If A person invests Rs. 8000 for 3 years on a certain rate of interest of simple interest and gets Rs. 3600 as interest. If he invests the same money on compound interest for Time 2 years at an equal rate of interest then find compounded interest if the interest is calculated on every 8 months.
(a)Rs. 2478
(b)Rs. 2196
(c)Rs. 2648
(d)Rs. 2772

Options

A. Rs. 2478
B. Rs. 2196
C. Rs. 2648
D. Rs. 2772

Answer

To solve the problem step by step, we will follow the outlined process to find the compound interest when the interest is calculated every 8 months.

### Step 1: Calculate the Rate of Interest from Simple Interest
We know the formula for Simple Interest (SI):
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest = Rs. 3600
- \(P\) = Principal = Rs. 8000
- \(T\) = Time = 3 years
- \(R\) = Rate of Interest (unknown)

Substituting the known values into the formula:
\[ 3600 = \frac{8000 \times R \times 3}{100} \]

### Step 2: Rearranging the Equation to Solve for R
Rearranging the equation gives:
\[ 3600 \times 100 = 8000 \times R \times 3 \]
\[ 360000 = 24000R \]
\[ R = \frac{360000}{24000} = 15\% \]

### Step 3: Determine the Compound Interest Rate for 8 Months
Since the interest is compounded every 8 months, we need to convert the annual rate into an 8-month rate.

To find the 8-month rate:
\[ \text{8-month rate} = \frac{15\%}{12} \times 8 = 10\% \]

### Step 4: Calculate the Number of Compounding Periods in 2 Years
In 2 years, we have:
\[ \text{Total months} = 2 \times 12 = 24 \text{ months} \]
The number of 8-month periods in 24 months:
\[ \text{Number of periods} = \frac{24}{8} = 3 \]

### Step 5: Calculate the Compound Interest
Using the compound interest formula:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Amount after interest
- \(P\) = Principal = Rs. 8000
- \(r\) = 10% (8-month rate)
- \(n\) = Number of compounding periods = 3

Substituting the values:
\[ A = 8000 \left(1 + \frac{10}{100}\right)^3 \]
\[ A = 8000 \left(1 + 0.1\right)^3 = 8000 \left(1.1\right)^3 \]

Calculating \(1.1^3\):
\[ 1.1^3 = 1.331 \]
Thus,
\[ A = 8000 \times 1.331 = 10648 \]

### Step 6: Calculate the Compound Interest
The compound interest (CI) is given by:
\[ CI = A - P \]
Substituting the values:
\[ CI = 10648 - 8000 = 2648 \]

### Final Answer
The compounded interest is Rs. 2648.