The difference between compound interest compounded annually and simple interest on a certain sum of money for 2 years at 5% per annum is Rs 12.50. What is the compound interest on this sum for 2 years with same rate?
(a)500
(b)512
(c)600
(d)512.5

Answer

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

### Step 1: Understand the relationship between Compound Interest (CI) and Simple Interest (SI)
The difference between the compound interest and simple interest for 2 years at a rate of 5% per annum is given as Rs 12.50.

### Step 2: Use the formula for Simple Interest
The formula for Simple Interest (SI) for 2 years is:
\[ SI = \frac{P \times r \times t}{100} \]
where:
- \( P \) = Principal amount
- \( r \) = Rate of interest (5%)
- \( t \) = Time (2 years)

### Step 3: Calculate Simple Interest
For 2 years at 5%:
\[ SI = \frac{P \times 5 \times 2}{100} = \frac{10P}{100} = \frac{P}{10} \]

### Step 4: Use the formula for Compound Interest
The formula for Compound Interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
For 2 years at 5%:
\[ CI = P \left(1 + \frac{5}{100}\right)^2 - P = P \left(1.05^2\right) - P \]
Calculating \(1.05^2\):
\[ 1.05^2 = 1.1025 \]
Thus,
\[ CI = P \times 1.1025 - P = P(1.1025 - 1) = P \times 0.1025 \]

### Step 5: Find the difference between CI and SI
The difference between CI and SI is given as Rs 12.50:
\[ CI - SI = P \times 0.1025 - \frac{P}{10} = 12.50 \]
Converting \(\frac{P}{10}\) to a decimal:
\[ \frac{P}{10} = 0.1P \]
Thus,
\[ P \times 0.1025 - 0.1P = 12.50 \]
This simplifies to:
\[ P \times (0.1025 - 0.1) = 12.50 \]
\[ P \times 0.0025 = 12.50 \]

### Step 6: Solve for Principal (P)
\[ P = \frac{12.50}{0.0025} = 5000 \]

### Step 7: Calculate Compound Interest for 2 years
Now we can calculate the compound interest using the principal \( P = 5000 \):
\[ CI = P \times 0.1025 = 5000 \times 0.1025 = 512.5 \]

### Final Answer
The compound interest on this sum for 2 years at the same rate is Rs 512.5.