I randomly chose a number and decided to make that the mean, mode, median and range.
Let us make the number I chose to be ?
To make the number ? the mean, mode, median and range, I would need to do something like this:
x , ? , ? , ? , x+?
where x is a number(it could be ?, it could be something else).
But for the actual questions:
A. Mode < Median < Mean
So the mode has to be smaller than the median, which is smaller than the mean.
So, if you do something like this(z is the median, and between x and z there has to be a mode, and x is the mode because the mode has to be smaller than the median):
x , x , y , ? , ?+(any number other than zero)
you can make x a very small number(smaller than median), ? a much larger number, and there is nothing stated about the range, so this can be a solution:
2, 2, 3, 8, 9 (mean 4.8)
B. Mode < Mean < Median
So the mode has to be smaller than the mean and the mean is smaller than the median.
The mode can't be after the median(mode is smaller) and the mean has to be smaller than the median.
Something like this would work:
x , x , y , y + 1 , y + 2
There was something saying we weren't allowed negatives, otherwise we could have done this...(x must be really really small):
-25, -25, 0, 1, 2 (mean is -9.4)
However, if we try with positive numbers(numbers larger than 0), we can't solve this:
1, 1, 2, 3, 4 (mean is 2.2)
C. Mean < Mode < Median
So the mean has to be smaller than the mode, which has to be smaller than the median.
Something like this would work:
x, x, y, y+1, y+2
The mean must be the smallest out of the averages, but let's try(with no negatives):
1, 1, 2, 3, 4(mean 2.2)
The mean has to be smaller than the mode. The mode can't be bigger than the median, so we can't do this as the median is the middle number and the numbers following it can't be the same as the median or more because the mean has to be less than the mode.
D. Mean < Median < Mode
So, the mean must be smaller than the median, which is smaller than the mode.
Something like this would work:
x , x+1 , y , y+1 , y+1
And with no negatives, the minimum we can make the mean is(with regards to all other averages):
1, 2, 3, 4, 4 (mean 2.8)
It works!
E. Median < Mode < Mean
So the median is smaller than the mode, which is smaller than the mean.
Something like this would work:
x , x+1 , y , y+1 , y+1
Let's try do this.
1, 2, 3, 4, 4 (mean 2.8)
It doesn't work.
F. Median < Mean < Mode
So the median is smaller than the mean, which is smaller than the mode.
Something like this would work:
x , x+1 , y , (any number bigger than y) , (any number bigger than y)
So let's try do this.
1, 2, 3, 900, 900(mean 361.2)
It works!