UGC NET Questions (Paper – 1)

Let the original numbers be 4x and 7x. After adding 8 to each, the new numbers are 4x + 8 and 7x + 8 and their ratio is 5:8. So (4x + 8)/(7x + 8) = 5/8. Cross-multiplying gives 8(4x + 8) = 5(7x + 8), which simplifies to 32x + 64 = 35x + 40 and then to 3x = 24, so x = 8. The original numbers are 32 and 56, and the larger is 56.

Option A:
Option A, 40, does not appear as either 4x or 7x when x = 8 and would not preserve the required ratio 4:7 together with its pair. If we assume 40 as the larger number, the smaller would be less than 40, leading to a ratio different from 4:7 and a different transformed ratio after adding 8. Thus 40 is inconsistent with the given conditions.

Option B:
Option B, 48, might be guessed by partial calculation but does not satisfy the equation (4x + 8)/(7x + 8) = 5/8 when treated as the larger number. With 48 as the larger term, the corresponding smaller term from ratio 4:7 would be non-integral or lead to a different new ratio after adding 8. Hence 48 is not correct.

Option C:
Option C is correct because with x = 8 the original numbers are 4 × 8 = 32 and 7 × 8 = 56, which are in the ratio 4:7. After adding 8, we get 40 and 64, whose ratio is 40:64 = 5:8, exactly matching the condition. This confirms 56 as the unique value for the larger original number.

Option D:
Option D, 64, is the larger number after the increment rather than before it in the correct solution. Treating 64 as the original larger number would produce a much bigger value after adding 8 and the resulting ratio would no longer become 5:8. Thus 64 does not align with the stated transformation.

Let the original numbers be 2x and 5x. After adding 6 to each, they become 2x + 6 and 5x + 6, and the new ratio is 1:2. Thus, (2x + 6)/(5x + 6) = 1/2. Cross-multiplying gives 2(2x + 6) = 5x + 6, which simplifies to 4x + 12 = 5x + 6 and then x = 6. Therefore, the original numbers are 12 and 30, and the smaller is 12.

Option A:
Option A is correct because it corresponds to 2x with x = 6 in the ratio 2:5. Substituting 12 and 30 back into the problem, we get (12 + 6):(30 + 6) = 18:36 = 1:2, which matches the given transformed ratio. This confirms that 12 is the smaller of the original pair.

Option B:
Option B, 18, would require the larger number to be 45 to maintain the original ratio 2:5. The pair 18 and 45 has sum 63, and after adding 6 to each we get 24 and 51, whose ratio is 24:51 = 8:17, not 1:2. Hence 18 does not satisfy the transformation condition.

Option C:
Option C, 20, when paired with 50 for the 2:5 ratio, leads to new numbers 26 and 56 after the equal addition of 6. Their ratio 26:56 simplifies to 13:28, which again is not 1:2. Thus, 20 cannot be the smaller original number.

Option D:
Option D, 24, would pair with 60 to give the 2:5 ratio. After adding 6 we obtain 30 and 66, and the ratio 30:66 reduces to 5:11, not 1:2. Therefore, 24 fails to meet the required ratio change after the addition.

Let the numbers be 4x and 7x. After adding 5 to each, they become 4x + 5 and 7x + 5, and their ratio is given as 3:5. Thus (4x + 5)/(7x + 5) = 3/5. Cross multiplying gives 5(4x + 5) = 3(7x + 5), which simplifies to 20x + 25 = 21x + 15, and hence x = 10. Therefore the numbers are 40 and 70, and the larger original number is 70.

Option A:
Option A, 40, is the smaller of the two original numbers when x = 10. It satisfies the initial ratio but does not answer the question, which specifically asks for the larger number. Moreover, choosing 40 as the larger number would contradict the ratio 4:7 where 4x must be smaller than 7x.

Option B:
Option B, 45, does not fit into any pair maintaining both the original ratio 4:7 and the new ratio 3:5 after adding 5. If we treated 45 as the larger original number, then the smaller would be 45 × (4/7), which is not an integer and fails the requirement for simple integer ratio terms. Thus 45 is an inconsistent choice.

Option C:
Option C is correct because 70 arises naturally as 7 × 10 when solving the given proportion. Substituting 40 and 70 back into the problem, we get (40 + 5):(70 + 5) = 45:75 = 3:5, which verifies the condition. This confirms that 70 is the true larger original number.

Option D:
Option D, 75, would require the smaller number to be 75 × (4/7) ≈ 42.86 to preserve the original ratio, which is not an integer and also fails to give the new ratio 3:5 after adding 5. Therefore 75 does not satisfy the structural requirements of the problem.