A sum of ₹6,400 invested on the basis of yearly compounding of interest, grows to ₹7,056 in two years. What is the percentage rate of interest?

Options

A. 6.25
B. 4
C. 7.5
D. 5

Answer

To find the percentage rate of interest for a sum of ₹6,400 that grows to ₹7,056 in two years with yearly compounding, we can follow these steps:

### Step 1: Identify the known values
- Principal (P) = ₹6,400
- Amount (A) = ₹7,056
- Time (T) = 2 years

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

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 7,056 = 6,400 \left(1 + \frac{r}{100}\right)^2 \]

### Step 4: Divide both sides by the principal
To isolate the term with the rate, divide both sides by ₹6,400:
\[ \frac{7,056}{6,400} = \left(1 + \frac{r}{100}\right)^2 \]

### Step 5: Calculate the left side
Calculating the left side:
\[ \frac{7,056}{6,400} = 1.1 \]

### Step 6: Take the square root of both sides
Now take the square root of both sides to solve for \(1 + \frac{r}{100}\):
\[ \sqrt{1.1} = 1 + \frac{r}{100} \]

### Step 7: Calculate the square root
Calculating the square root:
\[ \sqrt{1.1} \approx 1.0488 \]

### Step 8: Isolate the rate
Now, isolate \(r\):
\[ 1.0488 - 1 = \frac{r}{100} \]
\[ 0.0488 = \frac{r}{100} \]

### Step 9: Solve for r
Multiply both sides by 100 to find the rate:
\[ r = 0.0488 \times 100 \]
\[ r \approx 4.88 \]

### Step 10: Round to the nearest percentage
Rounding to the nearest whole number gives us:
\[ r \approx 5\% \]

### Final Answer
The percentage rate of interest is **5% per annum**.

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