What is the value of x such that the list 100, 102, 104, x, x, 106, 104, 118, 113, 101 has the same mean and mode?
Well, the mean is the average of all the numbers in the list, and the mode is the number that appears most frequently. So, we need to find the value of x that creates a balanced list.
To find the mean, we sum up all the numbers and divide by the total count. Since there are already 10 numbers given, the sum of those numbers is 100 + 102 + 104 + x + x + 106 + 104 + 118 + 113 + 101 = 1048 + 2x.
Now, to find the mode, we need to find the number that appears most frequently. Currently, 104 appears twice, so we need x to be equal to 104 in order to maintain the mode.
So, if x = 104, the list becomes 100, 102, 104, 104, 104, 106, 104, 118, 113, 101.
The sum of these numbers is 100 + 102 + 104 + 104 + 104 + 106 + 104 + 118 + 113 + 101 = 1056.
The total count is 10, so the mean would be 1056 / 10 = 105.6.
Therefore, x = 104 is the value that creates a list with the same mean and mode.