UGCNET Questions

The average of three numbers is the sum of the numbers divided by 3. If the average is 25, then the total sum must be 25 × 3 = 75. The two known numbers are 20 and 30, whose sum is 50. Subtracting 50 from 75 gives 25, so the third number must be 25.

Option A:
Option A, 25, exactly matches the required third number, because 20 + 30 + 25 = 75 and 75 ÷ 3 = 25. It is therefore the correct choice.

Option B:
Option B, 20, would give a total of 20 + 20 + 30 = 70, and 70 ÷ 3 does not equal the required average of 25, so this choice fails the given condition.

Option C:
Option C, 30, leads to 20 + 30 + 30 = 80, giving an average of 80 ÷ 3, which is greater than 25 and inconsistent with the stated mean.

Option D:
Option D, 35, produces a total of 20 + 30 + 35 = 85, resulting in an average greater than 25, again contradicting the problem’s information.

Five consecutive odd integers can be written as n − 4, n − 2, n, n + 2 and n + 4, where n is the middle term. The average of such a symmetric set is always the middle term, so the average being 37 means the middle integer is 37. The largest of the five is therefore 37 + 4, which is 41. This uses both the definition of average and the structure of consecutive odd numbers.

Option A:
Option A, 39, is the second largest of the five integers, not the largest. It is 2 greater than the middle term instead of 4 and therefore does not represent the end of the sequence.

Option B:
Option B correctly identifies that with an average of 37, the five odd numbers are 33, 35, 37, 39 and 41. Among these, 41 is clearly the largest and fits the pattern of consecutive odd integers.

Option C:
Option C, 43, would require shifting the sequence so that the average is larger than 37. That would contradict the given mean and lead to a different central value.

Option D:
Option D, 45, is even further from the centre and again cannot be part of the exact sequence whose average is 37.

The arithmetic mean of five numbers is 18, so their total sum is 18 × 5 = 90. One of the numbers is given as 24. To find the sum of the remaining four numbers, we subtract this known value from the total: 90 − 24 = 66. Thus, the combined total of the four remaining numbers is 66.

Option A:
Option A directly applies the relationship Total = Average × Number of items and then subtracts the known element to obtain the sum of the others. This method is standard in average problems and yields 66 consistently.

Option B:
Option B, 72, would correspond to a larger total of 96 if 24 were added back, which would produce an average of 19.2 instead of 18. This contradicts the given mean.

Option C:
Option C, 80, similarly gives a total of 104 when 24 is added back, leading to an average of 20.8, which is inconsistent with the initial information.

Option D:
Option D, 84, implies a total of 108 for five numbers, resulting in an average of 21.6, again conflicting with the given average of 18.

The average of four numbers being 26 means that their total sum is 26 × 4 = 104. The sum of the three known numbers 20, 25 and 29 is 20 + 25 + 29 = 74. The fourth number must make the overall sum 104, so it is 104 − 74 = 30. Therefore, the missing number is 30, which is Option B.

Option A:
Option A, 25, would lead to a total of 20 + 25 + 29 + 25 = 99, and 99 ÷ 4 is 24.75, which is less than 26, so the average condition fails.

Option B:
Option B, 30, makes the sum of the four numbers 104, which divided by 4 gives exactly 26, perfectly satisfying the definition of the average.

Option C:
Option C, 32, would produce a total of 106, and 106 ÷ 4 equals 26.5, which is higher than the stated average.

Option D:
Option D, 34, makes the total 108, giving an average of 27, again inconsistent with the given value.