Interest ratio from rate and time product – UGC NET Question

For a fixed principal P, simple interest is proportional to the product of rate and time, SI ∝ R × T. In the first case, R × T = 9 × 4 = 36. In the second case, R × T = 6 × 7 = 42. Thus, the ratio of the first interest to the second is 36:42, which simplifies to 6:7 on dividing both terms by 6.

Option A:
Option A, 3:4, might be a rough guess comparing 9% and 12% or other partial values, but it does not arise from the exact products 36 and 42. Simplifying 36:42 gives 6:7, not 3:4, so 3:4 misrepresents the proportion of interest amounts.

Option B:
Option B is correct because it strictly follows the rule that simple interest is directly proportional to R × T when principal is constant. Using 36 and 42 as the proportional measures and simplifying leads uniquely to 6:7. This ratio precisely captures the slight advantage of the second scheme’s longer duration and different rate.

Option C:
Option C, 4:5, would require the ratio 36:45, but the second product is 42, not 45. Therefore, this option does not align with the actual calculation of rate–time products.

Option D:
Option D, 5:6, simplifies to 30:36, which again does not match 36:42. Using 5:6 would imply different underlying R × T values than those given, so it cannot represent the true ratio between the two simple interests.