The simple interest for certain in 2 yr at 4% pa is ? 80. What will be the compound interest for the same sum, if conditions of rate and time period are same ?

Options

A. ? 91.60
B. ? 81.60
C. ? 71.60
D. ? 80

Answer

To solve the problem step-by-step, we will follow the process of calculating the principal amount using simple interest and then finding the compound interest based on that principal.

### Step 1: Understand the given information
- Simple Interest (SI) = ₹80
- Rate (R) = 4% 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:
\[ 80 = \frac{P \times 4 \times 2}{100} \]

### Step 3: Simplify the equation
First, calculate the multiplication in the numerator:
\[ 80 = \frac{P \times 8}{100} \]
Now, multiply both sides by 100 to eliminate the fraction:
\[ 8000 = P \times 8 \]

### Step 4: Solve for P
Now, divide both sides by 8:
\[ P = \frac{8000}{8} = 1000 \]
So, the principal amount (P) is ₹1000.

### Step 5: Calculate the Amount (A) using the Compound Interest formula
The formula for the Amount (A) in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where \(n\) is the number of years. Substituting the values:
\[ A = 1000 \left(1 + \frac{4}{100}\right)^2 \]
This simplifies to:
\[ A = 1000 \left(1 + 0.04\right)^2 = 1000 \left(1.04\right)^2 \]

### Step 6: Calculate \( (1.04)^2 \)
Calculating \( (1.04)^2 \):
\[ (1.04)^2 = 1.0816 \]
Now substitute this back into the equation for A:
\[ A = 1000 \times 1.0816 = 1081.6 \]

### Step 7: Calculate the Compound Interest (CI)
The formula for Compound Interest is:
\[ CI = A - P \]
Substituting the values we found:
\[ CI = 1081.6 - 1000 = 81.6 \]
So, the Compound Interest is ₹81.6.

### Final Answer
The compound interest for the same sum, at the same rate and time period, is **₹81.6**.

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