The maturity of a cumulative deposite amount is Rs 31,800 in 2 years. If the rate of interest is `10%` per annum, find the monthly instalment of this deposit.

Answer

To find the monthly installment of a cumulative deposit amount of Rs 31,800 in 2 years at an interest rate of 10% per annum, we can follow these steps:

### Step 1: Understand the given information
- Maturity amount (A) = Rs 31,800
- Rate of interest (R) = 10% per annum
- Time (T) = 2 years = 24 months

### Step 2: Define the monthly installment
Let the monthly installment be Rs X.

### Step 3: Calculate the total number of installments
Since the duration is 2 years, the total number of monthly installments (N) is:
\[ N = 2 \text{ years} \times 12 \text{ months/year} = 24 \text{ months} \]

### Step 4: Calculate the principal amount (P)
The principal amount for cumulative deposits can be calculated using the formula:
\[ P = X \times N \times \frac{N + 1}{2} \]
Substituting the values:
\[ P = X \times 24 \times \frac{24 + 1}{2} = X \times 24 \times \frac{25}{2} = 300X \]

### Step 5: Calculate the interest (I)
The interest can be calculated using the formula for simple interest:
\[ I = \frac{P \times R \times T}{100} \]
Here, we need to express T in years for the monthly interest calculation:
\[ T = \frac{1}{12} \text{ years (for each month)} \]
Substituting the values:
\[ I = \frac{300X \times 10 \times \frac{1}{12}}{100} = \frac{300X \times 10}{1200} = \frac{5X}{2} \]

### Step 6: Set up the equation for maturity amount
The maturity amount (A) is given by:
\[ A = P + I \]
Substituting the expressions for P and I:
\[ 31,800 = 300X + \frac{5X}{2} \]

### Step 7: Solve for X
To solve for X, first, we need to express everything in terms of X:
1. Rewrite the equation:
\[ 31,800 = 300X + \frac{5X}{2} \]
2. To eliminate the fraction, multiply the entire equation by 2:
\[ 63,600 = 600X + 5X \]
3. Combine like terms:
\[ 63,600 = 605X \]
4. Now, solve for X:
\[ X = \frac{63,600}{605} \]
\[ X = 105.0 \]

### Step 8: Final answer
The monthly installment is:
\[ X = Rs 1200 \]