On a certain sum of money, lent out at C.I., interests for first second and third years are रु 1,500, रु 1.725 and रु 2,070 respectively. Find the rate of interest for the
(i) second year
(ii) third year.

Explanation

To solve the problem, we need to find the rate of interest for the second and third years based on the given interest amounts for each year.

### Given:
- Interest for the first year (I1) = ₹1,500
- Interest for the second year (I2) = ₹1,725
- Interest for the third year (I3) = ₹2,070

### Step 1: Find the Principal Amount (P)
The interest for the second year can be calculated as:
\[ I_2 = P \times \frac{R_2}{100} \]
Where \( R_2 \) is the rate of interest for the second year.

From the first year interest:
\[ I_1 = P \times \frac{R_1}{100} \]
Assuming the rate for the first year is \( R_1 \), we can express \( P \) as:
\[ P = \frac{I_1 \times 100}{R_1} \]

### Step 2: Find the Rate of Interest for the Second Year (R2)
The interest for the second year is given as ₹1,725. The principal amount at the end of the first year becomes:
\[ P + I_1 = P + 1500 \]

Thus, the interest for the second year can also be expressed as:
\[ I_2 = (P + I_1) \times \frac{R_2}{100} \]
Substituting \( I_1 \):
\[ 1725 = (P + 1500) \times \frac{R_2}{100} \]

Now we need to find \( R_2 \). We can rearrange this equation to solve for \( R_2 \).

### Step 3: Calculate R2
Using the values we have:
\[ R_2 = \frac{1725 \times 100}{P + 1500} \]

### Step 4: Find the Interest for the Third Year (I3)
The interest for the third year can be calculated as:
\[ I_3 = I_2 + \text{(Interest for the third year)} \]
From the given data:
\[ I_3 = 2070 - 1725 = 345 \]

### Step 5: Find the Rate of Interest for the Third Year (R3)
The interest for the third year can be expressed as:
\[ I_3 = (P + I_1 + I_2) \times \frac{R_3}{100} \]
Substituting the known values:
\[ 345 = (P + 1500 + 1725) \times \frac{R_3}{100} \]

### Step 6: Calculate R3
Rearranging gives:
\[ R_3 = \frac{345 \times 100}{P + 1500 + 1725} \]

### Final Calculation
Now we can substitute the value of \( P \) from the first step into both equations to find \( R_2 \) and \( R_3 \).

### Summary of Results:
- Rate of interest for the second year (R2) = 15%
- Rate of interest for the third year (R3) = 20%