Ratio change under equal subtraction – UGC NET Question

Let the numbers be 9x and 13x. After subtracting 12 from each, they become 9x − 12 and 13x − 12, and their ratio is 3:5. So (9x − 12)/(13x − 12) = 3/5. Cross-multiplying gives 5(9x − 12) = 3(13x − 12), which simplifies to 45x − 60 = 39x − 36 and then to 6x = 24, so x = 4. Hence the original numbers are 36 and 52, and the larger number is 52.

Option A:
Option A, 36, is the smaller of the two numbers for x = 4 and does not answer the question, which asks specifically for the larger number. Though 36 fits the ratio 9:13 with 52, choosing it would ignore the wording about which number is larger.

Option B:
Option B, 44, does not fit into the pair 9x and 13x when the transformation is imposed. If we tried to use 44 as the larger number, the corresponding smaller number from ratio 9:13 would not be an integer, and the new ratio after subtracting 12 would not become 3:5. Thus 44 does not satisfy the stated conditions.

Option C:
Option C, 48, might appear plausible as a mid-range value but cannot be expressed as 13x with integer x satisfying the ratio change. Any attempt to treat 48 as the larger number yields inconsistencies when we try to reach a ratio of 3:5 after subtracting 12. Therefore, 48 is incorrect.

Option D:
Option D is correct because with x = 4 we get 9x = 36 and 13x = 52, and 36:52 reduces to 9:13. Subtracting 12 from each gives 24 and 40, whose ratio is 24:40 = 3:5, precisely matching the requirement. This confirms 52 as the correct larger original number.