To solve this problem, we need to find the original principal sum of money using the compound interest formula. Here are the step-by-step details:
1. Identify the given information:
- The amount after 2 years (A₂) is Rs. 1530.
- The amount after 3 years (A₃) is Rs. 1545.
2. Calculate the annual compound interest rate (r):
- Given that the sum grows from Rs. 1530 to Rs. 1545 in one year,
- The interest rate can be found using the formula:
[tex]\[
\left(1 + r\right) = \frac{A₃}{A₂}
\][/tex]
- Plugging in the values:
[tex]\[
1 + r = \frac{1545}{1530} \approx 1.0098039215686274
\][/tex]
- Solving for r:
[tex]\[
r \approx 1.0098039215686274 - 1 = 0.0098039215686274 \approx 0.98\%
\][/tex]
3. Calculate the principal amount (P):
- We will use the compound interest formula for year 2:
[tex]\[
A₂ = P \left(1 + r\right)^2
\][/tex]
- Rearrange to solve for P:
[tex]\[
P = \frac{A₂}{\left(1 + r\right)^2}
\][/tex]
- Substitute the values:
[tex]\[
P = \frac{1530}{\left(1.0098039215686274\right)^2}
\][/tex]
- Simplifying this results in:
[tex]\[
P \approx \frac{1530}{1.019723254} \approx 1500.43547931002
\][/tex]
4. Round the principal amount to the nearest whole number:
- The principal amount is approximately Rs. 1500.43547931002, which, when rounded, results in Rs. 1500.
Thus, the original sum of money was:
c. Rs. 1500
