A sum of money amounts to Rs. 9800 after 5 years and Rs. 12005 after 8 year at the same rate of simple interest. The rate of interest per annum is:

A) 5% b) 12% D) 15%

C) 12%

E) None of these

Let's solve this step by step using the formula for simple interest:

The formula for simple interest is:

\[
A = P + \left( \frac{P \cdot r \cdot t}{100} \right)
\]
Where:
- \(A\) is the total amount after interest,
- \(P\) is the principal (initial amount),
- \(r\) is the rate of interest,
- \(t\) is the time in years.

We are given:
- After 5 years, the amount is Rs. 9800.
- After 8 years, the amount is Rs. 12005.

### Step 1: Calculate the interest for the 3-year period
The difference between the amounts after 8 years and 5 years is due to the interest earned over 3 years.

\[
\text{Interest for 3 years} = 12005 - 9800 = 2205
\]

This Rs. 2205 is the simple interest for 3 years on the principal.

### Step 2: Find the annual interest
The simple interest for 1 year will be:

\[
\text{Interest for 1 year} = \frac{2205}{3} = 735
\]

So, the interest earned per year is Rs. 735.

### Step 3: Use the annual interest to calculate the principal
The interest for 5 years will be:

\[
\text{Interest for 5 years} = 735 \times 5 = 3675
\]

We know the total amount after 5 years is Rs. 9800, so:

\[
9800 = P + 3675
\]

Thus, the principal \(P\) is:

\[
P = 9800 - 3675 = 6125
\]

### Step 4: Calculate the rate of interest
We now know the principal \(P = 6125\) and the interest for 1 year is Rs. 735.

Using the simple interest formula, we can calculate the rate \(r\):

\[
\frac{P \cdot r \cdot t}{100} = \text{Interest for 1 year}
\]

Substitute the values:

\[
\frac{6125 \cdot r \cdot 1}{100} = 735
\]

Solve for \(r\):

\[
6125 \cdot r = 73500
\]
\[
r = \frac{73500}{6125} = 12\%
\]

### Final Answer:
The rate of interest per annum is **12%**.

---

Would you like a detailed explanation of any step or have any questions?

Here are some related questions for further practice:
1. What is the formula for compound interest and how does it differ from simple interest?
2. If the rate of interest were compounded annually, what would the total amount be after 5 years?
3. How does the principal affect the interest earned in simple interest calculations?
4. What happens to the interest earned if the time period is halved but the rate remains the same?
5. How would the problem change if the rate of interest varied annually?

**Tip:** Always remember that in simple interest, the interest is calculated on the original principal amount throughout the loan or investment period.

A sum of money amounts to Rs. 9800 after 5 years and Rs. 12005 after 8 years at the same rate of simple interest. The rate of interest per annum is:
A) 5% b) 12% D) 15% C) 12% E) None of these

Find the rate of interest per annum when a sum of money amounts to Rs. 9800 after 5 years and Rs. 12005 after 8 years using simple interest calculations. Detailed step-by-step solution included. Suitable for students in Grades 9-10.

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