Ratio change under equal addition – UGC NET Question

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.