A sum invested at 8% p.a. amounts to Rs.20280 at the end of one year, when the interest is compounded half yearly. What will be the simple interest on the same sum for `4(2)/(5)` years at double the earlier rate of interest ?

Options

A. `Rs. 14500`
B. `Rs. 13200`
C. `Rs. 14200`
D. `Rs. 13800`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the Problem
We need to find the principal amount (P) that was invested at an interest rate of 8% per annum, compounded half-yearly, which amounts to Rs. 20,280 at the end of one year. After finding the principal, we will calculate the simple interest for a time period of \(4 \frac{2}{5}\) years at double the interest rate.

### Step 2: Calculate the Effective Rate and Time for Compounding
Since the interest is compounded half-yearly, we need to adjust the rate and time:
- The annual interest rate is 8%, so the half-yearly rate is:
\[ \text{Half-yearly rate} = \frac{8\%}{2} = 4\% \]
- The time for one year in half-yearly terms is:
\[ \text{Time} = 1 \text{ year} = 2 \text{ half-years} \]

### Step 3: Use the Compound Interest Formula
The formula for the amount (A) when compounded is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
Where:
- \(A\) = Amount after time \(n\)
- \(P\) = Principal
- \(r\) = Rate of interest
- \(n\) = Number of compounding periods

Substituting the known values:
\[ 20280 = P \left(1 + \frac{4}{100}\right)^2 \]
\[ 20280 = P \left(1.04\right)^2 \]
\[ 20280 = P \times 1.0816 \]

### Step 4: Solve for Principal (P)
Rearranging the equation to find \(P\):
\[ P = \frac{20280}{1.0816} \]
Calculating \(P\):
\[ P = 18750 \]

### Step 5: Calculate the Simple Interest
Now we need to calculate the simple interest for \(4 \frac{2}{5}\) years at double the earlier rate of interest. The new rate will be:
\[ \text{New Rate} = 2 \times 8\% = 16\% \]
Convert \(4 \frac{2}{5}\) years to an improper fraction:
\[ 4 \frac{2}{5} = \frac{22}{5} \text{ years} \]

Using the formula for Simple Interest (SI):
\[ SI = \frac{P \times r \times t}{100} \]
Substituting the values:
\[ SI = \frac{18750 \times 16 \times \frac{22}{5}}{100} \]

### Step 6: Calculate Simple Interest
Calculating the SI:
\[ SI = \frac{18750 \times 16 \times 22}{500} \]
\[ SI = \frac{6600000}{500} = 13200 \]

### Final Answer
The simple interest on the same sum for \(4 \frac{2}{5}\) years at double the earlier rate of interest is Rs. 13,200.

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