If the difference between the compound interest and simple interest on a sum of money at 5% pa for 2 yr is ? 16, then find the simple interest .

Options

A. ? 840
B. ? 932
C. ? 520
D. ? 640

Answer

To solve the problem step by step, we will follow the mathematical principles of compound interest and simple interest.

### Step 1: Understand the given information
We know:
- The difference between Compound Interest (CI) and Simple Interest (SI) for 2 years at a rate of 5% is 16.
- We need to find the Simple Interest.

### Step 2: Formula for Simple Interest
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest (5% in this case)
- \( T \) = Time period (2 years)

### Step 3: Calculate Simple Interest in terms of Principal
Substituting the values into the formula:
\[ \text{SI} = \frac{P \times 5 \times 2}{100} \]
\[ \text{SI} = \frac{10P}{100} \]
\[ \text{SI} = \frac{P}{10} \]

### Step 4: Formula for Compound Interest
The formula for Compound Interest is:
\[ \text{CI} = A - P \]
Where \( A \) is the amount after time \( T \):
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Substituting the values:
\[ A = P \left(1 + \frac{5}{100}\right)^2 \]
\[ A = P \left(1.05\right)^2 \]
\[ A = P \times 1.1025 \]

### Step 5: Calculate Compound Interest
Now, substituting \( A \) back into the CI formula:
\[ \text{CI} = P \times 1.1025 - P \]
\[ \text{CI} = P(1.1025 - 1) \]
\[ \text{CI} = P \times 0.1025 \]

### Step 6: Find the difference between CI and SI
The difference between CI and SI is given as 16:
\[ \text{CI} - \text{SI} = 16 \]
Substituting the values of CI and SI:
\[ P \times 0.1025 - \frac{P}{10} = 16 \]

### Step 7: Simplify the equation
Convert \(\frac{P}{10}\) to a decimal:
\[ P \times 0.1025 - P \times 0.1 = 16 \]
Now, factor out \( P \):
\[ P(0.1025 - 0.1) = 16 \]
\[ P(0.0025) = 16 \]

### Step 8: Solve for Principal \( P \)
Divide both sides by 0.0025:
\[ P = \frac{16}{0.0025} \]
\[ P = 6400 \]

### Step 9: Calculate Simple Interest
Now, substitute \( P \) back into the SI formula:
\[ \text{SI} = \frac{P}{10} \]
\[ \text{SI} = \frac{6400}{10} \]
\[ \text{SI} = 640 \]

### Final Answer
The Simple Interest is **640**.

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