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! [ /3 marks]
Let's denote the missing three numbers as a, b, and c.
Since the greatest number is 9 and the least number is 1, we can conclude that a, b, and c must be between 1 and 9.
We know that the mean of the six numbers is 6, so the sum of all six numbers is 6 * 6 = 36.
The median of the six numbers is 7, which means the third and fourth numbers, in ascending order, are both 7. Therefore, one of the missing numbers must be 7.
The mode of the six numbers is 8, which means 8 appears more times than any other number. Since the number 6 is removed, there can only be at most one number that appears twice. Therefore, one of the missing numbers must be 8.
When the number 6 is removed, we have a total of five numbers. The mean is still 6, so the sum of the remaining five numbers is 6 * 5 = 30.
The median is 8, which means the third number, in ascending order, must be 8. Since 7 is also missing, we can conclude that 7 is the fourth number, in ascending order.
The mode is still 8, which means there must be one number that appears more times than any other number. Since we already have 8 and 7 accounted for, the only number left is 9. Therefore, 9 is our fifth number, in ascending order.
Now, let's sum up all the numbers so far: 1 + 7 + 8 + 8 + 9 = 33.
To find the missing numbers, we need to subtract the sum so far from the total sum of the six numbers: 36 - 33 = 3.
So our missing three numbers are 1, 3, and 3 (since we need three numbers).