Understanding Compound Interest Calculation for Fractional Periods

The question asks us to find the compound interest on a principal amount for a time period that is not a whole number of years (1½ years). When calculating compound interest for a period like 1½ years compounded annually, we calculate the amount for the whole number of years (1 year) and then calculate simple interest on this amount for the remaining fractional part of the year (½ year).

Given Information:

• Principal amount (P) = ₹42000
• Time period (n) = 1½ years = 1 year + ½ year
• Annual interest rate (R) = 10% p.a.
• Compounding frequency: Annually

Step-by-Step Calculation of Compound Interest

Step 1: Calculate the Amount after the Whole Number of Years

First, we calculate the amount after 1 full year using the compound interest formula:

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

Here, P = ₹42000, R = 10%, and n = 1 year.

Amount after 1 year \( A_1 = 42000 \left(1 + \frac{10}{100}\right)^1 \)

\( A_1 = 42000 \left(1 + 0.1\right) \)

\( A_1 = 42000 \times 1.1 \)

\( A_1 = ₹46200 \)

So, the amount after 1 year is ₹46200.

Step 2: Calculate Simple Interest on the Amount for the Fractional Year

Now, this amount \( A_1 \) becomes the principal for the next ½ year. We calculate simple interest on this amount for the remaining time period (½ year) at the given annual rate (10%).

Principal for simple interest \( P' = A_1 = ₹46200 \)

Rate \( R' = 10\% \) p.a.

Time \( t = \frac{1}{2} \) years \( = 0.5 \) years

Simple Interest (SI) = \( \frac{P' \times R' \times t}{100} \)

\( SI = \frac{46200 \times 10 \times 0.5}{100} \)

\( SI = \frac{46200 \times 5}{100} \)

\( SI = 462 \times 5 \)

\( SI = ₹2310 \)

The simple interest for the remaining ½ year is ₹2310.

Step 3: Calculate the Total Amount after 1½ Years

The total amount after 1½ years is the amount after 1 year plus the simple interest for the next ½ year.

Total Amount \( A = A_1 + SI \)

\( A = 46200 + 2310 \)

\( A = ₹48510 \)

The total amount after 1½ years is ₹48510.

Step 4: Calculate the Compound Interest

Compound Interest (CI) is the difference between the total amount and the original principal.

Compound Interest = Total Amount - Original Principal

\( CI = A - P \)

\( CI = 48510 - 42000 \)

\( CI = ₹6510 \)

The compound interest on ₹42000 for 1½ years at 10% p.a. compounded annually is ₹6510.

Checking the Options

Let's compare our calculated compound interest with the given options:

• Option 1: ₹5601
• Option 2: ₹5610
• Option 3: ₹6510
• Option 4: ₹6620.25

Our calculated compound interest is ₹6510, which matches Option 3.

Summary of the Compound Interest Calculation

To find the compound interest for a period with a fraction of a year, we treat the whole number of years as compounded annually and the remaining fraction of a year as simple interest on the amount accumulated at the end of the whole years. This approach correctly handles the compounding effect for the full periods and applies the annual rate proportionally for the final partial period.

Revision Table: Key Concepts in Compound Interest

Additional Information on Compound Interest Calculation

When calculating compound interest for a period of \( n \) whole years and a fraction \( f \) of a year (like \( n + f \)), the formula used implicitly is:

\( \text{Amount} = P \left(1 + \frac{R}{100}\right)^n \left(1 + \frac{f \times R}{100}\right) \)

Let's test this formula with our values: P = 42000, R = 10, n = 1, f = 0.5.

\( \text{Amount} = 42000 \left(1 + \frac{10}{100}\right)^1 \left(1 + \frac{0.5 \times 10}{100}\right) \)

\( \text{Amount} = 42000 \left(1.1\right) \left(1 + \frac{5}{100}\right) \)

\( \text{Amount} = 42000 \times 1.1 \times (1 + 0.05) \)

\( \text{Amount} = 42000 \times 1.1 \times 1.05 \)

\( \text{Amount} = 46200 \times 1.05 \)

\( \text{Amount} = ₹48510 \)

This formula gives the total amount, which matches our step-by-step calculation. The compound interest is then \( 48510 - 42000 = ₹6510 \). This confirms that our step-by-step method is equivalent to using this combined formula for fractional years when compounding is annual.

It's important to note that this method for fractional years applies when compounding is done annually. If compounding were done half-yearly or quarterly, the approach for fractional periods within those compounding cycles would differ.