A man has a Recurring Deposit Account in a bank for `3(1)/(2)` years. If the rate of interest is `12%` per annum and the man gets `₹ 10, 206` on maturity. Find the value of monthly instalments.

Answer

To solve the problem step by step, we will follow the method of calculating the monthly installments for the recurring deposit account.

### Step 1: Understand the given information
- Duration of the deposit: \(3 \frac{1}{2}\) years
- Rate of interest: \(12\%\) per annum
- Maturity amount: ₹10,206

### Step 2: Convert the duration into months
The duration in years is \(3 \frac{1}{2} = 3.5\) years. To convert this into months:
\[ \text{Number of months} = 3.5 \times 12 = 42 \text{ months} \]

### Step 3: Define the variables
Let \(Y\) be the monthly installment amount. The total number of months \(n = 42\).

### Step 4: Calculate the Simple Interest (SI)
The formula for Simple Interest in the case of recurring deposits is:
\[ SI = \frac{P \cdot n(n + 1)}{2 \cdot 12} \cdot \frac{r}{100} \]
Where:
- \(P\) is the monthly installment (which is \(Y\)),
- \(n\) is the number of months (which is 42),
- \(r\) is the rate of interest (which is 12%).

Substituting the values into the formula:
\[ SI = \frac{Y \cdot 42 \cdot (42 + 1)}{2 \cdot 12} \cdot \frac{12}{100} \]
\[ SI = \frac{Y \cdot 42 \cdot 43}{2 \cdot 12} \cdot \frac{12}{100} \]
\[ SI = \frac{Y \cdot 42 \cdot 43}{2 \cdot 100} \]
\[ SI = \frac{Y \cdot 903}{100} = 9.03Y \]

### Step 5: Write the equation for maturity amount
The maturity amount is the sum of the total principal and the Simple Interest:
\[ \text{Maturity Amount} = \text{Principal} + SI \]
The total principal for 42 months is \(42Y\), thus:
\[ 10,206 = 42Y + 9.03Y \]
\[ 10,206 = 51.03Y \]

### Step 6: Solve for \(Y\)
To find \(Y\):
\[ Y = \frac{10,206}{51.03} \]
Calculating \(Y\):
\[ Y = 200 \]

### Conclusion
The value of the monthly installments is ₹200.

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