If compound interest on some amount at the rate of 4% interest for 9 months is 303.01 then find the principle amount if interest is compounded quarterly?

Options

A. ₹ 10000
B. ₹ 11000
C. ₹ 12000
D. ₹9500

Answer

To find the principal amount given the compound interest, we can follow these steps:

### Step 1: Understand the given information
- Compound Interest (CI) = 303.01
- Rate of interest (R) = 4% per annum
- Time (T) = 9 months
- Compounding frequency = Quarterly

### Step 2: Convert time into years
Since the rate is per annum, we need to convert 9 months into years.
\[ T = \frac{9}{12} = 0.75 \text{ years} \]

### Step 3: Determine the quarterly rate
Since the interest is compounded quarterly, we need to divide the annual rate by 4 (the number of quarters in a year).
\[ \text{Quarterly Rate} = \frac{4\%}{4} = 1\% \]

### Step 4: Determine the number of compounding periods
Since the interest is compounded quarterly and the time is 9 months, we need to find out how many quarters are in 9 months.
\[ \text{Number of Quarters} = \frac{9}{3} = 3 \]

### Step 5: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Total amount after interest
- \(P\) = Principal amount
- \(r\) = Rate of interest per period
- \(n\) = Number of compounding periods

We know that:
\[ CI = A - P \]
So,
\[ A = P + CI \]
Substituting this into the compound interest formula gives:
\[ P + CI = P \left(1 + \frac{r}{100}\right)^n \]

### Step 6: Substitute known values
Substituting the known values into the equation:
\[ P + 303.01 = P \left(1 + \frac{1}{100}\right)^3 \]
Calculating \( \left(1 + \frac{1}{100}\right)^3 \):
\[ \left(1 + 0.01\right)^3 = 1.01^3 \]
Calculating \(1.01^3\):
\[ 1.01^3 \approx 1.030301 \]

### Step 7: Substitute this back into the equation
Now we have:
\[ P + 303.01 = P \cdot 1.030301 \]

### Step 8: Rearranging the equation
Rearranging gives:
\[ P \cdot 1.030301 - P = 303.01 \]
\[ P(1.030301 - 1) = 303.01 \]
\[ P(0.030301) = 303.01 \]

### Step 9: Solve for P
Now, divide both sides by 0.030301:
\[ P = \frac{303.01}{0.030301} \approx 10000 \]

### Conclusion
Thus, the principal amount is approximately **10000**.

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