At the same rate of simple interest and on the same principal, what will be the ratio of the simple interest earned in 4 years to the simple interest earned in 6 years?

Introduction / Context:
This question focuses on the proportional nature of simple interest with respect to time. When the principal and rate remain the same, simple interest grows linearly with time. Therefore, the interest earned for different time periods is directly proportional to those time periods. Here we compare interest over 4 years and 6 years and find the ratio of the two amounts of interest.

Given Data / Assumptions:
- Principal P is the same in both cases. - Annual simple interest rate R is the same in both cases. - Time periods considered are T1 = 4 years and T2 = 6 years. - Interest is calculated using simple interest only. - Required ratio is SI for 4 years : SI for 6 years.

Concept / Approach:
Simple interest is given by SI = (P * R * T) / 100. For a fixed principal and rate, SI is directly proportional to time T. This means SI1 / SI2 = T1 / T2. Hence, the ratio of simple interest for 4 years and 6 years is simply the ratio of 4 to 6. Simplifying this ratio gives the final answer without needing to know the actual principal or rate values.

Step-by-Step Solution:
Step 1: Write SI1 = (P * R * T1) / 100 for T1 = 4 years. Step 2: Write SI2 = (P * R * T2) / 100 for T2 = 6 years. Step 3: Compute SI1 / SI2 = [(P * R * 4) / 100] / [(P * R * 6) / 100]. Step 4: Cancel common factors P, R, and 100 to obtain SI1 / SI2 = 4 / 6. Step 5: Simplify the fraction 4 / 6 by dividing numerator and denominator by 2 to get 2 / 3. Step 6: Therefore, the ratio of simple interest for 4 years to that for 6 years is 2 : 3.

Verification / Alternative check:
Choose a convenient principal and rate to verify. For example, let P = Rs 100 and R = 10 percent per annum. For 4 years, SI1 = (100 * 10 * 4) / 100 = Rs 40. For 6 years, SI2 = (100 * 10 * 6) / 100 = Rs 60. The ratio SI1 : SI2 = 40 : 60, which simplifies to 2 : 3, confirming the conceptual reasoning.

Why Other Options Are Wrong:
A ratio of 1 : 2 would mean the interest over 6 years is exactly double that over 4 years, which is not correct because 6 is not double 4. A ratio of 2 : 1 would incorrectly imply that interest is larger for 4 years than for 6 years. A ratio of 3 : 2 is the inverse of the correct ratio. Only 2 : 3 matches the proportional relationship between 4 years and 6 years.

Common Pitfalls:
Learners may mistakenly think of compound interest behaviour and assume a non linear relation with time. Others might try to assign arbitrary values to principal and rate without realising that these cancel out. Remembering that simple interest is directly proportional to time for fixed principal and rate simplifies many ratio questions.

Final Answer:
The ratio of simple interest for 4 years to that for 6 years is 2 : 3.