UGC NET Questions (Paper – 1)

Let the numbers be 6x and 7x. Their sum is given as 6x + 7x = 13x = 325. Solving this, we get x = 325/13 = 25. Therefore the two numbers are 6 × 25 = 150 and 7 × 25 = 175. The smaller number is 150.

Option A:
Option A is correct because it emerges directly from solving the equation based on the ratio and the sum. Substituting 150 and 175 back into the problem, their sum is 325 and their ratio is 150:175, which reduces to 6:7 on dividing by 25. This confirms the consistency of the solution.

Option B:
Option B, 175, is the larger of the two numbers. The question explicitly asks for the smaller number, so choosing 175 would ignore this requirement. Although 175 is part of the correct pair, it does not answer the query.

Option C:
Option C, 200, would imply that the other number is 125 to maintain the sum 325. The ratio 125:200 simplifies to 5:8, not 6:7, so this pair does not match the stated ratio. Therefore 200 cannot be the smaller number satisfying both conditions.

Option D:
Option D, 225, paired with 100, would give a sum of 325 but a ratio of 100:225, which simplifies to 4:9. This is different from the required 6:7, so 225 is not a valid candidate for the smaller number under the given constraints.

Let the two numbers be 4x and 5x. Their sum is 4x + 5x = 9x, which is given as 81. Solving 9x = 81 gives x = 9. Therefore, the numbers are 4 × 9 = 36 and 5 × 9 = 45, and the larger number is clearly 45. This uses the standard technique of expressing numbers with a common factor and solving a single linear equation.

Option A:
Option A, 36, is the smaller of the two numbers in the pair 36 and 45. While 36 does fit the ratio 4:5 when paired with 45, the question specifically asks for the larger number, so choosing 36 would ignore the direction of comparison. It therefore does not answer what is being asked.

Option B:
Option B is correct because it corresponds to 5x with x = 9, the larger term in the ratio 4:5. When we check, 36 + 45 = 81 and 36:45 simplifies to 4:5 on dividing both numbers by 9. This double verification confirms that 45 is the only number that fits both the ratio and sum conditions as the larger value.

Option C:
Option C, 40, might seem plausible as a mid-range value, but if 40 were the larger number, the smaller would have to be 40 × (4/5) = 32 to maintain the ratio 4:5. The pair 32 and 40 has a sum of 72, not 81, so this option fails the sum condition.

Option D:
Option D, 48, would give a smaller number of 48 × (4/5) = 38.4 if the ratio 4:5 were preserved, which is not even an integer. Moreover, 38.4 + 48 ≠ 81, so this choice violates both the integer requirement and the sum requirement.