A sum of Rs 400 would become Rs. 441 after 2 years at r% compound interest , find the value of 'r' :

Options

A. `10%`
B. `5%`
C. `15%`
D. `20%`

Answer

To solve the problem step by step, we will use the formula for compound interest and manipulate it to find the rate of interest \( r \).

### Step 1: Identify the given values
We have:
- Principal (P) = Rs. 400
- Amount (A) = Rs. 441
- Time (T) = 2 years

### Step 2: Write the formula for compound interest
The formula for the amount \( A \) in compound interest is given by:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \( A \) = Amount after time \( n \)
- \( P \) = Principal amount
- \( r \) = Rate of interest per annum
- \( n \) = Number of years

### Step 3: Substitute the known values into the formula
Substituting the known values into the formula, we get:
\[ 441 = 400 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Divide both sides by 400
To isolate the term with \( r \), divide both sides by 400:
\[ \frac{441}{400} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Simplify the left side
Calculating the left side:
\[ \frac{441}{400} = 1.1025 \]
Thus, we have:
\[ 1.1025 = \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Take the square root of both sides
Taking the square root of both sides gives:
\[ \sqrt{1.1025} = 1 + \frac{r}{100} \]
Calculating the square root:
\[ 1.05 = 1 + \frac{r}{100} \]

### Step 7: Isolate \( r \)
Now, subtract 1 from both sides:
\[ 0.05 = \frac{r}{100} \]

### Step 8: Multiply both sides by 100
To find \( r \), multiply both sides by 100:
\[ r = 0.05 \times 100 = 5 \]

### Step 9: Conclusion
Thus, the rate of interest \( r \) is:
\[ \boxed{5\%} \]

---