In how much time (in months), the compound interest, compounded annually, on Rs.30,000 at `7%` per annum is Rs.4,347?

Options

A. 12
B. 24
C. 36
D. 42

Answer

To solve the problem step by step, we need to find the time (in months) for which the compound interest on Rs. 30,000 at an annual interest rate of 7% amounts to Rs. 4,347.

### Step 1: Identify the given values
- Principal (P) = Rs. 30,000
- Compound Interest (CI) = Rs. 4,347
- Rate of Interest (R) = 7% per annum
- Time (T) = ? (in years)

### Step 2: Calculate the total amount (A)
The total amount (A) can be calculated by adding the principal and the compound interest:
\[ A = P + CI \]
\[ A = 30,000 + 4,347 = 34,347 \]

### Step 3: Use the compound interest formula
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the known values into the formula:
\[ 34,347 = 30,000 \left(1 + \frac{7}{100}\right)^T \]
\[ 34,347 = 30,000 \left(1.07\right)^T \]

### Step 4: Divide both sides by the principal
To isolate the term with T, divide both sides by 30,000:
\[ \frac{34,347}{30,000} = (1.07)^T \]
Calculating the left side:
\[ \frac{34,347}{30,000} \approx 1.1449 \]
So we have:
\[ 1.1449 = (1.07)^T \]

### Step 5: Take logarithm on both sides
To solve for T, we can take the logarithm of both sides:
\[ \log(1.1449) = T \cdot \log(1.07) \]
Now, calculate the logarithms:
\[ T = \frac{\log(1.1449)}{\log(1.07)} \]

### Step 6: Calculate the values
Using a calculator:
- \(\log(1.1449) \approx 0.0607\)
- \(\log(1.07) \approx 0.0291\)

Now substituting these values:
\[ T \approx \frac{0.0607}{0.0291} \approx 2.08 \text{ years} \]

### Step 7: Convert years to months
Since the question asks for the time in months, we convert years to months:
\[ T \text{ (in months)} = 2.08 \times 12 \approx 24.96 \text{ months} \]
Rounding to the nearest whole number, we get:
\[ T \approx 25 \text{ months} \]

### Final Answer
The time in months is approximately **25 months**.

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