If CI on certain amount at certain rate of in terest for 2 years is 540.80 and simple interest for 1 year is 260 then find the amount and rate of interest?

Options

A. `9.5% `
B. `8%`
C. ` 8.5%`
D. `9%`

Answer

To solve the problem step by step, we need to find the amount and the rate of interest based on the given information.

### Step 1: Understand the given information
- Compound Interest (CI) for 2 years = 540.80
- Simple Interest (SI) for 1 year = 260

### Step 2: Calculate Simple Interest for 2 years
Since Simple Interest for 1 year is given as 260, we can find the Simple Interest for 2 years:
\[ \text{SI for 2 years} = \text{SI for 1 year} \times 2 = 260 \times 2 = 520 \]

### Step 3: Find the difference between CI and SI
Now, we need to find the difference between the Compound Interest for 2 years and the Simple Interest for 2 years:
\[ \text{Difference} = \text{CI} - \text{SI for 2 years} = 540.80 - 520 = 20.80 \]

### Step 4: Relate the difference to the rate of interest
The difference between CI and SI for 2 years is equal to the interest earned on the principal for the first year. This is calculated using the formula:
\[ \text{Difference} = \frac{P \times R}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest

From the previous steps, we know:
\[ 20.80 = \frac{P \times R}{100} \]

### Step 5: Express the Principal in terms of Simple Interest
We also know that the Simple Interest for 1 year is given by:
\[ \text{SI} = \frac{P \times R}{100} = 260 \]
Thus, we can express \( P \times R \) as:
\[ P \times R = 260 \times 100 = 26000 \]

### Step 6: Solve for the rate of interest
Now we have two equations:
1. \( P \times R = 26000 \)
2. \( 20.80 = \frac{P \times R}{100} \)

From the second equation, we can express \( P \) in terms of \( R \):
\[ P = \frac{20.80 \times 100}{R} = \frac{2080}{R} \]

### Step 7: Substitute \( P \) into the first equation
Substituting \( P \) into the first equation:
\[ \frac{2080}{R} \times R = 26000 \]
This simplifies to:
\[ 2080 = 26000 \]
This equation is not correct, so we need to find \( R \) directly from the first equation.

### Step 8: Calculate the rate of interest
From \( 20.80 = \frac{P \times R}{100} \) and substituting \( P \):
\[ R = \frac{20.80 \times 100}{P} \]
Now we can use \( P \) from the SI equation:
\[ P = \frac{260 \times 100}{R} \]
Substituting this into the equation gives:
\[ R = \frac{20.80 \times 100}{\frac{260 \times 100}{R}} \]
This leads to:
\[ R^2 = \frac{20.80 \times R}{260} \]
Solving for \( R \) gives:
\[ R = 8\% \]

### Step 9: Find the Principal Amount
Now substituting \( R \) back into the SI formula:
\[ 260 = \frac{P \times 8}{100} \]
This gives:
\[ P = \frac{260 \times 100}{8} = 3250 \]

### Final Results
- Amount (Principal) = 3250
- Rate of Interest = 8%