The ratio of the amount for two years under Cl annually and for one year under SI is 6 : 5 . When the rate of interest is same, then the value of rate of interest is :

Options

A. `12.5%`
B. `18%`
C. `20%`
D. `16.66%`

Answer

To solve the problem, we need to find the rate of interest (R) given the ratio of the amounts for two years under compound interest (CI) and for one year under simple interest (SI) is 6:5.

### Step-by-Step Solution:

1. **Understand the Formulas**:
- The formula for the amount under Compound Interest (CI) for 2 years is:
\[ A_{CI} = P \left(1 + \frac{R}{100}\right)^2 \]
- The formula for the amount under Simple Interest (SI) for 1 year is:
\[ A_{SI} = P + \left(P \cdot \frac{R}{100} \cdot 1\right) = P \left(1 + \frac{R}{100}\right) \]

2. **Set Up the Ratio**:
- According to the problem, the ratio of the amounts is given as:
\[ \frac{A_{CI}}{A_{SI}} = \frac{6}{5} \]
- Substituting the formulas into the ratio gives:
\[ \frac{P \left(1 + \frac{R}{100}\right)^2}{P \left(1 + \frac{R}{100}\right)} = \frac{6}{5} \]
- We can cancel \(P\) from both sides (assuming \(P \neq 0\)):
\[ \frac{1 + \frac{R}{100}}{1} = \frac{6}{5} \]

3. **Simplify the Equation**:
- This simplifies to:
\[ 1 + \frac{R}{100} = \frac{6}{5} \]

4. **Isolate R**:
- Subtract 1 from both sides:
\[ \frac{R}{100} = \frac{6}{5} - 1 \]
- Convert 1 to a fraction with a denominator of 5:
\[ \frac{R}{100} = \frac{6}{5} - \frac{5}{5} = \frac{1}{5} \]

5. **Solve for R**:
- Multiply both sides by 100 to find R:
\[ R = 100 \cdot \frac{1}{5} = 20 \]

### Final Answer:
The value of the rate of interest (R) is **20%**.