The compound interest on a certain sum for 2 years at `10%` per annum is `Rs. 525`. The simple interest on the same sum for double the time at half the rate per cent per annum is:

Options

A. `Rs. 520`
B. `Rs. 550`
C. `Rs. 500 `
D. `Rs. 515`

Answer

To solve the problem step by step, we need to find the principal amount first using the given compound interest, and then calculate the simple interest based on the new conditions.

### Step 1: Understand the given information
- Compound Interest (CI) for 2 years = Rs. 525
- Rate of interest (R) = 10% per annum
- Time (T) = 2 years

### Step 2: Use the formula for Compound Interest
The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \(A\) = Final amount
- \(P\) = Principal amount
- \(R\) = Rate of interest
- \(T\) = Time in years

The compound interest can also be expressed as:
\[ CI = A - P \]
Thus, we can rewrite the formula as:
\[ A = P + CI \]

### Step 3: Substitute the values into the formula
We know:
\[ CI = 525 \]
So, we can write:
\[ A = P + 525 \]
Substituting this into the compound interest formula:
\[ P + 525 = P \left(1 + \frac{10}{100}\right)^2 \]
This simplifies to:
\[ P + 525 = P \left(1 + 0.1\right)^2 \]
\[ P + 525 = P \left(1.1\right)^2 \]
\[ P + 525 = P \cdot 1.21 \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation:
\[ P \cdot 1.21 - P = 525 \]
\[ P(1.21 - 1) = 525 \]
\[ P \cdot 0.21 = 525 \]

### Step 5: Solve for Principal (P)
Now, we can solve for \(P\):
\[ P = \frac{525}{0.21} \]
Calculating this gives:
\[ P = 2500 \]

### Step 6: Calculate Simple Interest
Now we need to find the simple interest for double the time (4 years) at half the rate (5%):
- New Time (T) = 4 years
- New Rate (R) = 5%

Using the formula for Simple Interest (SI):
\[ SI = \frac{P \cdot R \cdot T}{100} \]
Substituting the values:
\[ SI = \frac{2500 \cdot 5 \cdot 4}{100} \]
Calculating this gives:
\[ SI = \frac{50000}{100} = 500 \]

### Final Answer
The simple interest on the same sum for double the time at half the rate per annum is **Rs. 500**.

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