Deepa has a `4` - year recurring deposit account in a bank and deposits `₹ 1,800` per month. If she gets `₹ 1,08,450` at the time of maturity, find the rate of interest.

Answer

To find the rate of interest for Deepa's recurring deposit account, we can follow these steps:

### Step 1: Identify the given values
- Monthly deposit (P) = ₹1,800
- Duration of the deposit = 4 years
- Total maturity amount = ₹1,08,450

### Step 2: Calculate the total number of months
Since Deepa deposits money for 4 years, we need to convert this into months:
\[ \text{Number of months (n)} = 4 \times 12 = 48 \text{ months} \]

### Step 3: Use the formula for Simple Interest (SI) for Recurring Deposits
The formula for the Simple Interest (SI) for a recurring deposit is:
\[ SI = P \times \frac{n(n + 1)}{2} \times \frac{r}{100 \times 12} \]
Where:
- \( P \) = monthly deposit
- \( n \) = number of months
- \( r \) = rate of interest per annum

### Step 4: Substitute the known values into the SI formula
Substituting \( P = 1800 \) and \( n = 48 \):
\[ SI = 1800 \times \frac{48 \times (48 + 1)}{2} \times \frac{r}{100 \times 12} \]
Calculating \( n(n + 1) \):
\[ 48 \times 49 = 2352 \]
Now substituting this back:
\[ SI = 1800 \times \frac{2352}{2} \times \frac{r}{1200} \]
Calculating \( \frac{2352}{2} = 1176 \):
\[ SI = 1800 \times 1176 \times \frac{r}{1200} \]

### Step 5: Simplify the SI expression
\[ SI = 1800 \times 1176 \div 1200 \times r \]
Calculating \( 1800 \div 1200 = 1.5 \):
\[ SI = 1.5 \times 1176 \times r \]
Calculating \( 1.5 \times 1176 = 1764 \):
\[ SI = 1764r \]

### Step 6: Relate SI to the maturity amount
The maturity amount (M) is given by:
\[ M = \text{Total deposits} + SI \]
The total deposits over 48 months:
\[ \text{Total deposits} = 1800 \times 48 = 86400 \]
Thus, we have:
\[ 1,08,450 = 86400 + 1764r \]

### Step 7: Solve for r
Rearranging the equation:
\[ 1764r = 1,08,450 - 86400 \]
Calculating the right side:
\[ 1,08,450 - 86400 = 22,050 \]
So:
\[ 1764r = 22,050 \]
Now, divide both sides by 1764:
\[ r = \frac{22,050}{1764} \]

### Step 8: Calculate r
Calculating the division:
\[ r \approx 12.5 \]

### Conclusion
The rate of interest \( r \) is approximately:
\[ \text{Rate of interest} = 12.5\% \text{ per annum} \]

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