The compound interest on `Rs. 30,000` at `7%` per annum for a certain time is `Rs. 4,347` . The time is

Options

A. 3 years
B. 4 years
C. 2 years
D. 2.5 years

Answer

To find the time for which the compound interest on Rs. 30,000 at 7% per annum is Rs. 4,347, we can follow these steps:

### Step 1: Understand the formula for Compound Interest
The formula for calculating the amount (A) after a certain time (t) with compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
where:
- \( A \) = Total amount after time \( t \)
- \( P \) = Principal amount (initial investment)
- \( r \) = Rate of interest per annum
- \( t \) = Time in years

### Step 2: Calculate the total amount (A)
Given that the compound interest (CI) is Rs. 4,347, we can find the total amount (A) by adding the principal (P) to the compound interest (CI):
\[ A = P + CI \]
\[ A = 30,000 + 4,347 = 34,347 \]

### Step 3: Set up the equation using the compound interest formula
Now we can substitute the values into the compound interest formula:
\[ 34,347 = 30,000 \left(1 + \frac{7}{100}\right)^t \]

### Step 4: Simplify the equation
First, simplify \( 1 + \frac{7}{100} \):
\[ 1 + \frac{7}{100} = 1.07 \]
Now the equation becomes:
\[ 34,347 = 30,000 \times (1.07)^t \]

### Step 5: Divide both sides by 30,000
To isolate \( (1.07)^t \):
\[ \frac{34,347}{30,000} = (1.07)^t \]
\[ 1.1449 = (1.07)^t \]

### Step 6: Take the logarithm of both sides
To solve for \( t \), take the logarithm:
\[ \log(1.1449) = t \cdot \log(1.07) \]

### Step 7: Solve for \( t \)
Now, we can isolate \( t \):
\[ t = \frac{\log(1.1449)}{\log(1.07)} \]

### Step 8: Calculate the values
Using a calculator:
- \( \log(1.1449) \approx 0.0607 \)
- \( \log(1.07) \approx 0.0291 \)

Now plug in these values:
\[ t \approx \frac{0.0607}{0.0291} \approx 2.08 \]

### Step 9: Round to the nearest whole number
Since time is usually expressed in whole years, we can round \( 2.08 \) to \( 2 \) years.

### Final Answer
The time is approximately **2 years**.

---