The principal amount which yields a compund interest of ? 208 in the second year at 4% is

Options

A. ? 5000
B. ? 10000
C. ? 130000
D. ? 6500

Answer

To solve the problem of finding the principal amount that yields a compound interest of ₹208 in the second year at a rate of 4%, we can follow these steps:

### Step 1: Understand the Concept of Compound Interest
In compound interest, the interest for each year is calculated on the principal plus the interest accumulated from previous years.

### Step 2: Identify the Rate and Interest
We are given:
- Rate of interest (R) = 4%
- Compound interest in the second year (CI) = ₹208

### Step 3: Calculate the Interest for the First Year
The interest for the first year can be calculated using the formula:
\[ \text{Interest for Year 1} = \frac{P \times R}{100} \]
Where P is the principal amount.

### Step 4: Calculate the Interest for the Second Year
The interest for the second year is calculated on the new principal, which is the original principal plus the interest from the first year:
\[ \text{Interest for Year 2} = \frac{(P + \text{Interest for Year 1}) \times R}{100} \]
This can also be expressed as:
\[ \text{Interest for Year 2} = \frac{P \times R}{100} + \frac{(P \times R^2)}{100^2} \]
Where \( R^2 \) is the rate squared.

### Step 5: Set Up the Equation
Since we know the interest for the second year is ₹208, we can set up the equation:
\[ \frac{P \times 4}{100} + \frac{(P \times 4^2)}{100^2} = 208 \]

### Step 6: Simplify the Equation
Substituting \( R = 4 \):
\[ \frac{P \times 4}{100} + \frac{(P \times 16)}{10000} = 208 \]
This simplifies to:
\[ \frac{4P}{100} + \frac{16P}{10000} = 208 \]

### Step 7: Find a Common Denominator
The common denominator is 10000:
\[ \frac{400P + 16P}{10000} = 208 \]
This simplifies to:
\[ \frac{416P}{10000} = 208 \]

### Step 8: Solve for P
Multiply both sides by 10000:
\[ 416P = 208 \times 10000 \]
\[ 416P = 2080000 \]
Now, divide both sides by 416:
\[ P = \frac{2080000}{416} \]
Calculating this gives:
\[ P = 5000 \]

### Final Answer
The principal amount is ₹5000.

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