Resultant ratio of sums of terms – UGC NET Question

From a:b = 2:3, let a = 2k and b = 3k. From b:c = 4:5, let b = 4m and c = 5m. Equating b gives 3k = 4m, so we choose k = 4 and m = 3 to get a = 8, b = 12 and c = 15. Then (a + b) = 20 and (b + c) = 27, so the required ratio is 20:27. This ratio is unique after we bring the two given ratios to a common middle term.

Option A:
Option A, 14:19, does not arise from any consistent choice of a, b and c that satisfies both 2:3 and 4:5 as the given ratios. If we attempt to back-calculate from 14:19, the implied values of a, b and c will violate at least one of the original relationships. Hence 14:19 is not compatible with the data.

Option B:
Option B is correct because it is computed after properly equalising the middle term b in both ratios. The method ensures that b plays the same role in both proportional relationships before forming (a + b) and (b + c). This leads precisely to 20:27 as the resulting ratio of the sums.

Option C:
Option C, 7:9, looks like a simplified ratio but does not match the actual numerical values 20 and 27 obtained from solving. Any attempt to assign a, b and c consistent with 7:9 will contradict the starting ratios 2:3 and 4:5. Therefore 7:9 cannot represent (a + b):(b + c).

Option D:
Option D, 3:4, is unrelated to the sums and instead resembles a typical simple ratio. When we compute (a + b) and (b + c) using the correct method, their ratio is not reducible to 3:4. Thus, 3:4 is not the correct resultant ratio.