Mr. Gupta opened a recurring deposit account in a bank. He deposited `₹ 2,500` per month for two years. At the time of maturity he got `₹ 67, 500` . Find :
(i) the total interest earned by Mr. Gupta
(ii) the rate of interest per annum.

Answer

To solve the problem step by step, we will break it down into two parts as requested.

### Step 1: Calculate the Total Principal Amount
Mr. Gupta deposits ₹ 2,500 per month for 2 years.

1. **Convert years to months:**
\[ \text{Total months} = 2 \text{ years} \times 12 \text{ months/year} = 24 \text{ months} \]

2. **Calculate the total principal amount:**
\[ \text{Principal Amount} = \text{Monthly Deposit} \times \text{Total Months} = 2500 \times 24 = ₹ 60,000 \]

### Step 2: Calculate the Total Interest Earned
The maturity amount received by Mr. Gupta is ₹ 67,500.

1. **Use the formula for maturity amount:**
\[ \text{Maturity Amount} = \text{Principal Amount} + \text{Interest Earned} \]

2. **Rearranging the formula to find interest:**
\[ \text{Interest Earned} = \text{Maturity Amount} - \text{Principal Amount} = 67,500 - 60,000 = ₹ 7,500 \]

### Step 3: Calculate the Rate of Interest
Using the formula for simple interest, we can find the rate of interest.

1. **Simple Interest Formula:**
\[ \text{Interest} = \frac{\text{Principal} \times n \times (n + 1) \times r}{24 \times 100} \]
Where:
- Principal = ₹ 2,500 (monthly deposit)
- \( n \) = 24 (number of months)
- Interest = ₹ 7,500

2. **Substituting the values into the formula:**
\[ 7500 = \frac{2500 \times 24 \times 25 \times r}{24 \times 100} \]

3. **Simplifying the equation:**
- The \( 24 \) cancels out:
\[ 7500 = \frac{2500 \times 25 \times r}{100} \]
- Multiply both sides by 100:
\[ 750000 = 2500 \times 25 \times r \]
- Simplifying further:
\[ 750000 = 62500 \times r \]
- Now, divide both sides by 62500:
\[ r = \frac{750000}{62500} = 12 \]

### Final Answers
(i) The total interest earned by Mr. Gupta is **₹ 7,500**.
(ii) The rate of interest per annum is **12%**.