A sum of Rs. 2400 amount to Rs. 2646 in 2 years. Find the rate of compound interest per annum?

Answer

To find the rate of compound interest per annum when a sum of Rs. 2400 amounts to Rs. 2646 in 2 years, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = Rs. 2400
- Amount (A) = Rs. 2646
- Time (T) = 2 years

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

### Step 3: Substitute the known values into the formula
Substituting the values we have:
\[ 2646 = 2400 \left(1 + \frac{R}{100}\right)^2 \]

### Step 4: Simplify the equation
To isolate the term with R, we first divide both sides by 2400:
\[ \frac{2646}{2400} = \left(1 + \frac{R}{100}\right)^2 \]

Calculating the left side:
\[ \frac{2646}{2400} = 1.1025 \]
So we have:
\[ 1.1025 = \left(1 + \frac{R}{100}\right)^2 \]

### Step 5: 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 6: Isolate R
Now, subtract 1 from both sides:
\[ 1.05 - 1 = \frac{R}{100} \]
This simplifies to:
\[ 0.05 = \frac{R}{100} \]

### Step 7: Solve for R
To find R, multiply both sides by 100:
\[ R = 0.05 \times 100 \]
\[ R = 5 \]

### Conclusion
The rate of compound interest per annum is **5%**.

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