Subject: mean median and mode
Name: rick

Question:
I have five number places and the mean and median are both "0". The mode is greater than the mean. The sum of the absolute values of all of the numbers is 20. The smallest number is 4 less than the next smallest number. What are the five numbers. (negative integers can be used) All of the numbers are integers.

Hi Rick,

Since the median is the middle number when they are sorted from smallest to largest, the middle number is zero. If zero appears twice in the list then, since the mode is larger than zero, the other three numbers must all have the same valus and be larger than zero. In this case the mean could not be zero. Thus zero must appear exactly once in the list of five numbers.
   Since the mode is larger than zero at least two of the four non-zero numbers are larger than zero. If three of the numbers are larger than zero then, since smallest number is 4 less than the next smallest number the numbers would have to be

-4, 0, k, m, n

where k, m and n are positive integers (possibly all equal). Since the mean is zero

k + m + n + (-4) = 0 or
k + m + n = 4

Since the sum of the absolute values is 20

k + m + n + 4 = 20 or
k + m + n = 16

This is clearly impossible so there must be exactly two of the numbers that are greater than zero, and since the mode is greater than zero these two numbers must have the same value. Hence the five numbers must be

-a, -b, 0, c, c

where a, b, and c are positive integers.
   Since the mean is zero

-a -b + c + c = 0 or
a + b = 2c

Since the sum of the absolute values is 20

a + b + 2c = 20

But a + b = 2c so 4c = 20, that is c = 5. Also a + b = 10 and -b is 4 more than -a so the four numbers are

-7, -3, 0, 5, 5

Cheers,
Penny

 

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