find all possible values of x such that, when the five numbers x, 14, 11, 6, 17 are arranged in order, the mean is equal to the median
To find all possible values of x such that the mean is equal to the median when the five numbers are arranged in order, we need to consider the different scenarios that would satisfy this condition.
First, let's arrange the numbers in order from smallest to largest: 6, 11, 14, 17, x.
Next, we need to determine when the mean (denoted by M) and the median (denoted by Me) will be equal:
Scenario 1: x is the middle number (Me)
In this case, the median would be x. To find the mean, we need to add up all five numbers and divide by 5:
M = (6 + 11 + 14 + 17 + x) / 5.
Since the mean (M) and median (Me) are equal, we can write the equation:
x = (6 + 11 + 14 + 17 + x) / 5.
To solve for x, we can multiply both sides of the equation by 5:
5x = 6 + 11 + 14 + 17 + x.
Combining like terms:
5x = 48 + x.
We can subtract x from both sides of the equation:
4x = 48.
Finally, divide both sides by 4 to isolate x:
x = 12.
Thus, in Scenario 1, the possible value of x is 12.
Scenario 2: x is one of the middle numbers (Me)
In this case, the median would be the average of the two middle numbers. Since the numbers are arranged in ascending order, the median would be (11 + 14) / 2 = 25 / 2 = 12.5.
Again, we need to find the mean:
M = (6 + 11 + 14 + 17 + x) / 5.
Setting the mean equal to the median, we get:
12.5 = (6 + 11 + 14 + 17 + x) / 5.
Multiplying both sides by 5:
62.5 = 48 + x.
Subtracting 48 from both sides:
14.5 = x.
Thus, in Scenario 2, the possible value of x is 14.5.
Therefore, the possible values of x that satisfy the condition where the mean is equal to the median are 12 and 14.5.