Mr. Richard has a recurring deposit account in a post office for `3` years at `7.5%` p.a. simple interest . If he gets `₹ 8,325` as interest at the time of maturity, find :
(i) the monthly instalment.
(ii) the amount of maturity.

Answer

To solve the problem step by step, we will first find the monthly installment and then the maturity amount.

### Step 1: Identify the given values
- Simple Interest (SI) = ₹ 8,325
- Time (T) = 3 years
- Rate of Interest (R) = 7.5% per annum

### Step 2: Convert time into months
Since the recurring deposit is for 3 years, we convert this into months:
\[ \text{Number of months (n)} = 3 \text{ years} \times 12 \text{ months/year} = 36 \text{ months} \]

### Step 3: Use the formula for Simple Interest
The formula for Simple Interest in the case of recurring deposits is:
\[ SI = P \times \frac{n(n+1)}{2 \times 12} \times \frac{R}{100} \]
Where:
- \( P \) = monthly installment
- \( n \) = number of months
- \( R \) = rate of interest

### Step 4: Substitute the known values into the formula
We know:
- \( SI = 8,325 \)
- \( n = 36 \)
- \( R = 7.5 \)

Substituting these values into the formula:
\[ 8,325 = P \times \frac{36 \times (36 + 1)}{2 \times 12} \times \frac{7.5}{100} \]

### Step 5: Simplify the equation
Calculating \( n(n+1) \):
\[ 36 \times 37 = 1332 \]
Now substituting this back into the equation:
\[ 8,325 = P \times \frac{1332}{2 \times 12} \times \frac{7.5}{100} \]
Calculating \( 2 \times 12 = 24 \):
\[ 8,325 = P \times \frac{1332}{24} \times \frac{7.5}{100} \]
Calculating \( \frac{1332}{24} = 55.5 \):
\[ 8,325 = P \times 55.5 \times \frac{7.5}{100} \]
Calculating \( 55.5 \times \frac{7.5}{100} = 4.1625 \):
\[ 8,325 = P \times 4.1625 \]

### Step 6: Solve for P (monthly installment)
\[ P = \frac{8,325}{4.1625} \approx 2,000 \]

### Step 7: Calculate the maturity amount
The maturity amount (M) is given by:
\[ M = P \times n + SI \]
Substituting the values:
\[ M = 2,000 \times 36 + 8,325 \]
Calculating:
\[ M = 72,000 + 8,325 = 80,325 \]

### Final Answers:
(i) The monthly installment is ₹ 2,000.
(ii) The amount of maturity is ₹ 80,325.

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