The rate of interest on a sum of money is `4%` per annum for the first 2 years. `6%` per annum for the next 4 years and `8%` per annum for the period beyond 6 years. If the simple interest accured by the sum for a total period of 9 years is Rs. 1120. What is the sum?

Options

A. Rs. 1500
B. Rs. 2000
C. Rs. 2500
D. Rs. 4000

Answer

To find the sum of money (principal) based on the given rates of interest and the total simple interest accrued over 9 years, we can follow these steps:

### Step 1: Breakdown the Interest Calculation by Time Periods
1. For the first 2 years, the interest rate is 4% per annum.
2. For the next 4 years (years 3 to 6), the interest rate is 6% per annum.
3. For the last 3 years (years 7 to 9), the interest rate is 8% per annum.

### Step 2: Calculate the Interest for Each Period
Let the principal amount be \( P \).

- **Interest for the first 2 years:**
\[ \text{SI}_1 = P \times \frac{4}{100} \times 2 = \frac{8P}{100} \]

- **Interest for the next 4 years:**
\[ \text{SI}_2 = P \times \frac{6}{100} \times 4 = \frac{24P}{100} \]

- **Interest for the last 3 years:**
\[ \text{SI}_3 = P \times \frac{8}{100} \times 3 = \frac{24P}{100} \]

### Step 3: Sum Up the Total Interest
The total simple interest accrued over 9 years is the sum of the interests from all three periods:
\[ \text{Total SI} = \text{SI}_1 + \text{SI}_2 + \text{SI}_3 \]
\[ \text{Total SI} = \frac{8P}{100} + \frac{24P}{100} + \frac{24P}{100} = \frac{56P}{100} \]

### Step 4: Set Up the Equation
According to the problem, the total simple interest accrued over 9 years is Rs. 1120:
\[ \frac{56P}{100} = 1120 \]

### Step 5: Solve for P
To find \( P \), we can rearrange the equation:
\[ 56P = 1120 \times 100 \]
\[ 56P = 112000 \]
\[ P = \frac{112000}{56} \]
\[ P = 2000 \]

### Conclusion
The sum (principal amount) is Rs. 2000.

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