A certain list consists of 3 different numbers. Does the median of the 3 numbers equal the average (arithmetic mean) of the 3 numbers?(1) The range of the 3 numbers is equal to twice the difference between the greatest number and the median.(2) The sum of the 3 numbers is equal to 3 times one of the numbers.

Answer with explanation:A certain list consists of 3 different numbers. Does the median of the 3 numbers equal the average (arithmetic mean) of the 3 numbers?Let the three numbers be x, y, and z, where x < y < z. Now, the median will be y and the average will be [tex](x+y+z)/3[/tex]We have to tell if [tex]y=(x+y+z)/3[/tex] or [tex]2y=x+z[/tex](1) The range of the 3 numbers is equal to twice the difference between the greatest number and the median.Range is the largest number minus the smallest number of the set,Here range =  z - x. We are given in the statement that [tex]z-x=2(z-y)[/tex]Solving this we get;[tex]z-x=2z-2y[/tex]or [tex]2y=x+ z[/tex] So, this condition is fulfilled.(2) The sum of the 3 numbers is equal to 3 times one of the numbers.The sum of 3 numbers cannot be equal to 3 times the smallest number or 3 times largest number as the given numbers are distinct and they cannot be equal to mean. So, median = mean.[tex]x+y+z=3y[/tex]or [tex]x+z=2y[/tex]So, this condition is fulfilled.