Six numbers are all less than 10. The greatest number is 9, and the least number is 1. The mean of these numbers is 6, the median is 7, and the mode is 8. When the number 6 is removed, the mean of the remaining five numbers is 6, the median is 8, and the mode is 8. What three numbers are missing? SHOW YOUR WORK TO RECEIVE FULL MARKS! [
Let's call the missing numbers a, b, c.
From the given information, we know that the numbers are 1, a, b, c, 6, 9.
To find the mean, we can sum up all the numbers and divide by the total count:
(1 + a + b + c + 6 + 9) / 6 = 6
17 + a + b + c = 36
a + b + c = 19 ...........(1)
To find the median, we arrange the numbers in order: 1, a, b, c, 6, 9.
Since the median is 7, we have b = 7.
To find the mode, we look for the number that appears most frequently.
Since the mode is 8, we must have either a = 8 or c = 8.
Now, let's consider the second set of numbers after removing 6.
The numbers become 1, a, b, c, 9.
To find the mean, we sum up all the numbers and divide by the total count:
(1 + a + b + c + 9) / 5 = 6
a + b + c + 10 = 30
a + b + c = 20 ...........(2)
To find the median, we arrange the numbers in order: 1, a, b, c, 9.
Since the median is 8, we have either a = 8 or c = 8.
To find the mode, we look for the number that appears most frequently.
Since the mode is 8, we again have either a = 8 or c = 8.
Now, we have two cases to consider:
Case 1: a = 8
From equation (1), we have 8 + b + c = 19
Since b = 7, we substitute it into the equation:
8 + 7 + c = 19
c = 4
So the missing numbers are 4, 7, and 8.
Case 2: c = 8
From equation (1), we have a + b + 8 = 19
Since b = 7, we substitute it into the equation:
a + 7 + 8 = 19
a = 4
So the missing numbers are 4, 7, and 8.
Therefore, the three missing numbers are 4, 7, and 8.