Gopal has a cumulative deposit account and deposits `₹ 900` per month for a period of `4` years. If he gets `₹ 52,020` at the time of maturity, find the rate of interest.

Answer

To find the rate of interest for Gopal's cumulative deposit account, we can follow these steps:

### Step 1: Identify the given values
- Monthly deposit (P) = ₹900
- Total duration = 4 years
- Maturity amount (A) = ₹52,020

### Step 2: Calculate the total number of months
Total number of months (N) = 4 years × 12 months/year = 48 months.

### Step 3: Use the formula for Simple Interest (SI) in a recurring deposit
The formula for Simple Interest in a recurring deposit account is:
\[ SI = \frac{P \times N \times (N + 1)}{2 \times 12} \times \frac{r}{100} \]
Where:
- P = Monthly deposit
- N = Total number of months
- r = Rate of interest (which we need to find)

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:
\[ SI = \frac{900 \times 48 \times (48 + 1)}{2 \times 12} \times \frac{r}{100} \]

### Step 5: Simplify the equation
Calculating:
- \( N + 1 = 48 + 1 = 49 \)
- So, \( SI = \frac{900 \times 48 \times 49}{2 \times 12} \times \frac{r}{100} \)

Now, simplify:
- \( \frac{48}{12} = 4 \)
- Therefore, \( SI = \frac{900 \times 49 \times 4}{2} \times \frac{r}{100} \)
- \( SI = 900 \times 98 \times \frac{r}{100} \)
- \( SI = 882 \times r \)

### Step 6: Relate SI to the maturity amount
The maturity amount (A) is given by:
\[ A = P \times N + SI \]
Substituting the known values:
\[ 52,020 = 900 \times 48 + 882 \times r \]

### Step 7: Calculate the total principal amount
Calculating \( 900 \times 48 \):
- \( 900 \times 48 = 43,200 \)

### Step 8: Substitute back into the equation
Now, substituting this back into the equation:
\[ 52,020 = 43,200 + 882 \times r \]

### Step 9: Isolate r
Rearranging gives:
\[ 882 \times r = 52,020 - 43,200 \]
\[ 882 \times r = 8,820 \]

### Step 10: Solve for r
Now, divide both sides by 882:
\[ r = \frac{8,820}{882} \]
\[ r = 10\% \]

### Conclusion
The rate of interest is **10%**.

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