On a certain sum of money, invested at the rate of 10 percent per annum compounded annually, the interest for the first year plus the interest for the third year is रु 2,652. Find the sum.

Explanation

To solve the problem step by step, we need to find the principal sum of money (P) given that the interest for the first year plus the interest for the third year equals ₹2,652 at a rate of 10% compounded annually.

### Step 1: Calculate the interest for the first year (I1)
The interest for the first year can be calculated using the formula:
\[ I_1 = P \times \frac{r}{100} \]
Where:
- \( P \) is the principal amount
- \( r \) is the rate of interest (10%)

So,
\[ I_1 = P \times \frac{10}{100} = 0.1P \]

### Step 2: Calculate the amount after the first year
The amount after the first year (A1) is:
\[ A_1 = P + I_1 = P + 0.1P = 1.1P \]

### Step 3: Calculate the interest for the second year (I2)
The interest for the second year is calculated on the amount after the first year:
\[ I_2 = A_1 \times \frac{r}{100} = 1.1P \times \frac{10}{100} = 0.11P \]

### Step 4: Calculate the amount after the second year
The amount after the second year (A2) is:
\[ A_2 = A_1 + I_2 = 1.1P + 0.11P = 1.21P \]

### Step 5: Calculate the interest for the third year (I3)
The interest for the third year is calculated on the amount after the second year:
\[ I_3 = A_2 \times \frac{r}{100} = 1.21P \times \frac{10}{100} = 0.121P \]

### Step 6: Set up the equation for the total interest
According to the problem, the sum of the interest for the first year and the third year is equal to ₹2,652:
\[ I_1 + I_3 = 2652 \]
Substituting the values we calculated:
\[ 0.1P + 0.121P = 2652 \]
\[ 0.221P = 2652 \]

### Step 7: Solve for P
Now, we can solve for P:
\[ P = \frac{2652}{0.221} \]
Calculating this gives:
\[ P = 12000 \]

Thus, the principal amount is ₹12,000.

### Final Answer
The sum of money (principal) is ₹12,000.

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