Compound ratio using two given ratios – UGC NET Question

From x:y = 4:7, we write x = 4k and y = 7k. From y:z = 21:10, we write y = 21m and z = 10m. Equating y gives 7k = 21m, so k = 3m. Substituting back, x = 4k = 12m and z = 10m, so x:z = 12m:10m = 6:5. Thus, 6:5 is the correct compound ratio of x to z.

Option A:
Option A, 4:5, would require x to be 4 units when z is 5 units, which is not consistent with x = 12m and z = 10m derived from solving the two given ratios simultaneously. If we force 4:5, we cannot maintain both x:y = 4:7 and y:z = 21:10 at the same time.

Option B:
Option B is correct because it results from a systematic elimination of the common term y and then simplifying the resulting x:z ratio. The use of a common multiple for y ensures that the structure of both original ratios is respected. This guaranteed alignment leads uniquely to 6:5.

Option C:
Option C, 8:5, would make x comparatively smaller than 12m when measured against z = 10m, breaking the proportional link with y. No consistent choice of scaling parameters k and m will give x:z = 8:5 without contradicting the given ratios.

Option D:
Option D, 12:7, mixes the numerator from x = 12m with an unrelated denominator. A ratio of 12:7 would not maintain the numerical relationships defined by 4:7 and 21:10 and therefore fails the internal consistency check.