In a set of five distinct positive integers, 1, 3, 8, 12 and x, for how many values of x will the mean of the set be greater than or equal to the median of the set?
Question
In a set of five distinct positive integers, 1, 3, 8, 12 and x, for how many values of x will the mean of the set be greater than or equal to the median of the set?
Solution
To solve this problem, we first need to understand what the mean and median are. The mean is the average of all numbers in a set, and the median is the middle number when the set is arranged in ascending order.
Step 1: Calculate the median In the given set, the numbers are already in ascending order: 1, 3, 8, 12, and x. Since we have 5 numbers, the median will be the third number, which is 8.
Step 2: Calculate the mean The mean is the sum of all numbers divided by the count of numbers. So, the mean of this set is (1 + 3 + 8 + 12 + x) / 5 = (24 + x) / 5.
Step 3: Set up the inequality We want the mean to be greater than or equal to the median, so we set up the following inequality: (24 + x) / 5 >= 8.
Step 4: Solve the inequality To solve the inequality, we first multiply both sides by 5 to get rid of the fraction: 24 + x >= 40. Then, we subtract 24 from both sides to solve for x: x >= 16.
So, the values of x that make the mean greater than or equal to the median are all integers greater than or equal to 16. Since x must be a positive integer and distinct from the other numbers in the set, the possible values for x are 16, 17, 18, 19, 20, 21, and so on. There are infinitely many such values, so the answer is "infinitely many values".
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