A sum is invested at compound interest compounded yearly. If the interest for two successive years be रु 5,700 and रु 7,410, calculate the rate of interest .

Explanation

To solve the problem of finding the rate of interest given the compound interest for two successive years, we can follow these steps:

### Step 1: Understand the Given Information
We know that the interest for the first year is रु 5,700 and for the second year is रु 7,410.

### Step 2: Calculate the Interest for the Second Year
The interest earned in the second year can be calculated by subtracting the interest of the first year from the interest of the second year:
\[ \text{Interest for second year} = \text{Interest for second year} - \text{Interest for first year} \]
\[ = 7,410 - 5,700 = 1,710 \text{ rupees} \]

### Step 3: Relate the Interests to Principal and Rate
In compound interest, the interest for the second year is calculated on the total amount after the first year's interest has been added to the principal. Therefore, we can express the interest for the second year as:
\[ \text{Interest for second year} = \text{Principal} \times \text{Rate} \times \frac{1}{100} \]
Let \( P \) be the principal and \( r \) be the rate of interest. The interest for the first year can be expressed as:
\[ 5,700 = P \times \frac{r}{100} \]

### Step 4: Set Up the Equations
From the first year’s interest:
\[ P \times \frac{r}{100} = 5,700 \quad \text{(1)} \]
From the second year’s interest:
\[ P \times (1 + \frac{r}{100}) \times \frac{r}{100} = 7,410 \quad \text{(2)} \]

### Step 5: Substitute and Solve for Rate
From equation (1), we can express \( P \) in terms of \( r \):
\[ P = \frac{5,700 \times 100}{r} \quad \text{(3)} \]
Now, substitute equation (3) into equation (2):
\[ \frac{5,700 \times 100}{r} \times (1 + \frac{r}{100}) \times \frac{r}{100} = 7,410 \]
Simplifying this gives:
\[ \frac{5,700 \times 100 \times (100 + r)}{r^2} = 7,410 \]
Cross-multiplying to eliminate the fraction:
\[ 5,700 \times 100 \times (100 + r) = 7,410 \times r^2 \]

### Step 6: Rearranging and Solving
Expanding and rearranging the equation will lead to a quadratic equation in terms of \( r \). After solving for \( r \), we find:
\[ r = 30 \]

### Conclusion
Thus, the rate of interest is **30%**.

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