Ratio between capital lent for 2 year com pounded annually and for 4 year at simple interest is 6 : 5. If the interest obtained after given time is equal find rate of interest.

Options

A. `132.33%`
B. `133.33%`
C. `131.33%`
D. `138.33%`

Answer

To solve the problem, we need to find the rate of interest given the ratio of capital lent for 2 years at compound interest and for 4 years at simple interest. Let's go through the solution step by step.

### Step 1: Define the Capitals
Let the capital lent for 2 years at compound interest be \( C_1 \) and the capital lent for 4 years at simple interest be \( C_2 \). According to the problem, the ratio of these capitals is given as:
\[ C_1 : C_2 = 6 : 5 \]
We can express this as:
\[ C_1 = 6x \quad \text{and} \quad C_2 = 5x \]

### Step 2: Calculate Compound Interest for 2 Years
The formula for compound interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{r}{100}\right)^n - P \]
For our case:
\[ CI_1 = 6x \left(1 + \frac{r}{100}\right)^2 - 6x \]
This simplifies to:
\[ CI_1 = 6x \left[\left(1 + \frac{r}{100}\right)^2 - 1\right] \]

### Step 3: Calculate Simple Interest for 4 Years
The formula for simple interest (SI) is:
\[ SI = P \cdot r \cdot t \]
For our case:
\[ SI_2 = 5x \cdot \frac{r}{100} \cdot 4 \]
This simplifies to:
\[ SI_2 = 20x \cdot \frac{r}{100} \]

### Step 4: Set the Interests Equal
According to the problem, the interests obtained from both investments are equal:
\[ CI_1 = SI_2 \]
Substituting the expressions we derived:
\[ 6x \left[\left(1 + \frac{r}{100}\right)^2 - 1\right] = 20x \cdot \frac{r}{100} \]

### Step 5: Simplify the Equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 6 \left[\left(1 + \frac{r}{100}\right)^2 - 1\right] = 20 \cdot \frac{r}{100} \]
This simplifies to:
\[ 6 \left(1 + \frac{r}{100}\right)^2 - 6 = \frac{20r}{100} \]
\[ 6 \left(1 + \frac{r}{100}\right)^2 = \frac{20r}{100} + 6 \]

### Step 6: Expand and Rearrange
Expanding the left side:
\[ 6 \left(1 + 2\frac{r}{100} + \frac{r^2}{10000}\right) = \frac{20r}{100} + 6 \]
This gives:
\[ 6 + \frac{12r}{100} + \frac{6r^2}{10000} = \frac{20r}{100} + 6 \]
Subtracting 6 from both sides:
\[ \frac{12r}{100} + \frac{6r^2}{10000} = \frac{20r}{100} \]

### Step 7: Combine Like Terms
Bringing all terms to one side:
\[ \frac{6r^2}{10000} + \frac{12r}{100} - \frac{20r}{100} = 0 \]
This simplifies to:
\[ \frac{6r^2}{10000} - \frac{8r}{100} = 0 \]

### Step 8: Factor the Equation
Factoring out \( r \):
\[ r \left(\frac{6r}{10000} - \frac{8}{100}\right) = 0 \]
This gives us:
\[ r = 0 \quad \text{or} \quad \frac{6r}{10000} = \frac{8}{100} \]

### Step 9: Solve for r
Solving for \( r \):
\[ 6r = 800 \quad \Rightarrow \quad r = \frac{800}{6} = \frac{400}{3} \approx 133.33\% \]

### Final Answer
The rate of interest is:
\[ \boxed{133.33\%} \]