In how many years will Rs. 400 amount to Rs. 441 at 5% compound interest?

Answer

To solve the problem of how many years it will take for Rs. 400 to amount to Rs. 441 at a 5% compound interest rate, we can follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 400
- Amount (A) = Rs. 441
- Rate of interest (r) = 5%

### Step 2: Use the formula for compound interest
The formula for compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \( A \) is the amount after time \( t \)
- \( P \) is the principal amount
- \( r \) is the rate of interest
- \( t \) is the time in years

### Step 3: Substitute the known values into the formula
We can substitute the values we have into the formula:
\[ 441 = 400 \left(1 + \frac{5}{100}\right)^t \]

### Step 4: Simplify the equation
First, simplify \( 1 + \frac{5}{100} \):
\[ 1 + \frac{5}{100} = 1 + 0.05 = 1.05 \]
Now, the equation becomes:
\[ 441 = 400 \times (1.05)^t \]

### Step 5: Divide both sides by 400
To isolate \( (1.05)^t \), divide both sides by 400:
\[ \frac{441}{400} = (1.05)^t \]
Calculating \( \frac{441}{400} \):
\[ \frac{441}{400} = 1.1025 \]
So, we have:
\[ 1.1025 = (1.05)^t \]

### Step 6: Take logarithm of both sides
To solve for \( t \), we can take the logarithm of both sides:
\[ \log(1.1025) = t \cdot \log(1.05) \]

### Step 7: Solve for \( t \)
Now, we can solve for \( t \):
\[ t = \frac{\log(1.1025)}{\log(1.05)} \]

### Step 8: Calculate the values
Using a calculator:
- \( \log(1.1025) \approx 0.0414 \)
- \( \log(1.05) \approx 0.0212 \)

Now substitute these values into the equation:
\[ t \approx \frac{0.0414}{0.0212} \approx 1.95 \]

### Step 9: Round to the nearest whole number
Since we typically round to the nearest whole number in this context, we can say:
\[ t \approx 2 \text{ years} \]

### Final Answer
It will take approximately **2 years** for Rs. 400 to amount to Rs. 441 at a 5% compound interest rate.

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