Bunga Majemuk ~ Matematika Wajib Kelas XI

Introduction

This tutorial provides a step-by-step guide on compound interest, a critical concept in mathematics, particularly for students in grade XI. Understanding compound interest is essential for real-world applications, such as saving money, investment strategies, and loan calculations. This guide will explain how to derive the formula and solve related problems.

Step 1: Understanding Compound Interest

• Compound interest refers to the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods.
• This differs from simple interest, where interest is only calculated on the original principal.

Key Concepts

• Principal (P): The initial amount of money invested or borrowed.
• Rate (r): The annual interest rate (in decimal form).
• Time (t): The number of years the money is invested or borrowed.
• Compound Frequency (n): The number of times that interest is compounded per year.

Step 2: Deriving the Compound Interest Formula

The formula for compound interest can be derived as follows:

Compound Interest Formula

[ 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).
• ( r ) = annual nominal interest rate (as a decimal).
• ( n ) = number of times that interest is compounded per year.
• ( t ) = the time the money is invested or borrowed for, in years.

Example of Derivation

1. The interest is added to the principal after each compounding period.
2. For each period, the new principal becomes the previous amount plus interest.

Step 3: Solving Compound Interest Problems

When solving problems, follow these steps to apply the formula:

1. Identify the Variables:

   • Determine the values of ( P ), ( r ), ( n ), and ( t ) from the problem statement.

2. Convert Rate:

   • If the rate is given in percentage, convert it to decimal by dividing by 100.

3. Plug in the Values:

   • Substitute the identified values into the compound interest formula.

4. Calculate:

   • Use a calculator to perform the exponentiation and multiplication to find ( A ).

Example Problem

Suppose you invest $1,000 at an annual interest rate of 5% compounded quarterly for 3 years.

1. Identify Variables:

   • ( P = 1000 )
   • ( r = 0.05 )
   • ( n = 4 ) (quarterly)
   • ( t = 3 )

2. Plug in Values: [ A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} ]

3. Calculate:

   • The calculation becomes: [ A = 1000 \left(1 + 0.0125\right)^{12} ] [ A = 1000 (1.0125)^{12} ]
   • Use a calculator to find ( A ).

Conclusion

Understanding and applying the concept of compound interest is crucial for financial literacy. By following the steps outlined in this tutorial, you can confidently derive the compound interest formula and solve related problems. Practice with various problems to strengthen your understanding, and explore applications in real-life financial scenarios, such as savings accounts and investment plans.