The compound interest on a certain sum at a certain rate of interest for the second year and third year is Rs 21780 and Rs. 23958 respectively. What is the rate of interest ?

Options

A. `6%`
B. `12%`
C. `10%`
D. `15%`

Answer

To find the rate of interest based on the compound interest for the second and third years, we can follow these steps:

### Step 1: Define Variables
Let the principal amount be \( P \) and the rate of interest be \( R \% \).

### Step 2: Write the Compound Interest Formulas
The compound interest for the second year can be expressed as:
\[ \text{CI for 2nd year} = A_2 - A_1 \]
Where:
- \( A_2 = P \left(1 + \frac{R}{100}\right)^2 \) (Amount after 2 years)
- \( A_1 = P \left(1 + \frac{R}{100}\right)^1 \) (Amount after 1 year)

Thus, we have:
\[ A_2 - A_1 = P \left(1 + \frac{R}{100}\right)^2 - P \left(1 + \frac{R}{100}\right) = 21780 \]

### Step 3: Simplify the Equation
Factoring out \( P \):
\[ P \left(\left(1 + \frac{R}{100}\right)^2 - \left(1 + \frac{R}{100}\right)\right) = 21780 \]
This simplifies to:
\[ P \left(1 + \frac{R}{100}\right) \left(\frac{R}{100}\right) = 21780 \quad \text{(Equation 1)} \]

### Step 4: Write the Equation for the Third Year
The compound interest for the third year can be expressed as:
\[ \text{CI for 3rd year} = A_3 - A_2 \]
Where:
- \( A_3 = P \left(1 + \frac{R}{100}\right)^3 \)

Thus, we have:
\[ A_3 - A_2 = P \left(1 + \frac{R}{100}\right)^3 - P \left(1 + \frac{R}{100}\right)^2 = 23958 \]

### Step 5: Simplify the Third Year Equation
Factoring out \( P \):
\[ P \left(\left(1 + \frac{R}{100}\right)^3 - \left(1 + \frac{R}{100}\right)^2\right) = 23958 \]
This simplifies to:
\[ P \left(1 + \frac{R}{100}\right)^2 \left(\frac{R}{100}\right) = 23958 \quad \text{(Equation 2)} \]

### Step 6: Divide the Two Equations
Now, divide Equation 2 by Equation 1:
\[ \frac{P \left(1 + \frac{R}{100}\right)^2 \left(\frac{R}{100}\right)}{P \left(1 + \frac{R}{100}\right) \left(\frac{R}{100}\right)} = \frac{23958}{21780} \]
This simplifies to:
\[ \frac{1 + \frac{R}{100}}{1} = \frac{23958}{21780} \]

### Step 7: Calculate the Ratio
Calculating the ratio:
\[ \frac{23958}{21780} \approx 1.1 \]
So we have:
\[ 1 + \frac{R}{100} = 1.1 \]

### Step 8: Solve for R
Subtract 1 from both sides:
\[ \frac{R}{100} = 0.1 \]
Multiply by 100:
\[ R = 10 \]

### Conclusion
The rate of interest is \( 10\% \).

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