A list of ten distinct, positive integers has a median of 5.5. What is the smallest possible average of the ten positive integers? Explain your ansswer in complete sentences.
The smallest possible average of ten distinct, positive integers with a median of 5.5 is 4.95. This can be achieved by having the first five integers be 1, 2, 3, 4, and 5, and the last five be 6, 7, 8, 9, and 10. To calculate the average, we add all ten integers together and divide by 10: (1+2+3+4+5+6+7+8+9+10)/10 = 55/10 = 5.5. Since we want the smallest possible average, we need to minimize the sum of the ten integers. By having the first five integers start at 1 and increase by 1 up to 5, and having the last five integers start at 6 and increase by 1 up to 10, we achieve the smallest possible sum of 1+2+3+4+5+6+7+8+9+10 = 55, and therefore the smallest possible average of 4.95.