Calculating Compound Interest Compounded 8-Monthly

This problem involves calculating compound interest when the compounding frequency is different from the standard annual or semi-annual periods. Here, the interest is compounded every 8 months.

To solve this, we first need to determine the interest rate per compounding period and the total number of compounding periods over the given time frame.

Understanding 8-Monthly Compounding

• The annual interest rate is 12% p.a.
• The interest is compounded every 8 months.
• The total time period is 2 years.

Step-by-Step Compound Interest Calculation

Here's how we break down the calculation:

Step 1: Determine the Rate per Compounding Period

The annual rate is 12%. An 8-month period is $\frac{8}{12}$ of a year. So, the interest rate for an 8-month period is:

Rate per period $= \text{Annual Rate} \times \frac{\text{Compounding Period in Months}}{12}$

Rate per period $= 12\% \times \frac{8}{12} = 1\% \times 8 = 8\%$.

In decimal form, this rate is $0.08$.

Step 2: Determine the Total Number of Compounding Periods

The total time is 2 years. Since interest is compounded every 8 months, we find how many 8-month periods are there in 2 years (24 months):

Total periods $= \frac{\text{Total Time in Months}}{\text{Compounding Period in Months}}$

Total periods $= \frac{24 \text{ months}}{8 \text{ months/period}} = 3 \text{ periods}$.

Step 3: Use the Compound Interest Formula

The formula for the amount ($A$) after compound interest is:

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

Where:

• $P$ is the principal amount
• $r$ is the annual interest rate (as a decimal)
• $n$ is the number of times interest is compounded per year
• $t$ is the time in years

Alternatively, using the rate per period and total number of periods:

$$A = P (1 + R)^N$$

Where:

• $P = 62,500$ Rs.
• $R =$ Rate per period $= 8\% = 0.08$
• $N =$ Total number of periods $= 3$

Step 4: Calculate the Amount

Substitute the values into the formula:

$$A = 62,500 (1 + 0.08)^3$$

$$A = 62,500 (1.08)^3$$

Calculate $(1.08)^3$:

$$(1.08)^2 = 1.08 \times 1.08 = 1.1664$$

$$(1.08)^3 = 1.1664 \times 1.08 = 1.259712$$

Now calculate the amount $A$:

$$A = 62,500 \times 1.259712$$

$$A = 78,732$$

Step 5: Calculate the Compound Interest

The compound interest (CI) is the total amount minus the principal:

$$CI = A - P$$

$$CI = 78,732 - 62,500$$

$$CI = 16,232$$

So, the compound interest is Rs. 16,232.

Revision Table: Compound Interest Terms

Additional Information on Compound Interest Calculations

When dealing with compound interest, especially with frequencies other than annual, the key is consistency in units. The interest rate must be for the same period as the compounding frequency, and the total time must be converted into the total number of such periods.

• For half-yearly compounding, the rate is annual rate / 2, and periods are years * 2.
• For quarterly compounding, the rate is annual rate / 4, and periods are years * 4.
• For monthly compounding, the rate is annual rate / 12, and periods are years * 12.
• For 8-monthly compounding, as shown, the rate is annual rate * (8/12), and periods are total months / 8.

Understanding the relationship between the annual rate, compounding frequency, and the rate/number of periods is crucial for accurate compound interest calculations.