A set of numbers from 1 to 36 (both inclusive) are split into 6 groups...
Understanding the Problem
To find the smallest and largest possible averages of the medians of 6 groups of numbers from 1 to 36, we first recognize that medians in a group of 6 numbers are the average of the 3rd and 4th smallest numbers when they are arranged in order.
Finding the Smallest Average (x)
To minimize the average of the medians (P, Q, R, S, T, U), we should aim to form groups with the smallest possible numbers:
- Group 1: {1, 2, 3, 4, 5, 6} → Median = (3 + 4)/2 = 3.5
- Group 2: {7, 8, 9, 10, 11, 12} → Median = (9 + 10)/2 = 9.5
- Group 3: {13, 14, 15, 16, 17, 18} → Median = (15 + 16)/2 = 15.5
- Group 4: {19, 20, 21, 22, 23, 24} → Median = (21 + 22)/2 = 21.5
- Group 5: {25, 26, 27, 28, 29, 30} → Median = (27 + 28)/2 = 27.5
- Group 6: {31, 32, 33, 34, 35, 36} → Median = (33 + 34)/2 = 33.5
The average of the medians:
- x = (3.5 + 9.5 + 15.5 + 21.5 + 27.5 + 33.5) / 6 = 18.5
Finding the Largest Average (x)
To maximize the average of the medians, we should form groups with the largest possible numbers:
- Group 1: {31, 32, 33, 34, 35, 36} → Median = (33 + 34)/2 = 33.5
- Group 2: {25, 26, 27, 28, 29, 30} → Median = (27 + 28)/2 = 27.5
- Group 3: {19, 20, 21, 22, 23, 24} → Median = (21 + 22)/2 = 21.5
- Group 4: {13, 14, 15, 16, 17, 18} → Median = (15 + 16)/2 = 15.5
- Group 5: {7, 8, 9, 10, 11, 12} → Median = (9 + 10)/2 = 9.5
- Group 6: {1, 2, 3, 4, 5, 6} → Median = (3 + 4)/2 = 3.5
The average of the medians:
- x = (33.5 + 27.5 + 21.5 + 15.5 + 9.5 + 3.5) /