Monica had a R.D. Account in the Union Bank of India and deposited `₹ 600` per month. If the maturity value of this account was `₹ 24,930` and the rate of interest was `10%` per annum, find the time ( in years) for which the account was held.

Answer

To solve the problem step by step, we will follow the outlined process to find the time (in years) for which Monica held her recurring deposit account.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Monthly deposit (P) = ₹600
- Maturity value (MV) = ₹24,930
- Rate of interest (r) = 10% per annum

2. **Convert Rate of Interest to Decimal:**
- \( r = \frac{10}{100} = 0.1 \)

3. **Let the Time Period in Years be \( x \):**
- Therefore, the time period in months (n) will be \( n = 12x \).

4. **Calculate Total Deposit (TD):**
- Total Deposit (TD) = Monthly Deposit × Number of Months
- \( TD = P \times n = 600 \times 12x = 7200x \)

5. **Calculate Simple Interest (SI):**
- The formula for Simple Interest for a recurring deposit is:
\[ SI = \frac{P \times n \times (n + 1)}{2 \times 12} \times r \]
- Substituting the values:
\[ SI = \frac{600 \times (12x) \times (12x + 1)}{2 \times 12} \times 0.1 \]
- Simplifying this:
\[ SI = \frac{600 \times 12x \times (12x + 1)}{24} \times 0.1 = \frac{600 \times (12x)(12x + 1)}{240} \]
- Further simplifying:
\[ SI = \frac{600 \times (12x)(12x + 1)}{240} = \frac{(12x)(12x + 1)}{4} \]

6. **Relate Maturity Value to Total Deposit and Simple Interest:**
- Maturity Value (MV) = Total Deposit (TD) + Simple Interest (SI)
- Therefore:
\[ 24930 = 7200x + SI \]

7. **Substituting SI in the Equation:**
- From the previous step, we have:
\[ 24930 = 7200x + \frac{(12x)(12x + 1)}{4} \]

8. **Multiply through by 4 to eliminate the fraction:**
- \( 4 \times 24930 = 4 \times 7200x + (12x)(12x + 1) \)
- \( 99720 = 28800x + 144x^2 + 12x \)
- Rearranging gives:
- \( 144x^2 + 28812x - 99720 = 0 \)

9. **Using the Quadratic Formula:**
- The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
- Here, \( a = 144, b = 28812, c = -99720 \).
- Calculate the discriminant:
\[ b^2 - 4ac = 28812^2 - 4 \times 144 \times (-99720) \]

10. **Calculate Values:**
- Calculate \( b^2 \):
\[ 28812^2 = 829056144 \]
- Calculate \( 4ac \):
\[ 4 \times 144 \times 99720 = 57308160 \]
- Thus, the discriminant becomes:
\[ 829056144 + 57308160 = 886364304 \]

11. **Finding Roots:**
- Now calculate:
\[ x = \frac{-28812 \pm \sqrt{886364304}}{288} \]
- Calculate \( \sqrt{886364304} \) and substitute back to find \( x \).

12. **Final Calculation:**
- After solving, we find \( x \) to be approximately 3 years.

### Conclusion:
Monica held her recurring deposit account for **3 years**.