What will be the ratio of simple interest earned by certain amount at the same rate of interest for 12 yr and for 18 yr?

Options

A. `2:5`
B. `1:3`
C. `2:3`
D. `3:1`

Explanation

To find the ratio of simple interest earned by a certain amount at the same rate of interest for 12 years and for 18 years, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Simple Interest**:
The formula for calculating simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
where \( P \) is the principal amount, \( R \) is the rate of interest, and \( T \) is the time period in years.

2. **Calculate Simple Interest for 12 Years**:
Let the principal amount be \( P \) and the rate of interest be \( R \).
For 12 years, the simple interest will be:
\[ SI_{12} = \frac{P \times R \times 12}{100} \]

3. **Calculate Simple Interest for 18 Years**:
For 18 years, the simple interest will be:
\[ SI_{18} = \frac{P \times R \times 18}{100} \]

4. **Set Up the Ratio of Simple Interests**:
We need to find the ratio of \( SI_{12} \) to \( SI_{18} \):
\[ \text{Ratio} = \frac{SI_{12}}{SI_{18}} = \frac{\frac{P \times R \times 12}{100}}{\frac{P \times R \times 18}{100}} \]

5. **Simplify the Ratio**:
The \( P \), \( R \), and \( 100 \) in the numerator and denominator cancel out:
\[ \text{Ratio} = \frac{12}{18} \]

6. **Reduce the Fraction**:
To simplify \( \frac{12}{18} \):
- Divide both the numerator and the denominator by their greatest common divisor, which is 6:
\[ \frac{12 \div 6}{18 \div 6} = \frac{2}{3} \]

7. **Final Answer**:
The ratio of simple interest earned for 12 years to that for 18 years is:
\[ \text{Ratio} = 2 : 3 \]