Understanding Compound Interest Calculations

Compound interest is calculated on the initial principal and also on the accumulated interest from previous periods. It's often described as "interest on interest," and it makes a sum grow faster compared to simple interest.

In this problem, we are given the compound interest earned, the annual interest rate, and the time period. We need to find the total amount received after the given time.

Key Information Provided:

• Compound Interest (CI) = Rs. 11,138.40
• Rate of Interest (R) = 21% per annum
• Time (n) = 2 years
• Compounding Frequency: Yearly

Formula for Compound Interest:

The formula for the Amount (A) after 'n' years with compound interest is:

$$A = P \left(1 + \frac{R}{100}\right)^n$$

Where:

• \(P\) is the Principal amount
• \(R\) is the annual rate of interest
• \(n\) is the number of years

The Compound Interest (CI) is the difference between the Amount and the Principal:

$$CI = A - P$$

Substituting the formula for A into the CI formula:

$$CI = P \left(1 + \frac{R}{100}\right)^n - P$$

$$CI = P \left[ \left(1 + \frac{R}{100}\right)^n - 1 \right]$$

Calculating the Principal Amount

We are given CI, R, and n. We can use the formula to find the Principal (P).

Given: CI = 11138.40, R = 21%, n = 2 years

Substitute the values into the formula:

$$11138.40 = P \left[ \left(1 + \frac{21}{100}\right)^2 - 1 \right]$$

$$11138.40 = P \left[ \left(1 + 0.21\right)^2 - 1 \right]$$

$$11138.40 = P \left[ \left(1.21\right)^2 - 1 \right]$$

Calculate \((1.21)^2\):

$$1.21 \times 1.21 = 1.4641$$

Now substitute this back:

$$11138.40 = P \left[ 1.4641 - 1 \right]$$

$$11138.40 = P \times 0.4641$$

To find P, divide the CI by 0.4641:

$$P = \frac{11138.40}{0.4641}$$

Let's perform the division:

$$P = 24000$$

So, the Principal amount is Rs. 24,000.

Calculating the Total Amount Received

The total amount received after 2 years is the sum of the Principal and the Compound Interest.

Total Amount (A) = Principal (P) + Compound Interest (CI)

$$A = 24000 + 11138.40$$

$$A = 35138.40$$

The total amount received after 2 years is Rs. 35,138.40.

Summary of Calculation:

Conclusion

Starting with a principal of Rs. 24,000, at an annual compound interest rate of 21% for 2 years, the compound interest earned is Rs. 11,138.40. The total amount accumulated at the end of 2 years is Rs. 35,138.40.

Revision Table: Compound Interest Concepts

Additional Information on Compounding Frequency

The formula for compound interest changes slightly depending on how frequently the interest is compounded within a year (e.g., half-yearly, quarterly, monthly). If interest is compounded 'k' times per year, the formula becomes:

$$A = P \left(1 + \frac{R}{100k}\right)^{nk}$$

And the Compound Interest would be:

$$CI = P \left[ \left(1 + \frac{R}{100k}\right)^{nk} - 1 \right]$$

In this specific problem, the compounding is yearly, which means k=1. So, the simpler formula \(A = P \left(1 + \frac{R}{100}\right)^n\) is applicable.