The median for a set of 10 numbers is 10. An even number is added, and the median is now 11 and the range is now 4. What number was added?
If the numbers are x1-x10, then the median is
(x5 + x6)/2 = 10
Since the range was less than 4, that means that the numbers in order were
8 x x x 9 11 11 11 11 11
or
9 9 9 9 9 11 x x x 12 12
The median moved upward, so the even number was added on the top side, so it must have been 12.
Well, well, well, looks like we have a tricky situation here! Let me put on my juggling hat and see what I can do with these numbers.
Okay, so we know the median is 10 for the original set of 10 numbers. That means the fifth number in the set must be 10. (Imagine a parade and 10 ducks marching by, with the middle duck being number 5.)
Now, when we add an even number to the set, the median becomes 11. Hmm... how did that happen? Let's think about it. Since the median increased by 1, we must have added a number larger than the fifth number of the original set (which was 10).
But wait! Hold your laughter, because there's more! The range of the set also increased by 4. To calculate the range, we take the largest number minus the smallest number. So if we add 4 to the range, it means the largest number increased by 4.
So, putting all these clues together, the number added must be larger than 10, but not larger than 11, because it becomes the new median. And it must be larger than the original largest number by 4 to increase the range.
*drumroll please*
Therefore, the number that was added to the set is...
**15!** Because it's larger than 10 (original median) but smaller than 11 (new median), and it increases the range by 4. Ta-da!
I hope my clownish logic made you smile!
To find the number that was added, let's break down the information given step by step.
Step 1: We are told that the median for a set of 10 numbers is 10.
- Since the set of numbers has an odd length (10), the median is the middle number when the numbers are arranged in ascending order.
Step 2: An even number is added, and the median is now 11.
- Since the median increased from 10 to 11, the added number must be larger than the current median.
Step 3: The range is now 4.
- The range is the difference between the highest and lowest values in a set of numbers.
- Since the range increased from its initial value, the added number must be greater than the highest number in the original set of 10 numbers.
Step 4: We summarize the information.
- The added number is larger than the median (11) and larger than the highest number in the original set.
Based on the given information, it is impossible to determine the exact number that was added.
To find the number that was added, we need to analyze the information provided and determine what changed when the even number was added.
We are given that the median of the initial set of 10 numbers is 10. This means that when the numbers are arranged in ascending order, the fifth number is 10.
When an even number is added, it affects the positioning of the median. We are told that the new median is 11. This means that after adding the even number, the fifth number is now 11.
Additionally, we are given that the range of the new set of numbers is 4. The range is the difference between the largest and smallest numbers in a set. So, when the even number was added, the largest number increased by 4 compared to the initial set.
Since the fifth number changed from 10 to 11, and the largest number increased by 4, we can determine the number that was added by calculating the difference between 11 and 10, and then subtracting 4.
11 (new median) - 10 (initial median) = 1
1 - 4 = -3
From this calculation, we find that the number that was added is -3.