The maturity value of a cumulative deposit account is Rs 1,20,400. If each monthly instalment for this account is Rs 1,600 and the rate of interest is 10% per year, find the time for which the account was held.

Answer

To find the time for which the account was held, we can follow these steps:

### Step 1: Identify the given values
- Maturity Value (A) = Rs. 1,20,400
- Monthly Installment (P) = Rs. 1,600
- Rate of Interest (r) = 10% per annum

### Step 2: Let the time period be n months
We need to find the value of n.

### Step 3: Calculate the principal for n months
The total principal deposited over n months is:
\[ P \times n = 1600n \]

### Step 4: Calculate the interest earned
The interest for each monthly installment can be calculated using the formula:
\[ \text{Interest} = P \times r \times t \]
Since each installment earns interest for a different period, we need to consider the time each installment is held.

The total interest earned can be calculated as:
\[ \text{Total Interest} = \frac{P \times r \times n(n + 1)}{2400} \]
Where:
- \( P = 1600 \)
- \( r = 10 \) (as a percentage)
- \( n \) is the number of months

### Step 5: Substitute the values into the interest formula
Substituting the values we have:
\[ \text{Total Interest} = \frac{1600 \times 10 \times n(n + 1)}{2400} \]
This simplifies to:
\[ \text{Total Interest} = \frac{16000n(n + 1)}{2400} = \frac{20n(n + 1)}{3} \]

### Step 6: Set up the equation for maturity value
The maturity value (A) is given by:
\[ A = \text{Total Principal} + \text{Total Interest} \]
Thus:
\[ 1,20,400 = 1600n + \frac{20n(n + 1)}{3} \]

### Step 7: Clear the fraction by multiplying through by 3
To eliminate the fraction, multiply the entire equation by 3:
\[ 3 \times 1,20,400 = 3 \times 1600n + 20n(n + 1) \]
This gives:
\[ 3,61,200 = 4800n + 20n(n + 1) \]

### Step 8: Rearrange the equation
Rearranging gives:
\[ 20n^2 + 20n + 4800n - 3,61,200 = 0 \]
\[ 20n^2 + 4820n - 3,61,200 = 0 \]

### Step 9: Simplify the equation
Dividing the entire equation by 20:
\[ n^2 + 241n - 18060 = 0 \]

### Step 10: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to -18060 and add to 241. The factors are:
\[ (n + 301)(n - 60) = 0 \]

### Step 11: Solve for n
Setting each factor to zero gives:
1. \( n + 301 = 0 \) → \( n = -301 \) (not possible)
2. \( n - 60 = 0 \) → \( n = 60 \)

### Conclusion
Thus, the time for which the account was held is:
\[ n = 60 \text{ months} \]
Which is equivalent to:
\[ 5 \text{ years} \]

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