If the amount received at the end of 2nd and 3rd year at compound Interest on a certain Principal is Rs 9,600 and Rs 10,272 respectively, what is the rate of interest (in %)?

Options

A. 7
B. 8
C. 6
D. 5

Answer

To solve the problem step by step, we will use the information given about the amounts received at the end of the 2nd and 3rd years to find the rate of interest.

### Step 1: Define the Variables
Let:
- \( P \) = Principal amount
- \( r \) = Rate of interest (in decimal form)

### Step 2: Write the Amount Formulas
The amount at the end of the 2nd year is given by:
\[ A_2 = P(1 + r)^2 \]
The amount at the end of the 3rd year is given by:
\[ A_3 = P(1 + r)^3 \]

### Step 3: Set Up the Equations
From the problem, we know:
\[ A_2 = 9600 \quad \text{(Amount at the end of 2nd year)} \]
\[ A_3 = 10272 \quad \text{(Amount at the end of 3rd year)} \]

### Step 4: Write the Equations
Using the formulas from Step 2:
1. \( P(1 + r)^2 = 9600 \) (Equation 1)
2. \( P(1 + r)^3 = 10272 \) (Equation 2)

### Step 5: Divide the Two Equations
To eliminate \( P \), we can divide Equation 2 by Equation 1:
\[ \frac{P(1 + r)^3}{P(1 + r)^2} = \frac{10272}{9600} \]
This simplifies to:
\[ 1 + r = \frac{10272}{9600} \]

### Step 6: Calculate \( 1 + r \)
Calculating the right side:
\[ 1 + r = \frac{10272}{9600} = 1.0700 \]

### Step 7: Solve for \( r \)
Now, we can find \( r \):
\[ r = 1.0700 - 1 = 0.0700 \]

### Step 8: Convert \( r \) to Percentage
To convert \( r \) to a percentage:
\[ \text{Rate of interest} = r \times 100 = 0.0700 \times 100 = 7\% \]

### Final Answer
The rate of interest is **7%**.

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