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According to yearly compound interest, a sum of money amounts to Rs. 1530 in 2 years and Rs. 1545 in 3 years. What was the sum?

A. Rs. 1445
B. Rs. 1495
C. Rs. 1500
D. Rs. 1520


Sagot :

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