If CI on certain amount at certain rate of in terest for 2 years is 410 and simple interest for 2 years is 400 then find the amount and rate of interest?

Options

A. A. ₹ 5000,4%
B. B. ₹4500,4%
C. C. ₹4000,5%
D. D. ₹4500,5%

Answer

To solve the problem step by step, we will use the relationships between Compound Interest (CI), Simple Interest (SI), the principal amount (P), the rate of interest (R), and the time period (T).

### Step 1: Understanding the given information
- CI for 2 years = 410
- SI for 2 years = 400

### Step 2: Find the difference between CI and SI
The difference between Compound Interest and Simple Interest for 2 years can be expressed as:
\[ \text{CI} - \text{SI} = P \times \left(\frac{R}{100}\right)^2 \]
Where \( P \) is the principal amount and \( R \) is the rate of interest.

From the problem:
\[ 410 - 400 = P \times \left(\frac{R}{100}\right)^2 \]
\[ 10 = P \times \left(\frac{R}{100}\right)^2 \] (Equation 1)

### Step 3: Calculate the Simple Interest for 1 year
The Simple Interest for 2 years is given as 400, so the Simple Interest for 1 year (SI) is:
\[ \text{SI for 1 year} = \frac{400}{2} = 200 \]

### Step 4: Relate SI to Principal and Rate
The formula for Simple Interest is:
\[ \text{SI} = P \times \left(\frac{R}{100}\right) \]
For 1 year:
\[ 200 = P \times \left(\frac{R}{100}\right) \] (Equation 2)

### Step 5: Solve the equations
From Equation 2, we can express \( P \) in terms of \( R \):
\[ P = \frac{200 \times 100}{R} = \frac{20000}{R} \]

### Step 6: Substitute \( P \) in Equation 1
Substituting \( P \) in Equation 1:
\[ 10 = \left(\frac{20000}{R}\right) \times \left(\frac{R}{100}\right)^2 \]
\[ 10 = \frac{20000 \times R}{R^2 \times 10000} \]
\[ 10 = \frac{20000}{10000 \times R} \]
\[ 10R = 2 \]
\[ R = \frac{2}{10} = 0.2 \text{ (This is incorrect; let's simplify correctly)} \]
\[ R = 5 \]

### Step 7: Find the Principal Amount \( P \)
Now substituting \( R = 5 \) back into Equation 2:
\[ 200 = P \times \left(\frac{5}{100}\right) \]
\[ 200 = \frac{P \times 5}{100} \]
\[ 200 = \frac{P}{20} \]
\[ P = 200 \times 20 = 4000 \]

### Final Answers
- Amount (Principal) \( P = 4000 \)
- Rate of Interest \( R = 5\% \)