Raman deposited `₹7600` in a bank. He withdrew `₹3000` after `2` years. At the end of `7` years, he receives an amount of `₹7656`. Find the rate of interest.

Answer

To find the rate of interest in this problem, we will follow these steps:

### Step 1: Identify the given values
- Principal (P) = ₹7600
- Amount after 7 years (A) = ₹7656
- Amount withdrawn after 2 years = ₹3000
- Time (T) = 7 years

### Step 2: Calculate the remaining principal after withdrawal
After 2 years, Raman withdrew ₹3000 from his initial deposit of ₹7600.
Remaining Principal (P') = Initial Principal - Amount Withdrawn
\[ P' = 7600 - 3000 = ₹4600 \]

### Step 3: Calculate the total simple interest earned
The total amount received after 7 years is ₹7656. The total simple interest (SI) can be calculated using the formula:
\[ A = P + SI \]
Thus,
\[ SI = A - P' \]
Substituting the values we have:
\[ SI = 7656 - 4600 = ₹3056 \]

### Step 4: Calculate Simple Interest for the first 2 years
For the first 2 years, the simple interest is calculated on the initial principal (P = ₹7600):
Using the formula for Simple Interest:
\[ SI_1 = \frac{P \times R \times T}{100} \]
Substituting the values:
\[ SI_1 = \frac{7600 \times R \times 2}{100} = 152R \]

### Step 5: Calculate Simple Interest for the next 5 years
After 2 years, the remaining principal is ₹4600. The simple interest for the next 5 years is:
\[ SI_2 = \frac{P' \times R \times T}{100} \]
Substituting the values:
\[ SI_2 = \frac{4600 \times R \times 5}{100} = 230R \]

### Step 6: Set up the equation for total Simple Interest
The total Simple Interest is the sum of SI_1 and SI_2:
\[ SI = SI_1 + SI_2 \]
Thus,
\[ 3056 = 152R + 230R \]
Combining the terms gives:
\[ 3056 = 382R \]

### Step 7: Solve for R (Rate of Interest)
To find R, divide both sides by 382:
\[ R = \frac{3056}{382} \]
Calculating this gives:
\[ R = 8\% \]

### Final Answer
The rate of interest is **8%**.

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