Calculating Compound Interest: A Step-by-Step Guide

When dealing with financial matters, understanding and calculating compound interest is crucial. In this article, we will explore the process of determining the compound interest obtained after a given period using a specific example. We will use a formula and step-by-step calculations to derive the compound interest obtained after 12 years from a sum that yields Rs 5000 after 8 years at a 20% interest rate.

Introduction to Compound Interest

Compound interest is the interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit or loan. This is different from simple interest, which is calculated only on the principal amount. The formula for compound interest is as follows:

A P(1 r/n)^(nt)

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 Example: Compound Interest over 8 and 12 Years

Let's consider the example provided: a certain sum of money yields Rs 5000 as compound interest after 8 years at a 20% annual interest rate. We need to find the compound interest after 12 years.

Step 1: Identify the Given Values

Compound Interest (CI8) after 8 years Rs 5000 Annual Rate of Interest (r) 20% or 0.20 Time (t8) 8 years We need to find the CI after 12 years (t12)

Step 2: Use the Compound Interest Formula

The formula for compound interest can be derived from the formula for the amount (A):

C.I. A - P

Where:

A is the amount after n years P is the principal amount C.I. is the compound interest

Step 3: Calculate the Principal Amount (P)

First, we express the compound interest formula for 8 years:

CI8 P(1 r)n - P

Substituting the given values:

5000 P(1.208) - P

Using the value of 1.208:

1.208 ≈ 4.2998

Therefore:

5000 P(4.2998 - 1)

Which simplifies to:

5000 P(3.2998)

Solving for P:

P 5000 / 3.2998 ≈ 1512.89

Step 4: Calculate Compound Interest for 12 Years

Now that we have the principal amount, we can calculate the compound interest for 12 years:

C.I.12 P(1 r)t12 - P

Substituting the values:

C.I.12 1512.89(1.2012) - 1512.89

Using the value of 1.2012:

1.2012 ≈ 8.9161

Solving for C.I.12:

C.I.12 1512.89(8.9161) - 1512.89

This simplifies to:

C.I.12 ≈ 1512.89 * 7.9161 ≈ 11957.64

Thus, the compound interest obtained after 12 years is approximately Rs 11957.64.

Additional Calculations for Verification

Let's verify these calculations with an alternative approach:

Using the Principal and Interest Ratios

If we consider the values provided in the alternative solution:

C.I. P(1.208 - 1)

P can be derived by:

P 5000 / (1.208 - 1) ≈ 1515.24

Calculating C.I. for 12 Years

C.I.12 1515.24(1.2012) - 1515.24

Using 1.2012 ≈ 8.9161:

C.I.12 1515.24(8.9161 - 1) ≈ 11994.76

This method arrives at a similar result, confirming our initial calculations.

Conclusion

In this article, we've explored the process of calculating compound interest step by step, using a given example. By understanding and applying the compound interest formula, we can derive the compound interest obtained after 12 years from a sum that yields Rs 5000 after 8 years at a 20% interest rate. This method is essential for managing financial investments and loans, helping to understand the growth of money over time.