A sum of money invested at com pound interest amounts in 3 years to `Rs. 2,400` and in 4 years to `Rs. 2,520`. The interest rate per annum is:

Options

A. `5% `
B. `6% `
C. `10% `
D. `12% `

Answer

To find the interest rate per annum based on the given amounts for 3 years and 4 years, we can follow these steps:

### Step 1: Identify the amounts for the two consecutive years
- Amount after 3 years (A3) = Rs. 2,400
- Amount after 4 years (A4) = Rs. 2,520

### Step 2: Calculate the difference between the two amounts
- Difference (A4 - A3) = Rs. 2,520 - Rs. 2,400 = Rs. 120

### Step 3: Understand the relationship between the difference and the interest
The difference in amounts between two consecutive years is equal to the interest earned in the last year (which is the 4th year in this case).

### Step 4: Calculate the interest earned in the 4th year
- Interest for the 4th year = Rs. 120

### Step 5: Relate the interest to the principal amount
Let the principal amount (P) be the amount after 3 years (A3):
- A3 = P(1 + r)^3 = Rs. 2,400
Where r is the rate of interest expressed as a decimal.

### Step 6: Write the equation for the amount after 4 years
- A4 = P(1 + r)^4 = Rs. 2,520

### Step 7: Set up the equation for the interest
The interest for the 4th year can also be expressed as:
- Interest for the 4th year = A4 - A3 = P(1 + r)^4 - P(1 + r)^3
- This simplifies to: P(1 + r)^3 * r = Rs. 120

### Step 8: Substitute the value of P from A3
From A3, we have:
- P = Rs. 2,400 / (1 + r)^3

### Step 9: Substitute P into the interest equation
- (Rs. 2,400 / (1 + r)^3) * (1 + r)^3 * r = Rs. 120
- This simplifies to: Rs. 2,400 * r = Rs. 120

### Step 10: Solve for r
- r = Rs. 120 / Rs. 2,400
- r = 0.05

### Step 11: Convert r to percentage
- Rate of interest per annum = r * 100 = 0.05 * 100 = 5%

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

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