If compound interest on some amount is 21780 for 2nd year and 23958 for 3rd year then find the rate of interest?

Options

A. ` 11%`
B. `10%`
C. `13% `
D. `12%`

Answer

To find the rate of interest given the compound interest for the 2nd and 3rd years, we can follow these steps:

### Step 1: Understand the relationship between the compound interests
The compound interest for the 2nd year is given as 21,780 and for the 3rd year as 23,958. The compound interest for any year can be expressed in terms of the principal amount and the rate of interest.

### Step 2: Set up the equation
Let the rate of interest be \( R \% \). The amount at the end of the first year can be expressed as:
\[ A_1 = P(1 + \frac{R}{100}) \]
The amount at the end of the second year will be:
\[ A_2 = A_1(1 + \frac{R}{100}) = P(1 + \frac{R}{100})^2 \]
The compound interest for the second year is the difference between the amount at the end of the second year and the amount at the end of the first year:
\[ CI_2 = A_2 - A_1 = P(1 + \frac{R}{100})^2 - P(1 + \frac{R}{100}) \]
This simplifies to:
\[ CI_2 = P(1 + \frac{R}{100}) \left( (1 + \frac{R}{100}) - 1 \right) = P(1 + \frac{R}{100}) \cdot \frac{R}{100} \]

### Step 3: Write the equation for the 3rd year
Similarly, the compound interest for the 3rd year can be expressed as:
\[ CI_3 = A_3 - A_2 = P(1 + \frac{R}{100})^3 - P(1 + \frac{R}{100})^2 \]
This simplifies to:
\[ CI_3 = P(1 + \frac{R}{100})^2 \cdot \frac{R}{100} \]

### Step 4: Set up the ratio of the compound interests
From the problem, we have:
\[ CI_2 = 21,780 \]
\[ CI_3 = 23,958 \]
Now, we can set up the ratio:
\[ \frac{CI_3}{CI_2} = \frac{P(1 + \frac{R}{100})^2 \cdot \frac{R}{100}}{P(1 + \frac{R}{100}) \cdot \frac{R}{100}} = \frac{(1 + \frac{R}{100})^2}{(1 + \frac{R}{100})} = (1 + \frac{R}{100}) \]

### Step 5: Solve for the rate of interest
Now we can write:
\[ \frac{23,958}{21,780} = 1 + \frac{R}{100} \]
Calculating the left side:
\[ \frac{23,958}{21,780} \approx 1.1 \]
So we have:
\[ 1 + \frac{R}{100} = 1.1 \]
Subtracting 1 from both sides gives:
\[ \frac{R}{100} = 0.1 \]
Multiplying both sides by 100 gives:
\[ R = 10\% \]

### Final Answer
The rate of interest is **10%**.

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