In a set of five consecutive positive even integers, the ratio of the greatest integer to least integer is 2 is to 1. If these integers are arranged from lowest to highest, which is the middle integer in the list?
If x is the middle integer,
(x+4)/(x-4) = 2
12
To find the middle integer in the list of five consecutive positive even integers, we need to first determine the value of the smallest integer.
Let's assume the smallest integer is represented by "x." Since the integers are consecutive and even, the next four integers will be x+2, x+4, x+6, and x+8.
According to the given information, the ratio of the greatest integer to the least integer is 2:1. Therefore, we can set up the following equation:
(x+8) : x = 2 : 1
To solve for x, we can cross-multiply:
2x + 16 = x
Simplifying the equation, we subtract x from both sides:
x = -16
However, since we are dealing with positive even integers, a negative value is not possible. So, we need to adjust our equation.
Since the smallest positive integer is at least 2, we can set up a new equation:
(x+8) : 2 = 2 : 1
Cross-multiplying:
2(x+8) = 2
Simplifying and expanding:
2x + 16 = 2
Now, we subtract 16 from both sides:
2x = -14
Dividing both sides by 2, we find:
x = -7
Again, a negative value is not acceptable, so we need to adjust our equation.
Since the smallest even positive integer is at least 2, we set up another equation:
(x+8) : 2 = 2 : 1
Cross-multiplying:
2(x+8) = 4
Expanding and simplifying:
2x + 16 = 4
Subtracting 16 from both sides:
2x = -12
Dividing both sides by 2:
x = -6
Since the smallest even positive integer cannot be -6, we realize that we made an error.
We can try setting up a new equation:
(x+8) : (x+2) = 2 : 1
Cross-multiplying:
2(x+2) = (x+8)
Expanding and simplifying:
2x + 4 = x + 8
Subtracting x from both sides:
x + 4 = 8
Subtracting 4 from both sides:
x = 4
Thus, the smallest even positive integer is x = 4, and the five consecutive positive even integers are 4, 6, 8, 10, 12.
The middle integer in this list is 8.