A sum of `₹15,500` was deposited by Arnav in a bank. He deposited an additional `₹4000` after `3` years. At the end of `10` years, he receives an amount of `₹30,480`. Find the rate of interest.

Answer

To find the rate of interest, we can break down the problem step by step.

### Step 1: Identify the initial deposit and time period
Arnav initially deposited ₹15,500 for a period of 10 years.

### Step 2: Calculate the interest on the initial deposit
The formula for simple interest is:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount (initial deposit)
- \(R\) = Rate of interest (which we need to find)
- \(T\) = Time period in years

For the initial deposit of ₹15,500 over 10 years, the interest can be expressed as:
\[ \text{Interest from initial deposit} = \frac{15500 \times R \times 10}{100} = 1550R \]

### Step 3: Identify the additional deposit and its time period
Arnav deposited an additional ₹4,000 after 3 years. This means this amount will earn interest for 7 years (10 years - 3 years).

### Step 4: Calculate the interest on the additional deposit
Using the same formula, the interest from the additional deposit can be expressed as:
\[ \text{Interest from additional deposit} = \frac{4000 \times R \times 7}{100} = 280R \]

### Step 5: Write the equation for the total amount received
According to the problem, the total amount received at the end of 10 years is ₹30,480. Therefore, we can write the equation as:
\[ \text{Total Amount} = \text{Initial Deposit} + \text{Interest from Initial Deposit} + \text{Additional Deposit} + \text{Interest from Additional Deposit} \]
Substituting the values we have:
\[ 30480 = 15500 + 1550R + 4000 + 280R \]

### Step 6: Simplify the equation
Combine like terms:
\[ 30480 = 19500 + 1830R \]

### Step 7: Isolate the term with R
Subtract 19500 from both sides:
\[ 30480 - 19500 = 1830R \]
\[ 10980 = 1830R \]

### Step 8: Solve for R
Now, divide both sides by 1830 to find R:
\[ R = \frac{10980}{1830} = 6 \]

### Conclusion
The rate of interest \(R\) is **6%**.

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