Understanding the Compound Interest Problem

The question asks us to calculate the compound interest on a principal amount of Rs. 12,000 for a specific time period of 2 \( \frac{5}{8} \) years at an interest rate of 8% per annum, compounded annually. The challenge here is the time period which includes a fraction of a year.

When the time period for compound interest is given as a mix of full years and a fraction of a year, the calculation is done in two steps:

1. First, calculate the amount for the full number of years using the compound interest formula.
2. Then, calculate the simple interest on the amount obtained after the full years for the remaining fraction of the year.
3. The total amount is the sum of the amount after full years and the simple interest for the fractional period.
4. Finally, the compound interest is the total amount minus the original principal.

Step-by-Step Compound Interest Calculation

Given:

• Principal (P) = Rs. 12,000
• Rate (R) = 8% per annum
• Time (T) = 2 \( \frac{5}{8} \) years

Step 1: Calculate the Amount after 2 full years

We use the compound interest formula: \( A = P \left(1 + \frac{R}{100}\right)^n \), where n is the number of full years.

Here, P = 12000, R = 8, and n = 2.

Amount after 2 years \( (A_2) = 12000 \left(1 + \frac{8}{100}\right)^2 \)

\( A_2 = 12000 \left(1 + 0.08\right)^2 \)

\( A_2 = 12000 \left(1.08\right)^2 \)

\( A_2 = 12000 \times 1.1664 \)

\( A_2 = 13996.80 \)

So, the amount after 2 full years is Rs. 13,996.80.

Step 2: Calculate the Simple Interest for the remaining \( \frac{5}{8} \) year

Now, this amount (Rs. 13996.80) becomes the principal for the remaining fraction of the year, which is \( \frac{5}{8} \) years. We calculate simple interest for this period.

Simple Interest \( (SI) = \frac{P \times R \times T}{100} \)

Here, P = \( A_2 \) = 13996.80, R = 8, and T = \( \frac{5}{8} \).

\( SI_{\frac{5}{8}} = \frac{13996.80 \times 8 \times \frac{5}{8}}{100} \)

\( SI_{\frac{5}{8}} = \frac{13996.80 \times 5}{100} \)

\( SI_{\frac{5}{8}} = 139.968 \times 5 \)

\( SI_{\frac{5}{8}} = 699.84 \)

The simple interest for the remaining \( \frac{5}{8} \) year is Rs. 699.84.

Step 3: Calculate the Total Amount after \( 2 \frac{5}{8} \) years

The total amount at the end of the entire period is the sum of the amount after 2 years and the simple interest for the fractional year.

Total Amount \( (A_{total}) = A_2 + SI_{\frac{5}{8}} \)

\( A_{total} = 13996.80 + 699.84 \)

\( A_{total} = 14696.64 \)

The total amount after \( 2 \frac{5}{8} \) years is Rs. 14,696.64.

Step 4: Calculate the Compound Interest

The compound interest is the difference between the total amount and the original principal.

Compound Interest \( (CI) = A_{total} - P \)

\( CI = 14696.64 - 12000 \)

\( CI = 2696.64 \)

Step 5: Round to the nearest rupee

The calculated compound interest is Rs. 2696.64. Rounding this to the nearest rupee gives Rs. 2697.

Therefore, the compound interest on Rs. 12,000 for 2 \( \frac{5}{8} \) years at 8% p.a. compounded annually is approximately Rs. 2697.

Revision Table: Key Formulas

Additional Information on Compound Interest

Compound interest is often called "interest on interest". It's a powerful concept in finance because the interest earned in each period is added to the principal for the next period's calculation. This leads to exponential growth of the investment or debt over time, especially compared to simple interest.

When the interest is compounded annually, it means the interest is calculated and added to the principal once a year. If the compounding frequency is different (like half-yearly, quarterly, or monthly), the formula needs adjustment: the rate is divided by the number of compounding periods per year, and the time is multiplied by the number of compounding periods per year.

Calculating compound interest for periods involving fractions of a year under annual compounding is standard practice as demonstrated above. The logic is that compound interest is applied for the full periods, and simple interest is applied to the accumulated amount for the remaining partial period.