What is the compound interest on a sum of Rs 37500 for `1(1/3)` years at a rate of 12% p.a. if the interest is compounded 8- monthly?

Options

A. Rs 6440
B. Rs 6240
C. Rs 6420
D. Rs 6448

Answer

To calculate the compound interest on a sum of Rs 37500 for \(1 \frac{1}{3}\) years at a rate of 12% per annum, compounded 8-monthly, follow these steps:

### Step 1: Convert the time period into months
The total time period is \(1 \frac{1}{3}\) years, which can be converted into months:
\[ 1 \frac{1}{3} \text{ years} = 1 \text{ year} + \frac{1}{3} \text{ year} = 12 \text{ months} + 4 \text{ months} = 16 \text{ months} \]

### Step 2: Determine the number of compounding periods
Since the interest is compounded 8-monthly, we need to find out how many 8-month periods fit into 16 months:
\[ \text{Number of compounding periods} = \frac{16 \text{ months}}{8 \text{ months}} = 2 \]

### Step 3: Calculate the effective interest rate for each compounding period
The annual interest rate is 12%. Since the interest is compounded every 8 months, we need to find the interest rate for 8 months:
\[ \text{Effective interest rate for 8 months} = \frac{12\%}{3} = 4\% \]

### Step 4: Use the compound interest formula
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = the amount after time \(n\)
- \(P\) = principal amount (Rs 37500)
- \(r\) = rate of interest per compounding period (4%)
- \(n\) = number of compounding periods (2)

Substituting the values into the formula:
\[ A = 37500 \left(1 + \frac{4}{100}\right)^2 \]
\[ A = 37500 \left(1 + 0.04\right)^2 \]
\[ A = 37500 \left(1.04\right)^2 \]
\[ A = 37500 \times 1.0816 \]
\[ A = 40560 \]

### Step 5: Calculate the compound interest
The compound interest (CI) can be found by subtracting the principal from the total amount:
\[ CI = A - P \]
\[ CI = 40560 - 37500 \]
\[ CI = 3060 \]

### Final Answer
The compound interest on a sum of Rs 37500 for \(1 \frac{1}{3}\) years at a rate of 12% p.a. compounded 8-monthly is Rs 3060.

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