To solve this problem, let's break it down step by step, following an organized method:
### Step 1: Understand the Given Data
1. Rate of Interest: 4% per annum
2. Time Period: 2 years
3. Difference between Compound Interest and Simple Interest: Rs. 160
### Step 2: Define the Variables
- Let [tex]\( P \)[/tex] be the principal amount to be found.
### Step 3: Simple Interest Calculation
The formula for Simple Interest (SI) is:
[tex]\[ \text{SI} = P \times \text{rate} \times \text{time} \][/tex]
Given the rate is 4%, which is 0.04 in decimal, and time is 2 years:
[tex]\[ \text{SI} = P \times 0.04 \times 2 = 0.08P \][/tex]
### Step 4: Compound Interest Calculation
The formula for Compound Interest (CI) is:
[tex]\[ \text{CI} = P \left(1 + \text{rate}\right)^{\text{time}} - P \][/tex]
Given the rate is 4% (0.04) and time is 2 years:
[tex]\[ \text{CI} = P \left(1 + 0.04\right)^2 - P \][/tex]
[tex]\[ \text{CI} = P \left(1.04^2\right) - P \][/tex]
[tex]\[ \text{CI} = P \left(1.0816\right) - P \][/tex]
[tex]\[ \text{CI} = 1.0816P - P \][/tex]
[tex]\[ \text{CI} = 0.0816P \][/tex]
### Step 5: Difference Between Compound Interest and Simple Interest
According to the problem:
[tex]\[ \text{CI} - \text{SI} = 160 \][/tex]
[tex]\[ 0.0816P - 0.08P = 160 \][/tex]
### Step 6: Solve for the Principal Amount [tex]\( P \)[/tex]
[tex]\[ 0.0016P = 160 \][/tex]
[tex]\[ P = \frac{160}{0.0016} \][/tex]
[tex]\[ P = 100000 \][/tex]
### Step 7: Conclusion
The amount invested (in rupees) is [tex]\( \text{Rs.} 100,000 \)[/tex].
Therefore, the principal amount [tex]\( P \)[/tex] is 100000 rupees.
