The compound interest, calculated yearly, on a certain sum of money for the second year is रु 1,089 and for the third year it is? रु 1,197.90. Calculate the rate of interest and the sum of money.

Explanation

To solve the problem step by step, we will follow the given information and derive the necessary equations to find the rate of interest and the principal sum of money.

### Step 1: Identify the Given Information
- Compound Interest (CI) for the second year = Rs. 1,089
- Compound Interest (CI) for the third year = Rs. 1,197.90

### Step 2: Find the Difference in CI
The difference in compound interest between the third year and the second year can be calculated as follows:
\[ \text{Difference} = \text{CI for 3rd year} - \text{CI for 2nd year} = 1197.90 - 1089 = 108.90 \]

### Step 3: Relate the Difference to the Rate of Interest
The difference in compound interest between two successive years is equal to the interest on the amount at the end of the second year. We can use this difference to find the rate of interest.

Let:
- CI for 2nd year = Rs. 1,089
- Interest for 1 year = Rs. 108.90

Using the formula for interest:
\[ \text{Interest} = \frac{\text{Rate} \times \text{Principal} \times \text{Time}}{100} \]
Where:
- Interest = Rs. 108.90
- Principal = Rs. 1,089 (CI for 2nd year)
- Time = 1 year

Rearranging the formula to find the rate:
\[ \text{Rate} = \frac{\text{Interest} \times 100}{\text{Principal} \times \text{Time}} = \frac{108.90 \times 100}{1089 \times 1} \]
Calculating this gives:
\[ \text{Rate} = 10\% \]

### Step 4: Calculate the Principal Sum of Money
Now that we have the rate of interest, we can find the principal sum of money. Let the principal be \( x \).

Using the formula for compound interest:
1. Interest for the first year:
\[ \text{Interest}_1 = \frac{x \times 10 \times 1}{100} = \frac{10x}{100} = \frac{x}{10} \]

2. Amount after the first year:
\[ \text{Amount}_1 = x + \text{Interest}_1 = x + \frac{x}{10} = \frac{11x}{10} \]

3. Interest for the second year:
\[ \text{Interest}_2 = \frac{\text{Amount}_1 \times 10 \times 1}{100} = \frac{\frac{11x}{10} \times 10 \times 1}{100} = \frac{11x}{100} \]

We know that the CI for the second year is Rs. 1,089, so we can set up the equation:
\[ \frac{11x}{100} = 1089 \]

### Step 5: Solve for \( x \)
Multiplying both sides by 100:
\[ 11x = 108900 \]
Now, divide by 11:
\[ x = \frac{108900}{11} = 9900 \]

### Final Results
- The principal sum of money is Rs. 9,900.
- The rate of interest is 10%.