What will be the ratio of simple interest earned by certain amount at the same rate of interest for 6 years and that for 9 years.

Options

A. a. 1:2
B. b. 2:1
C. c. 2:2
D. d. 2:3

Answer

To solve the problem of finding the ratio of simple interest earned on a certain amount at the same rate of interest for 6 years and 9 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:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( t \) = Time in years

2. **Set Up the Problem**:
We need to find the ratio of simple interest for 6 years to that for 9 years. Let's denote:
- \( SI_6 \) = Simple Interest for 6 years
- \( SI_9 \) = Simple Interest for 9 years

3. **Calculate Simple Interest for 6 Years**:
Using the formula:
\[ SI_6 = \frac{P \times R \times 6}{100} \]

4. **Calculate Simple Interest for 9 Years**:
Similarly, for 9 years:
\[ SI_9 = \frac{P \times R \times 9}{100} \]

5. **Set Up the Ratio**:
Now, we need to find the ratio of \( SI_6 \) to \( SI_9 \):
\[ \text{Ratio} = \frac{SI_6}{SI_9} = \frac{\frac{P \times R \times 6}{100}}{\frac{P \times R \times 9}{100}} \]

6. **Simplify the Ratio**:
In the ratio, \( P \), \( R \), and \( 100 \) cancel out:
\[ \text{Ratio} = \frac{6}{9} \]

7. **Reduce the Fraction**:
To simplify \( \frac{6}{9} \):
- Divide both the numerator and the denominator by 3:
\[ \frac{6 \div 3}{9 \div 3} = \frac{2}{3} \]

8. **Final Answer**:
Thus, the ratio of simple interest earned for 6 years to that for 9 years is:
\[ \text{Ratio} = 2 : 3 \]