Finding numbers from sum and ratio – UGC NET Question

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.