What will be the ratio of simple interest earned by certain amount at the same rate of interest for 12 years and that for 18 years.

Options

A. a. 1:2
B. b. 2:1
C. c. 2:3
D. d. none of these

Answer

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

### Step-by-Step Solution:

1. **Understand the Formula for Simple Interest (SI)**:
The formula for calculating simple interest 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 in years.

2. **Calculate Simple Interest for 12 Years**:
For a time period of 12 years, the simple interest can be expressed as:
\[ SI_{12} = \frac{P \times R \times 12}{100} \]

3. **Calculate Simple Interest for 18 Years**:
For a time period of 18 years, the simple interest can be expressed as:
\[ SI_{18} = \frac{P \times R \times 18}{100} \]

4. **Set Up the Ratio**:
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**:
Since \( P \), \( R \), and \( 100 \) are common in both the numerator and the denominator, they cancel out:
\[ \text{Ratio} = \frac{12}{18} \]

6. **Reduce the Fraction**:
Now, we can simplify \( \frac{12}{18} \):
\[ \frac{12}{18} = \frac{2}{3} \]

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