The average of six numbers is 30. If the average of the first four is 25 and…

For a set of n numbers, the average equals the sum of all the values divided by n, so the sum equals the average multiplied by n. When two groups of numbers overlap in exactly one shared member, adding the sums of both groups counts that shared member twice, so subtracting the true total sum of all the numbers removes the double-count and isolates the shared member's value.

1. Total sum of all six numbers = average x count = 30 x 6 = 180.

2. Sum of the first four numbers = 25 x 4 = 100.

3. Sum of the last three numbers = 35 x 3 = 105.

4. Adding these two partial sums: 100 + 105 = 205. Since the fourth number belongs to both the first-four group and the last-three group, it has been counted twice in this combined total.

5. Subtracting the actual six-number total (180) from the combined subset total (205) removes the double-count and isolates the fourth number: 205 minus 180 = 25.

Cross-check: if the fourth number is 25, the first four numbers sum to 100 (average 25, matches) and the last three sum to 105 (average 35, matches); all six numbers together sum to 100 + 105 minus the shared 25, which is 180, giving an overall average of 180 divided by 6 = 30, matching the given data. The fourth number is confirmed as 25.