If compound interest on some amount is 210 for 2 years and 331 years for 3 years then find the rate of interest?

Options

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

Answer

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

### Step 1: Understand the Given Information
We know that:
- Compound Interest (CI) for 2 years = 210
- Compound Interest (CI) for 3 years = 331

### Step 2: Calculate the Interest for the Third Year
The interest for the third year can be calculated by subtracting the compound interest for 2 years from the compound interest for 3 years.

\[ \text{Interest for 3rd year} = \text{CI for 3 years} - \text{CI for 2 years} \]
\[ \text{Interest for 3rd year} = 331 - 210 = 121 \]

### Step 3: Identify the Interest for Each Year
Let:
- \( I_1 \) = Interest for the 1st year
- \( I_2 \) = Interest for the 2nd year
- \( I_3 \) = Interest for the 3rd year

From the information we have:
- \( I_1 + I_2 = 210 \) (for 2 years)
- \( I_3 = 121 \) (calculated from step 2)

### Step 4: Establish the Relationship Between the Interests
In compound interest, the interest for each year can be expressed in terms of a common ratio since they form a geometric progression (GP).

Let’s denote the rate of interest as \( r \). Then:
- \( I_1 = P \cdot r \)
- \( I_2 = P \cdot r(1 + r) \)
- \( I_3 = P \cdot r(1 + r)^2 \)

### Step 5: Set Up the Equations
From the above relationships:
1. \( I_1 + I_2 = 210 \)
2. \( I_3 = 121 \)

Using the GP property:
\[ \frac{I_2}{I_1} = \frac{I_3}{I_2} \]

Substituting the known values:
\[ \frac{I_2}{I_1} = \frac{121}{I_2} \]

Let \( I_1 = x \) and \( I_2 = 210 - x \):
\[ \frac{210 - x}{x} = \frac{121}{210 - x} \]

### Step 6: Cross Multiply and Solve for \( x \)
Cross multiplying gives:
\[ (210 - x)^2 = 121x \]
Expanding and rearranging:
\[ 44100 - 420x + x^2 = 121x \]
\[ x^2 - 541x + 44100 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -541, c = 44100 \):
\[ x = \frac{541 \pm \sqrt{(-541)^2 - 4 \cdot 1 \cdot 44100}}{2 \cdot 1} \]
Calculating the discriminant:
\[ \sqrt{292681 - 176400} = \sqrt{116281} = 341 \]
Thus:
\[ x = \frac{541 \pm 341}{2} \]
Calculating the two possible values:
1. \( x = \frac{882}{2} = 441 \)
2. \( x = \frac{200}{2} = 100 \)

### Step 8: Find the Rate of Interest
Using \( I_1 = 100 \) (the first year interest):
\[ I_1 = P \cdot r \implies r = \frac{I_1}{P} \]
Assuming \( P = 1000 \) (for simplicity):
\[ r = \frac{100}{1000} = 0.1 \implies r = 10\% \]

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

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