To find four numbers that fit all the rules, we can use the information given to create a system of equations:
1. The median is 5: Since we have four numbers, the median is the average of the two middle numbers. Let's call the two middle numbers x and y. Then, we have:
(x + y) / 2 = 5
2. The mode is 5: The mode is the number that appears most frequently. Since 5 is the mode, at least two of the four numbers must be 5. Let's say that the other two numbers are a and b.
3. The range is 1: The range is the difference between the largest and smallest numbers. Since the range is 1, we have:
max(x, y, a, b) - min(x, y, a, b) = 1
Now we can use these equations to solve for the four numbers:
From equation 1, we have:
x + y = 10
From equation 2, we know that two of the numbers are 5. Without loss of generality, let's assume that x = 5 and y = 5. Then, from equation 1, we have:
5 + 5 = 10
So, we still have:
a + b = 0
From equation 3, we know that the largest number minus the smallest number is 1. Since we have two 5's, either a or b must be the smallest number, and the other must be the largest number. Without loss of generality, let's assume that a is the smallest number and b is the largest number. Then, we have:
b - a = 1
Substituting x = 5 and y = 5 into the first equation, we get:
5 + 5 + a + b = 10
a + b = 0
So, we have:
b = -a
Substituting b = -a into the third equation, we get:
(-a) - a = 1
-2a = 1
a = -1/2
b = -a = 1/2
Therefore, the four numbers that fit all the rules are:
-1/2, 5, 5, 1/2.