A sum of money at compound interst amount in two years to 2809, and in three years to 2977.54. Find the original sum.

Options

A. ₹2000
B. ₹2100
C. ₹ 2200
D. ₹2500

Answer

To solve the problem step by step, we will follow the process of calculating the original sum using the information provided about the amounts at compound interest.

### Step 1: Identify the amounts
We know:
- Amount after 2 years (A2) = 2809
- Amount after 3 years (A3) = 2977.54

### Step 2: Calculate the interest for the third year
The interest earned in the third year can be calculated as:
\[ \text{Interest for 3rd year} = A3 - A2 \]
Substituting the values:
\[ \text{Interest for 3rd year} = 2977.54 - 2809 = 168.54 \]

### Step 3: Calculate the rate of interest
The rate of interest can be calculated using the formula:
\[ \text{Rate} = \frac{\text{Interest for 3rd year}}{A2} \times 100 \]
Substituting the values:
\[ \text{Rate} = \frac{168.54}{2809} \times 100 \]
Calculating this gives:
\[ \text{Rate} \approx 6\% \]

### Step 4: Use the amount formula to find the principal
The formula for the amount in compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- A = Amount after n years
- P = Principal (original sum)
- R = Rate of interest
- n = Number of years

For our case:
- A = 2809 (amount after 2 years)
- R = 6%
- n = 2

Substituting these values into the formula:
\[ 2809 = P \left(1 + \frac{6}{100}\right)^2 \]
This simplifies to:
\[ 2809 = P \left(1.06\right)^2 \]
Calculating \( (1.06)^2 \):
\[ (1.06)^2 = 1.1236 \]
So we have:
\[ 2809 = P \times 1.1236 \]

### Step 5: Solve for P
To find P, rearrange the equation:
\[ P = \frac{2809}{1.1236} \]
Calculating this gives:
\[ P \approx 2500 \]

### Final Answer
The original sum (principal) is approximately **2500**.

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