On certain amount, the rate of interest for first year is 10% and second year is 8% and difference between SI and CI for 2nd year is 64 then find the amount?

Options

A. ₹6400
B. ₹7200
C. ₹8000
D. ₹8800

Answer

To solve the problem step by step, we will follow the information given in the question and the calculations outlined in the video transcript.

### Step 1: Identify the given information
- Rate of interest for the first year (R1) = 10%
- Rate of interest for the second year (R2) = 8%
- Difference between Compound Interest (CI) and Simple Interest (SI) for the second year = 64

### Step 2: Calculate Simple Interest for the first year
The formula for Simple Interest (SI) for the first year is:
\[ SI_1 = \frac{P \times R_1 \times 1}{100} \]
Substituting the values:
\[ SI_1 = \frac{P \times 10 \times 1}{100} = \frac{10P}{100} = \frac{P}{10} \]

### Step 3: Calculate the Principal for the second year
At the beginning of the second year, the principal amount becomes:
\[ P + SI_1 = P + \frac{P}{10} = \frac{10P}{10} + \frac{P}{10} = \frac{11P}{10} \]

### Step 4: Calculate Simple Interest for the second year
The formula for Simple Interest (SI) for the second year is:
\[ SI_2 = \frac{P \times R_2 \times 1}{100} \]
Substituting the values:
\[ SI_2 = \frac{P \times 8 \times 1}{100} = \frac{8P}{100} = \frac{2P}{25} \]

### Step 5: Calculate Compound Interest for the second year
The Compound Interest (CI) for the second year is calculated based on the new principal:
\[ CI_2 = \frac{11P}{10} \times \frac{8}{100} = \frac{88P}{1000} = \frac{22P}{250} \]

### Step 6: Set up the equation for the difference between CI and SI
According to the problem, the difference between CI and SI for the second year is given as 64:
\[ CI_2 - SI_2 = 64 \]
Substituting the values we calculated:
\[ \frac{22P}{250} - \frac{2P}{25} = 64 \]

### Step 7: Simplify the equation
To simplify, convert \(\frac{2P}{25}\) to have a common denominator:
\[ \frac{2P}{25} = \frac{20P}{250} \]
Now substituting back into the equation:
\[ \frac{22P}{250} - \frac{20P}{250} = 64 \]
This simplifies to:
\[ \frac{2P}{250} = 64 \]

### Step 8: Solve for P
Multiply both sides by 250:
\[ 2P = 64 \times 250 \]
Calculating the right side:
\[ 2P = 16000 \]
Now divide by 2:
\[ P = 8000 \]

### Conclusion
The amount (Principal, P) is **8000**.

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