UGC NET Questions (Paper – 1)

Let the numbers be 5x and 8x. After subtracting 20 from each, they become 5x − 20 and 8x − 20, with ratio 3:5. So (5x − 20)/(8x − 20) = 3/5. Cross multiplying gives 5(5x − 20) = 3(8x − 20), which simplifies to 25x − 100 = 24x − 60 and hence x = 40. The original numbers are therefore 200 and 320, so the larger is 320.

Option A:
Option A, 200, is the smaller original number when x = 40. If we took 200 as the larger, it would contradict the initial ratio 5:8, in which 8x must exceed 5x. Substituting 200 as the larger in the equations would also fail to reproduce the ratio 3:5 after subtraction.

Option B:
Option B, 240, does not correspond to either 5x or 8x for the solved value x = 40. If we attempted to solve using 240 as one of the numbers, the resulting x would not satisfy the transformed ratio condition. Hence 240 is inconsistent with the problem’s constraints.

Option C:
Option C, 280, again fails to match the form 5x or 8x with x = 40 and does not produce the stated ratio 3:5 after subtracting 20 from both numbers. Any attempt to fit 280 into the ratio structure leads to contradictions with the proportional relationship.

Option D:
Option D is correct because 320 equals 8 × 40, matching the larger term in the original ratio 5:8. With numbers 200 and 320, subtracting 20 gives 180 and 300, and 180:300 simplifies to 3:5. This verifies that 320 is the correct larger original number.

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.

Let the original numbers be 7x and 4x. After subtracting 5 from each, they become 7x − 5 and 4x − 5, and their ratio is 3:1. Thus, (7x − 5)/(4x − 5) = 3/1. This gives 7x − 5 = 3(4x − 5) = 12x − 15, which simplifies to 10 = 5x and hence x = 2. Therefore, the original numbers are 14 and 8, and the larger is 14.

Option A:
Option A, 8, is the smaller of the two numbers in the correct solution. While it satisfies the original ratio when combined with 14, the question asks for the larger number. Choosing 8 would misinterpret which quantity the problem seeks.

Option B:
Option B is correct because 14 equals 7x with x = 2 and is the larger term in the ratio 7:4. Substituting 14 and 8 into the transformed condition, we get (14 − 5):(8 − 5) = 9:3 = 3:1, which exactly matches the new ratio, confirming that 14 is the correct larger original number.

Option C:
Option C, 16, would require the smaller number to be 16 × (4/7) ≈ 9.14 to preserve the ratio 7:4. This is not an integer and, even if approximated, would not yield a new ratio of 3:1 after subtracting 5. Thus, 16 cannot be the larger number that fits all conditions.

Option D:
Option D, 18, similarly leads to a non-integer partner of 18 × (4/7) ≈ 10.29 under the original ratio and fails to produce the required new ratio of 3:1 after the subtraction. Hence, 18 is inconsistent with the given information.