right)^{18}
\]
\[
A = 760 \left(1.035\right)^{18}
\]

Using a calculator:
\[
(1.035)^{18} \approx 1.8061112346
\]
\[
A \approx 760 \times 1.8061112346 \approx 1372.62
\]

#### Step 2: Calculate the interest earned
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 1372.62 - 760 \approx 612.62
\]

Answer:
\[
\boxed{612.62}
\]

---

Problem 2:

How much interest does a \$649 investment earn at 8% compounded semiannually over one year?

#### Given:
- \( P = 649 \)
- \( r = 0.08 \)
- \( n = 2 \)
- \( t = 1 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 649 \left(1 + \frac{0.08}{2}\right)^{2 \cdot 1}
\]
\[
A = 649 \left(1 + 0.04\right)^2
\]
\[
A = 649 \left(1.04\right)^2
\]

Using a calculator:
\[
(1.04)^2 = 1.0816
\]
\[
A \approx 649 \times 1.0816 \approx 703.86
\]

#### Step 2: Calculate the interest earned
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 703.86 - 649 \approx 54.86
\]

Answer:
\[
\boxed{54.86}
\]

---

Problem 3:

You take out a loan for \$494 at an interest rate of 7% compounded semiannually for eight years. What is the total amount that you will have at the end of the eight years?

#### Given:
- \( P = 494 \)
- \( r = 0.07 \)
- \( n = 2 \)
- \( t = 8 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 494 \left(1 + \frac{0.07}{2}\right)^{2 \cdot 8}
\]
\[
A = 494 \left(1 + 0.035\right)^{16}
\]
\[
A = 494 \left(1.035\right)^{16}
\]

Using a calculator:
\[
(1.035)^{16} \approx 1.748877828
\]
\[
A \approx 494 \times 1.748877828 \approx 864.50
\]

Answer:
\[
\boxed{864.50}
\]

---

Problem 4:

You take out a loan for \$438 at an interest rate of 6% compounded semiannually for seven years. What is the total amount that you will have at the end of the seven years?

#### Given:
- \( P = 438 \)
- \( r = 0.06 \)
- \( n = 2 \)
- \( t = 7 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 438 \left(1 + \frac{0.06}{2}\right)^{2 \cdot 7}
\]
\[
A = 438 \left(1 + 0.03\right)^{14}
\]
\[
A = 438 \left(1.03\right)^{14}
\]

Using a calculator:
\[
(1.03)^{14} \approx 1.538623955
\]
\[
A \approx 438 \times 1.538623955 \approx 674.00
\]

Answer:
\[
\boxed{674.00}
\]

---

Problem 5:

If you received \$116.63 on \$922 invested at a rate of 3% compounded semiannually, for how long did you invest the principal?

#### Given:
- \( A = 922 + 116.63 = 1038.63 \)
- \( P = 922 \)
- \( r = 0.03 \)
- \( n = 2 \)
- \( t = ? \)

#### Step 1: Use the compound interest formula to solve for \( t \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
1038.63 = 922 \left(1 + \frac{0.03}{2}\right)^{2t}
\]
\[
1038.63 = 922 \left(1 + 0.015\right)^{2t}
\]
\[
1038.63 = 922 \left(1.015\right)^{2t}
\]

#### Step 2: Isolate \( (1.015)^{2t} \)
\[
\frac{1038.63}{922} = (1.015)^{2t}
\]
\[
1.126507592 = (1.015)^{2t}
\]

#### Step 3: Take the natural logarithm of both sides
\[
\ln(1.126507592) = \ln((1.015)^{2t})
\]
\[
\ln(1.126507592) = 2t \cdot \ln(1.015)
\]

#### Step 4: Solve for \( t \)
\[
t = \frac{\ln(1.126507592)}{2 \cdot \ln(1.015)}
\]

Using a calculator:
\[
\ln(1.126507592) \approx 0.119391295
\]
\[
\ln(1.015) \approx 0.014925373
\]
\[
t = \frac{0.119391295}{2 \cdot 0.014925373} \approx \frac{0.119391295}{0.029850746} \approx 4
\]

Answer:
\[
\boxed{4}
\]

---

Problem 6:

If you put \$690 in a savings account that pays 8% compounded semiannually for one year, what is the amount of money you will have at the end of the one year?

#### Given:
- \( P = 690 \)
- \( r = 0.08 \)
- \( n = 2 \)
- \( t = 1 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 690 \left(1 + \frac{0.08}{2}\right)^{2 \cdot 1}
\]
\[
A = 690 \left(1 + 0.04\right)^2
\]
\[
A = 690 \left(1.04\right)^2
\]

Using a calculator:
\[
(1.04)^2 = 1.0816
\]
\[
A \approx 690 \times 1.0816 \approx 747.02
\]

Answer:
\[
\boxed{747.02}
\]

---

Problem 7:

If the balance at the end of eight years on an investment of \$429 that has been invested at a rate of 3% compounded semiannually is \$544.39, how much was the interest?

#### Given:
- \( P = 429 \)
- \( A = 544.39 \)

#### Step 1: Calculate the interest earned
\[
\text{Interest} = A - P
\]
\[
\text{Interest} = 544.39 - 429 \approx 115.39
\]

Answer:
\[
\boxed{115.39}
\]

---

Problem 8:

You take out a loan for \$907 at an interest rate of 9% compounded semiannually for nine years. What is the total amount that you will have at the end of the nine years?

#### Given:
- \( P = 907 \)
- \( r = 0.09 \)
- \( n = 2 \)
- \( t = 9 \)

#### Step 1: Calculate the total amount \( A \)
\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]
\[
A = 907 \left(1 + \frac{0.09}{2}\right)^{2 \cdot 9}
\]
\[
A = 907 \left(1 + 0.045\right)^{18}
\]
\[
A = 907 \left(1.045\right)^{18}
\]

Using a calculator:
\[
(1.045)^{18} \approx 2.229612997
\]
\[
A \approx 907 \times 2.229612997 \approx 2025.00
\]

Answer:
\[
\boxed{2025.00}
\]

---

Final Answers:

1. \(\boxed{612.62}\)
2. \(\boxed{54.86}\)
3. \(\boxed{864.50}\)
4. \(\boxed{674.00}\)
5. \(\boxed{4}\)
6. \(\boxed{747.02}\)
7. \(\boxed{115.39}\)
8. \(\boxed{2025.00}\)