In what time will Rs. 390625 amount to Rs. 456976 at 4 percent compound interest ?

Options

A. 2 years
B. 4 years
C. 3 years
D. 5 years

Answer

To solve the problem of finding the time it takes for Rs. 390625 to amount to Rs. 456976 at a compound interest rate of 4%, we can follow these steps:

### Step 1: Understand the Compound Interest Formula
The formula for calculating the amount (A) in compound interest is given by:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per period
- \( T \) = Time in years

### Step 2: Identify the Given Values
From the question, we have:
- Principal \( P = 390625 \)
- Amount \( A = 456976 \)
- Rate \( R = 4\% \)

### Step 3: Substitute the Values into the Formula
We can substitute the known values into the compound interest formula:
\[ 456976 = 390625 \left(1 + \frac{4}{100}\right)^T \]

### Step 4: Simplify the Equation
First, simplify \( 1 + \frac{4}{100} \):
\[ 1 + \frac{4}{100} = 1 + 0.04 = 1.04 \]
Now, the equation becomes:
\[ 456976 = 390625 \cdot (1.04)^T \]

### Step 5: Divide Both Sides by the Principal
To isolate \( (1.04)^T \), divide both sides by 390625:
\[ \frac{456976}{390625} = (1.04)^T \]

### Step 6: Calculate the Left Side
Now calculate the left side:
\[ \frac{456976}{390625} = 1.169 \] (approximately)

### Step 7: Set Up the Equation for \( T \)
Now we have:
\[ 1.169 = (1.04)^T \]

### Step 8: Take the Logarithm of Both Sides
To solve for \( T \), take the logarithm of both sides:
\[ \log(1.169) = T \cdot \log(1.04) \]

### Step 9: Solve for \( T \)
Now, rearranging gives:
\[ T = \frac{\log(1.169)}{\log(1.04)} \]

### Step 10: Calculate the Values
Using a calculator:
- \( \log(1.169) \approx 0.072 \)
- \( \log(1.04) \approx 0.017 \)

Thus,
\[ T \approx \frac{0.072}{0.017} \approx 4.24 \]

### Conclusion
Since we are looking for the time in years, we can round it to the nearest whole number. Therefore, \( T \approx 4 \) years.

### Final Answer
The time it will take for Rs. 390625 to amount to Rs. 456976 at 4% compound interest is approximately **4 years**.

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