Step 1: Understanding the Concept:
Compound interest (CI) is characterized by the fact that the interest earned at the end of a specific period is added to the principal to form the new principal for the next period.
This means that the interest is "compounded," leading to exponential growth of the initial investment over time.
In this specific problem, we are given the accumulated amounts for two consecutive years: the 2nd year and the 3rd year.
A vital property of compound interest is that the total amount at the end of any given year (Year \(n\)) serves as the starting principal for the subsequent year (Year \(n+1\)).
Therefore, the difference in the amount between the end of the 2nd year (\(A_2\)) and the end of the 3rd year (\(A_3\)) is purely the interest generated by the 2nd-year amount over exactly one year.
Since the time interval between the 2nd and 3rd year is exactly 1 year, we can calculate the rate of interest using the simple interest logic for that single-year gap.
Step 2: Key Formula or Approach:
1. Interest for the 3rd year (\(I\)) = Amount after 3 years (\(A_3\)) \(-\) Amount after 2 years (\(A_2\))
2. Rate of Interest (\(R\)) = \(\left( \frac{\text{Interest for the 3rd year}}{\text{Amount after 2 years}} \right) \times 100\)
This formula effectively treats the 2nd-year amount as the "Principal" and calculates the percentage growth that leads to the 3rd-year amount.
Step 3: Detailed Explanation:
Let's break down the numerical calculation based on the data provided in the question.
Given:
Amount after 2 years (\(A_2\)) = \(9600\)
Amount after 3 years (\(A_3\)) = \(11520\)
First, we must determine exactly how much interest was accrued specifically during the third year.
\[ \text{Interest (I)} = A_3 - A_2 \]
\[ \text{Interest (I)} = 11520 - 9600 \]
\[ \text{Interest (I)} = 1920 \]
This interest of \(1920\) was earned by the capital sum of \(9600\) that was present at the beginning of the 3rd year (which is the end of the 2nd year).
Now, we find what percentage of the principal (\(9600\)) this interest (\(1920\)) represents to find the annual rate (\(R\)).
\[ R = \left( \frac{1920}{9600} \right) \times 100 \]
To simplify this calculation, we can cancel the zeros from the numerator and the denominator:
\[ R = \left( \frac{192}{960} \right) \times 100 \]
By observing the numbers, we can see that \(960\) is a multiple of \(192\).
Calculating \(192 \times 5\):
\(192 \times 5 = (200 - 8) \times 5 = 1000 - 40 = 960\).
Therefore, the fraction \(\frac{192}{960}\) simplifies perfectly to \(\frac{1}{5}\).
\[ R = \frac{1}{5} \times 100 \]
\[ R = 20% \]
This indicates that the sum of money is growing at a rate of \(20%\) compounded annually.
Step 4: Final Answer:
The rate of interest per annum is 20%.