UGC NET Questions (Paper – 1)

The third proportional c to numbers a and b is defined by a:b = b:c. For a = 8 and b = 18, we have 8:18 = 18:c. Writing this as a fraction gives 8/18 = 18/c, so c = (18 × 18)/8 = 324/8 = 40.5. Therefore, 40.5 is the third proportional to 8 and 18 and satisfies 8:18 = 18:40.5 exactly.

Option A:
Option A, 32.4, corresponds to 324/10 and would come from dividing 324 by 10 rather than 8. Substituting 32.4 for c gives 8:18 ≠ 18:32.4, since 8/18 simplifies to 4/9 while 18/32.4 simplifies to a different value. Hence 32.4 does not fulfil the proportional condition.

Option B:
Option B, 36, gives 18:36 = 1:2, whereas 8:18 simplifies to 4:9. These are not equal ratios, so 36 cannot be the third proportional. The numerical pattern required by the definition of third proportional is not met by this value.

Option C:
Option C is correct because using c = 40.5 we get 18:40.5 = 18/(81/2) = 36/81 = 4/9, which matches 8:18 = 4:9. This equality of simplified ratios confirms that 40.5 is the unique third proportional to 8 and 18.

Option D:
Option D, 45, gives 18:45 = 2:5, which again differs from 4:9. While 45 is close numerically to 40.5, proportional relationships are sensitive to exact values, and this small change breaks the equality required by the definition. Therefore 45 is not the correct third proportional.

The third proportional c to numbers a and b is defined by a:b = b:c. For a = 5 and b = 20, we have 5:20 = 20:c. Writing this as a fraction gives 5/20 = 20/c, so c = (20 × 20)/5 = 400/5 = 80. Thus, 80 is the third proportional to 5 and 20.

Option A:
Option A, 40, produces the ratio 20:40 = 1:2, whereas 5:20 = 1:4. Since these ratios are not equal, 40 cannot be the third proportional under the definition a:b = b:c.

Option B:
Option B, 64, might emerge from mistakenly squaring only part of a fraction or choosing a convenient number, but 5:20 = 1:4 and 20:64 = 5:16, which do not match. Therefore, 64 does not satisfy the required proportional relationship.

Option C:
Option C is correct because with c = 80 we get 5:20 = 1:4 and 20:80 = 1:4. The equality of these ratios confirms that 80 is the valid third proportional to 5 and 20.

Option D:
Option D, 100, yields 20:100 = 1:5, which differs from 1:4, the simplified form of 5:20. While 100 is a round number, it does not preserve the proportional structure defined in the question.