If simple interest is 60 for 3 years and CI is 40.80 for 2 years at certain rate of interest on some amount then find the amount and rate of interest?

Options

A. ₹ 500, 4%
B. ₹ 250,8%
C. ₹ 800, 2.5%
D. ₹ 1000, 2%

Answer

To solve the problem step-by-step, we need to find the amount and the rate of interest based on the given simple interest (SI) and compound interest (CI).

### Step 1: Understand the given information
- Simple Interest (SI) for 3 years = 60
- Compound Interest (CI) for 2 years = 40.80

### Step 2: Calculate the rate of interest using the difference between CI and SI
The difference between the compound interest and simple interest can be calculated as follows:

\[ \text{Difference} = \text{CI} - \text{SI} \]

However, we need to express SI in terms of the same time period as CI. Since we have SI for 3 years, we can find the annual interest rate first.

### Step 3: Calculate the annual simple interest
The annual simple interest can be calculated as:

\[ \text{Annual SI} = \frac{\text{Total SI}}{\text{Number of Years}} = \frac{60}{3} = 20 \]

### Step 4: Calculate the total amount of interest earned in 2 years using SI
Using the annual simple interest, we can find the total simple interest for 2 years:

\[ \text{SI for 2 years} = 20 \times 2 = 40 \]

### Step 5: Find the difference between CI and SI for 2 years
Now we can calculate the difference between the compound interest and the simple interest for 2 years:

\[ \text{Difference} = \text{CI} - \text{SI for 2 years} = 40.80 - 40 = 0.80 \]

### Step 6: Relate the difference to the principal and rate
The difference between CI and SI can also be expressed as:

\[ \text{Difference} = \frac{P \times r^2}{100^2} \]

Where \( P \) is the principal amount and \( r \) is the rate of interest. We know that:

\[ 0.80 = \frac{P \times r^2}{100^2} \]

### Step 7: Calculate the rate of interest
From the previous calculations, we can also find the rate of interest. The annual SI is 20, which corresponds to:

\[ \text{SI} = \frac{P \times r \times t}{100} \]

For 3 years:

\[ 60 = \frac{P \times r \times 3}{100} \]

From this, we can express \( P \) in terms of \( r \):

\[ P = \frac{60 \times 100}{3r} = \frac{6000}{3r} = \frac{2000}{r} \]

### Step 8: Substitute \( P \) back to find \( r \)
Now substitute \( P \) back into the difference equation:

\[ 0.80 = \frac{\left(\frac{2000}{r}\right) \times r^2}{100^2} \]

This simplifies to:

\[ 0.80 = \frac{2000r}{10000} \]

\[ 0.80 = \frac{2r}{10} \]

\[ r = 4 \]

### Step 9: Calculate the principal amount
Now we can find \( P \):

\[ P = \frac{2000}{r} = \frac{2000}{4} = 500 \]

### Final Answers
- Amount (Principal) = 500
- Rate of Interest = 4%