If the simple interest on a sum of money for 2 years at `5%` per annum is Rs.60, what is the compound interest on the same at the same rate and for the same time?

Options

A. Rs.63.5
B. Rs.62
C. Rs.61.5
D. Rs.64

Answer

To solve the problem step by step, we will first find the principal amount using the given simple interest, and then we will calculate the compound interest using the principal, rate, and time.

### Step 1: Identify the given values
- Simple Interest (SI) = Rs. 60
- Rate (R) = 5% per annum
- Time (T) = 2 years

### Step 2: Use the formula for Simple Interest to find the Principal (P)
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 60 = \frac{P \times 5 \times 2}{100} \]

### Step 3: Rearranging the formula to find Principal (P)
Cross-multiplying gives:
\[ 60 \times 100 = P \times 5 \times 2 \]
\[ 6000 = P \times 10 \]
Now, divide both sides by 10:
\[ P = \frac{6000}{10} = 600 \]

### Step 4: Calculate the Compound Interest (CI)
The formula for Compound Interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where \(A\) is the amount. We will first calculate the amount (A):
Substituting the values we found:
\[ A = 600 \left(1 + \frac{5}{100}\right)^2 \]
\[ A = 600 \left(1 + 0.05\right)^2 \]
\[ A = 600 \left(1.05\right)^2 \]

### Step 5: Calculate \( (1.05)^2 \)
Calculating \( (1.05)^2 \):
\[ (1.05)^2 = 1.1025 \]
Now substitute back to find A:
\[ A = 600 \times 1.1025 = 661.5 \]

### Step 6: Find the Compound Interest (CI)
Now, to find the Compound Interest, we subtract the principal from the amount:
\[ CI = A - P \]
\[ CI = 661.5 - 600 = 61.5 \]

### Final Answer
The Compound Interest on the same principal at the same rate and for the same time is Rs. 61.5.

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