What will be the ratio of the simple interest earned on a fixed sum of money at the same rate of interest for 6 years and for 9 years under simple interest conditions?

Introduction / Context:
This question examines your understanding of how simple interest depends on time when the principal and rate are fixed. Under simple interest, the interest earned is directly proportional to the time period. Therefore, the ratio of interests for different time periods is simply the ratio of the corresponding times, assuming principal and rate remain the same.

Given Data / Assumptions:

Principal P is constant in both cases.
Rate of interest per annum r% is the same in both cases.
Two time periods: T1 = 6 years and T2 = 9 years.
Simple interest formula: SI = (P * r * T) / 100.
We must find the ratio SI_6years : SI_9years.

Concept / Approach:
Since P and r are fixed, simple interest is proportional only to T. Therefore, SI_1 / SI_2 = T1 / T2. By substituting T1 = 6 and T2 = 9, we find the ratio of the interests and simplify it to the lowest terms. There is no need to know the actual values of P or r.

Step-by-Step Solution:
Step 1: Express simple interest for a time T as SI_T = (P * r * T) / 100. Step 2: For 6 years, SI_6 = (P * r * 6) / 100. Step 3: For 9 years, SI_9 = (P * r * 9) / 100. Step 4: Compute the ratio SI_6 : SI_9. Step 5: SI_6 : SI_9 = (P * r * 6 / 100) : (P * r * 9 / 100). Step 6: Cancel common factors P, r, and 1 / 100. Step 7: The ratio reduces to 6 : 9. Step 8: Simplify 6 : 9 by dividing both sides by 3 to get 2 : 3.

Verification / Alternative check:
Assume a simple numerical example to verify. Let P = 100 and r = 10% per annum. Then SI_6 = (100 * 10 * 6) / 100 = 60 and SI_9 = (100 * 10 * 9) / 100 = 90. The ratio 60 : 90 simplifies to 2 : 3, confirming the general reasoning.

Why Other Options Are Wrong:
Ratios like 1 : 3, 1 : 4, or 3 : 4 do not match the direct proportionality between simple interest and time for 6 and 9 years. The option that says "Data is inadequate" is incorrect because we do not need actual values of principal or rate to determine the ratio. Only the time periods matter under simple interest, so the data given are completely sufficient.

Common Pitfalls:
Some learners mistakenly think that without knowing principal or rate, they cannot solve such problems. Others may confuse simple interest with compound interest, where the relationship between time and interest is not strictly linear. For simple interest, always remember that interest is directly proportional to time when other factors are fixed.

Final Answer:
The ratio of the simple interest for 6 years and 9 years is 2 : 3.