Master Compound Interest: Word Problems Worksheet & Answers
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Compound interest is an essential concept in finance that can significantly impact how we save and invest money. Understanding how to calculate compound interest is crucial for making informed decisions about personal finance. In this article, we will explore the concept of compound interest through various word problems, provide solutions to these problems, and highlight important notes to help deepen your understanding. 📈
What is Compound Interest? 🤔
Compound interest is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. In simpler terms, it means you earn interest on your interest! This can lead to exponential growth over time, making it a powerful tool for saving and investing.
The Compound Interest Formula
The formula to calculate compound interest is:
[ A = P \left(1 + \frac{r}{n}\right)^{nt} ]
Where:
- (A) = the amount of money accumulated after n years, including interest.
- (P) = the principal amount (the initial amount of money).
- (r) = the annual interest rate (decimal).
- (n) = the number of times that interest is compounded per year.
- (t) = the number of years the money is invested or borrowed.
Key Points to Remember 🔑
- Frequency of Compounding: The more frequently interest is compounded, the more total interest will be earned.
- Time Factor: The longer the money is invested, the more it will grow due to compound interest.
Word Problems on Compound Interest 📊
Let’s dive into some word problems involving compound interest. Each problem will be followed by a detailed explanation and solution.
Problem 1: Investment Growth
Question: Sarah invests $2,000 in an account that earns an annual interest rate of 5%, compounded annually. How much money will she have in the account after 10 years?
Solution: Using the compound interest formula:
- (P = 2000)
- (r = 0.05) (5% expressed as a decimal)
- (n = 1) (compounded annually)
- (t = 10)
Calculating:
[ A = 2000 \left(1 + \frac{0.05}{1}\right)^{1 \cdot 10} = 2000 \left(1 + 0.05\right)^{10} = 2000 \left(1.05\right)^{10} ]
Calculating (1.05^{10} \approx 1.62889):
[ A \approx 2000 \cdot 1.62889 \approx 3257.78 ]
Conclusion: After 10 years, Sarah will have approximately $3,257.78 in her account. 💵
Problem 2: Loan Amount
Question: John borrows $5,000 at an annual interest rate of 6% compounded quarterly. How much will he owe after 3 years?
Solution: Using the compound interest formula:
- (P = 5000)
- (r = 0.06)
- (n = 4) (compounded quarterly)
- (t = 3)
Calculating:
[ A = 5000 \left(1 + \frac{0.06}{4}\right)^{4 \cdot 3} = 5000 \left(1 + 0.015\right)^{12} ]
Calculating (1.015^{12} \approx 1.195618):
[ A \approx 5000 \cdot 1.195618 \approx 5978.09 ]
Conclusion: John will owe approximately $5,978.09 after 3 years. 💳
Problem 3: Monthly Savings
Question: Emily wants to save $100 each month in an account that earns an annual interest rate of 4% compounded monthly. How much will she have saved after 5 years?
Solution: In this case, we use the future value of an annuity formula because Emily is making monthly deposits.
[ FV = P \frac{(1 + r/n)^{nt} - 1}{(r/n)} ]
Where:
- (P = 100)
- (r = 0.04)
- (n = 12) (monthly compounding)
- (t = 5)
Calculating:
[ FV = 100 \frac{(1 + \frac{0.04}{12})^{12 \cdot 5} - 1}{(0.04/12)} ]
Calculating (1 + \frac{0.04}{12} = 1.00333333):
[ FV \approx 100 \frac{(1.00333333)^{60} - 1}{0.00333333} ]
Calculating ((1.00333333)^{60} \approx 1.221386):
[ FV \approx 100 \cdot \frac{1.221386 - 1}{0.00333333} \approx 100 \cdot 66.416 ]
Conclusion: Emily will have saved approximately $6,641.60 after 5 years. 💰
Practice Problems
To further test your understanding, here are a few additional problems for you to solve:
- Problem 4: Mark invests $1,500 at an interest rate of 8% compounded annually. How much will he have after 7 years?
- Problem 5: A loan of $10,000 has an interest rate of 5% compounded semi-annually. What will be the total owed after 4 years?
- Problem 6: Laura saves $200 every month for 3 years in an account that earns 3% interest compounded monthly. What will be her total savings?
Answers to Practice Problems
| Problem | Answer |
|---|---|
| 4 | Approximately $2,925.20 |
| 5 | Approximately $12,283.47 |
| 6 | Approximately $7,693.46 |
Important Notes 📝
- Practice is Key: The more you practice calculating compound interest, the more comfortable you'll become with the concept.
- Different Scenarios: Understand that there are different scenarios of compound interest (savings, loans, etc.), and each may require different formulas.
- Real-World Applications: Recognizing how compound interest works can help you make better financial decisions, whether you're saving for retirement, paying off debt, or investing.
Mastering compound interest through word problems not only enhances your mathematical skills but also empowers you with the knowledge needed for effective financial planning. Remember, understanding how your money can grow over time is a crucial aspect of achieving your financial goals! 🌟