Priyanka has a recurring deposite account of Rs 1000 per month at 10% per annum. If she gets Rs 5550 as interest at the time of maturity, find the total time for which the account was held.

Answer

To solve the problem step-by-step, we will follow the principles of simple interest and the specific formula for recurring deposits.

### Step 1: Understand the given information
- Monthly deposit (P) = Rs 1000
- Rate of interest (R) = 10% per annum
- Total interest earned (I) = Rs 5550

### Step 2: Use the formula for interest in a recurring deposit account
The formula for the interest earned on a recurring deposit account is given by:
\[ I = \frac{n(n + 1) \times P \times R}{2400} \]
where:
- \( n \) = number of months
- \( P \) = monthly deposit
- \( R \) = rate of interest (in percentage)

### Step 3: Substitute the known values into the formula
Substituting the values into the formula:
\[ 5550 = \frac{n(n + 1) \times 1000 \times 10}{2400} \]

### Step 4: Simplify the equation
First, simplify the right side:
\[ 5550 = \frac{10000n(n + 1)}{2400} \]
\[ 5550 = \frac{10000n(n + 1)}{2400} \]
\[ 5550 = \frac{10000}{2400} n(n + 1) \]
\[ 5550 = \frac{25}{6} n(n + 1) \]

### Step 5: Multiply both sides by 6 to eliminate the fraction
\[ 5550 \times 6 = 25n(n + 1) \]
\[ 33300 = 25n(n + 1) \]

### Step 6: Divide both sides by 25
\[ \frac{33300}{25} = n(n + 1) \]
\[ 1332 = n(n + 1) \]

### Step 7: Rearrange into a standard quadratic equation
Rearranging gives:
\[ n^2 + n - 1332 = 0 \]

### Step 8: Factor the quadratic equation
To factor the equation, we need two numbers that multiply to -1332 and add to 1. The factors are:
\[ (n - 36)(n + 37) = 0 \]

### Step 9: Solve for n
Setting each factor to zero gives:
1. \( n - 36 = 0 \) → \( n = 36 \)
2. \( n + 37 = 0 \) → \( n = -37 \) (not valid since n cannot be negative)

Thus, \( n = 36 \) months.

### Step 10: Convert months to years
To find the total time in years:
\[ \text{Time in years} = \frac{n}{12} = \frac{36}{12} = 3 \text{ years} \]

### Final Answer
The total time for which the account was held is **3 years**.

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