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 conceptual question focuses on the proportional nature of simple interest. It asks for the ratio of interest amounts over different time periods when both the principal and rate remain the same. It is meant to test understanding rather than arithmetic complexity.

Given Data / Assumptions:

• The same principal amount, P, is used for both time periods.
• The same annual simple interest rate, R%, applies in both cases.
• Time periods are T1 = 4 years and T2 = 6 years.
• We need the ratio SI4 : SI6, where SI4 is interest for 4 years and SI6 is interest for 6 years.

Concept / Approach:
Under simple interest, the interest is directly proportional to time when principal and rate are fixed. The formula is:
SI = (P * R * T) / 100If P and R are constant, then SI is proportional to T. Therefore:
SI4 / SI6 = T1 / T2 = 4 / 6We can then simplify this fraction to obtain the required ratio.

Step-by-Step Solution:
Step 1: Write the simple interest for each time period.SI4 = (P * R * 4) / 100SI6 = (P * R * 6) / 100Step 2: Form the ratio SI4 : SI6.SI4 / SI6 = [(P * R * 4) / 100] / [(P * R * 6) / 100]Step 3: Cancel common factors P, R, and 100.SI4 / SI6 = 4 / 6Step 4: Simplify the fraction.4 / 6 = 2 / 3So, SI4 : SI6 = 2 : 3

Verification / Alternative check:
We can assume any convenient principal and rate, say P = Rs 100 and R = 10%. For 4 years, SI4 = (100 * 10 * 4) / 100 = 40. For 6 years, SI6 = (100 * 10 * 6) / 100 = 60. The ratio 40 : 60 simplifies to 2 : 3, which confirms the result.

Why Other Options Are Wrong:

• 1 : 2 suggests that interest for 6 years is double that for 4 years, which is not correct.
• 2 : 1 suggests that interest for 4 years is double that for 6 years, which is clearly impossible.
• 3 : 2 reverses the correct ratio, giving a larger interest for 4 years than 6 years.

Common Pitfalls:
Some students misinterpret the direct proportionality and may think interest doubles whenever time increases by any amount, or they might try to plug in random numbers unnecessarily. The key is recognizing that under simple interest, the ratio of interest amounts is exactly the ratio of times when principal and rate are unchanged.

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