A person invests Rs 5,000 for three years at a certain rate of interest compounded annually. At the end of two years this sum amounts to Rs 6,272. Calculate the rate of interest per annum.

Answer

To solve the problem step by step, we will use the formula for compound interest. The formula is:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Identify the known values
From the problem, we have:
- Principal amount \( P = 5000 \) (Rs)
- Amount after 2 years \( A = 6272 \) (Rs)
- Time \( T = 2 \) years

### Step 2: Substitute the known values into the formula
Using the formula, we substitute the known values:

\[ 6272 = 5000 \left(1 + \frac{R}{100}\right)^2 \]

### Step 3: Simplify the equation
To isolate the term with \( R \), we first divide both sides by 5000:

\[ \frac{6272}{5000} = \left(1 + \frac{R}{100}\right)^2 \]

Calculating the left side:

\[ \frac{6272}{5000} = 1.2544 \]

So, we have:

\[ 1.2544 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Take the square root of both sides
To eliminate the square, we take the square root of both sides:

\[ \sqrt{1.2544} = 1 + \frac{R}{100} \]

Calculating the square root:

\[ 1.12 = 1 + \frac{R}{100} \]

### Step 5: Solve for \( R \)
Now, we isolate \( R \):

\[ \frac{R}{100} = 1.12 - 1 \]
\[ \frac{R}{100} = 0.12 \]

Multiplying both sides by 100 to find \( R \):

\[ R = 0.12 \times 100 = 12 \]

### Final Answer
The rate of interest per annum is \( R = 12\% \).

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