Radha borrowed a certain sum of simple interest from Laxmi incurring `6%` interest per annum for first 3 years, `8%` interest for the period next 5 years. If he paid Rs. 10080 as interest for a total period of 8 years. What sum had he borrowed?

Options

A. Rs. 18000
B. Rs. 14000
C. Rs. 20160
D. None of these

Answer

To find the sum that Radha borrowed, we can use the formula for simple interest:

\[ \text{Simple Interest (SI)} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (the sum borrowed)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 1: Calculate the interest for the first 3 years at 6%
For the first 3 years, the interest rate is 6%.

Using the formula:
\[ \text{SI}_1 = \frac{P \times 6 \times 3}{100} = \frac{18P}{100} \]

### Step 2: Calculate the interest for the next 5 years at 8%
For the next 5 years, the interest rate is 8%.

Using the formula:
\[ \text{SI}_2 = \frac{P \times 8 \times 5}{100} = \frac{40P}{100} \]

### Step 3: Combine the interests
The total interest paid over the 8 years is the sum of the two interests calculated above:
\[ \text{Total Interest} = \text{SI}_1 + \text{SI}_2 = \frac{18P}{100} + \frac{40P}{100} = \frac{58P}{100} \]

### Step 4: Set up the equation
According to the problem, Radha paid Rs. 10,080 as interest for the total period of 8 years. Therefore, we can set up the equation:
\[ \frac{58P}{100} = 10080 \]

### Step 5: Solve for \( P \)
To find \( P \), multiply both sides by 100:
\[ 58P = 10080 \times 100 \]
\[ 58P = 1008000 \]

Now, divide both sides by 58:
\[ P = \frac{1008000}{58} \]

Calculating this gives:
\[ P = 17379.31 \]

### Conclusion
Thus, the sum that Radha borrowed is approximately Rs. 17,379.31.

---