A sum of money amounts to Rs.98,000 after 5 years and Rs.1,20,050 after 8 years at the same rate of simple interest. The rate of interest per annum is:

Options

A. `15%`
B. `12%`
C. `8%`
D. `5%`

Answer

To find the rate of interest per annum given the amounts after 5 years and 8 years, we can follow these steps:

### Step 1: Identify the amounts and the time intervals
- Amount after 5 years (A1) = Rs. 98,000
- Amount after 8 years (A2) = Rs. 1,20,050
- Time interval between the two amounts = 8 years - 5 years = 3 years

### Step 2: Calculate the interest earned in the 3 years
The interest earned over the 3 years can be calculated as:
\[ \text{Interest} = A2 - A1 \]
\[ \text{Interest} = 1,20,050 - 98,000 = Rs. 22,050 \]

### Step 3: Calculate the simple interest per year
To find the simple interest per year, divide the total interest by the number of years:
\[ \text{Simple Interest per year} = \frac{\text{Total Interest}}{\text{Number of Years}} \]
\[ \text{Simple Interest per year} = \frac{22,050}{3} = Rs. 7,350 \]

### Step 4: Calculate the total simple interest for 5 years
Now, calculate the total simple interest for the 5 years:
\[ \text{Total Simple Interest for 5 years} = \text{Simple Interest per year} \times 5 \]
\[ \text{Total Simple Interest for 5 years} = 7,350 \times 5 = Rs. 36,750 \]

### Step 5: Calculate the principal amount
The principal amount (P) can be found by subtracting the total simple interest from the amount after 5 years:
\[ P = A1 - \text{Total Simple Interest for 5 years} \]
\[ P = 98,000 - 36,750 = Rs. 61,250 \]

### Step 6: Use the simple interest formula to find the rate
The formula for simple interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- SI = Simple Interest
- P = Principal
- R = Rate of interest per annum
- T = Time in years

Rearranging the formula to find R gives:
\[ R = \frac{SI \times 100}{P \times T} \]

Substituting the known values:
\[ R = \frac{36,750 \times 100}{61,250 \times 5} \]

### Step 7: Calculate the rate of interest
Calculating the above expression:
\[ R = \frac{36,750 \times 100}{306,250} \]
\[ R = \frac{3,675,000}{306,250} \]
\[ R = 12\% \]

Thus, the rate of interest per annum is **12%**.

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