A cummulative deposit account of monthly instalment of 3,600 at 9% p.a. simple interest earns an interest of 17,982. Find the number of instalments paid.

Answer

To solve the problem, we need to find the number of monthly installments paid into a cumulative deposit account that earns simple interest. Here’s how we can approach the solution step by step:

### Step 1: Understand the Problem
We have a cumulative deposit account where:
- Monthly installment (P) = 3,600
- Rate of interest (R) = 9% per annum
- Total interest earned (SI) = 17,982

We need to find the number of installments (n).

### Step 2: Calculate the Simple Interest for One Month
The formula for simple interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount
- \(R\) = Rate of interest
- \(T\) = Time in years

For one month, \(T = \frac{1}{12}\) years. Thus, we can substitute the values:
\[ SI_{1 \text{ month}} = \frac{3600 \times 9 \times \frac{1}{12}}{100} \]

### Step 3: Simplify the Calculation
Calculating the above expression:
\[ SI_{1 \text{ month}} = \frac{3600 \times 9}{1200} = \frac{32400}{1200} = 27 \]
So, the interest earned in the first month is 27.

### Step 4: Calculate Total Interest Earned Over n Months
Since this is a cumulative deposit account, the interest for each month increases cumulatively. Thus, for n months, the total interest can be represented as:
\[ SI_{total} = 27 \times (1 + 2 + 3 + ... + n) \]
The sum of the first n natural numbers is given by:
\[ \text{Sum} = \frac{n(n + 1)}{2} \]
So, we can write:
\[ SI_{total} = 27 \times \frac{n(n + 1)}{2} \]

### Step 5: Set Up the Equation
We know that the total interest earned is 17,982, so we set up the equation:
\[ 27 \times \frac{n(n + 1)}{2} = 17982 \]

### Step 6: Solve for n
First, simplify the equation:
\[ \frac{27n(n + 1)}{2} = 17982 \]
Multiply both sides by 2:
\[ 27n(n + 1) = 35964 \]
Now, divide both sides by 27:
\[ n(n + 1) = \frac{35964}{27} = 1332 \]
This simplifies to:
\[ n^2 + n - 1332 = 0 \]

### Step 7: Factor the Quadratic Equation
We can factor the quadratic equation:
\[ n^2 + n - 1332 = 0 \]
To factor, we look for two numbers that multiply to -1332 and add to 1. The factors are:
\[ (n - 36)(n + 37) = 0 \]
Setting each factor to zero gives:
\[ n - 36 = 0 \quad \text{or} \quad n + 37 = 0 \]
Thus, \(n = 36\) or \(n = -37\).

### Step 8: Conclusion
Since the number of installments cannot be negative, we take:
\[ n = 36 \]
Therefore, the number of installments paid is **36**.