### Problem 1
For the set [tex]\((30, 80, 50, 40, x)\)[/tex], we are given that the mean, mode, and median are all equal. We need to determine the value of [tex]\( x \)[/tex].
Step 1: Calculate the mean.
The mean of the set is given by:
[tex]\[
\text{Mean} = \frac{30 + 80 + 50 + 40 + x}{5}
\][/tex]
Step 2: Find the median.
To find the median, we need to consider the sorted order of the numbers. Let's assume [tex]\( x \)[/tex] is positioned in the sorted order [tex]\((30, 40, 50, 80, x)\)[/tex]. Depending on the value of [tex]\( x \)[/tex], it will be placed correctly. The median of five numbers is the third number in the sorted list.
Step 3: Considering mode.
For the mode to be equal to the mean and median, one of the numbers must be repeated. Thus, [tex]\( x \)[/tex] must be one of the existing numbers in the set. Let's try each possibility:
1. If [tex]\( x = 30 \)[/tex]:
[tex]\[
\text{Set} = \{30, 30, 40, 50, 80\}
\][/tex]
[tex]\[
\text{Mean} = \frac{30 + 30 + 40 + 50 + 80}{5} = \frac{230}{5} = 46
\][/tex]
[tex]\[
\text{Median} = 40
\][/tex]
Since mean ([tex]\(46\)[/tex]) ≠ median ([tex]\(40\)[/tex]), this is not a solution.
2. If [tex]\( x = 40 \)[/tex]:
[tex]\[
\text{Set} = \{30, 40, 40, 50, 80\}
\][/tex]
[tex]\[
\text{Mean} = \frac{30 + 40 + 40 + 50 + 80}{5} = \frac{240}{5} = 48
\][/tex]
[tex]\[
\text{Median} = 40
\][/tex]
Since mean ([tex]\(48\)[/tex]) ≠ median ([tex]\(40\)[/tex]), this is not a solution.
3. If [tex]\( x = 50 \)[/tex]:
[tex]\[
\text{Set} = \{30, 40, 50, 50, 80\}
\][/tex]
[tex]\[
\text{Mean} = \frac{30 + 40 + 50 + 50 + 80}{5} = \frac{250}{5} = 50
\][/tex]
[tex]\[
\text{Median} = 50
\][/tex]
Since mean ([tex]\(50\)[/tex]) = median ([tex]\(50\)[/tex]), this is a possible solution. Now we need to confirm if mode [tex]\((50)\)[/tex] is also [tex]\(50\)[/tex].
This set satisfies as all mean, median, and mode equal to [tex]\(50\)[/tex].
Therefore, [tex]\( x = 50 \)[/tex].
### Problem 2
We are given that the sum of a whole number and the next four consecutive whole numbers is 105. We need to find the result when the mean of the numbers is subtracted from the median of the numbers.
Step 1: Define the sequence of numbers.
Let the numbers be [tex]\( n, n+1, n+2, n+3, n+4 \)[/tex].
Step 2: Calculate the sum of the sequence.
The sum is given by:
[tex]\[
n + (n+1) + (n+2) + (n+3) + (n+4) = 105
\][/tex]
[tex]\[
5n + 10 = 105
\][/tex]
[tex]\[
5n = 95
\][/tex]
[tex]\[
n = 19
\][/tex]
So, the numbers are [tex]\( 19, 20, 21, 22, 23 \)[/tex].
Step 3: Find the mean.
The mean of these numbers is given by:
[tex]\[
\text{Mean} = \frac{19 + 20 + 21 + 22 + 23}{5} = \frac{105}{5} = 21
\][/tex]
Step 4: Find the median.
The median of the numbers [tex]\(19, 20, 21, 22, 23\)[/tex] is the middle value:
[tex]\[
\text{Median} = 21
\][/tex]
Step 5: Find the result when the mean is subtracted from the median.
[tex]\[
\text{Result} = \text{Median} - \text{Mean} = 21 - 21 = 0
\][/tex]
Thus, the result is [tex]\( 0 \)[/tex].
### Summary of Answers
1. [tex]\( x = 50 \)[/tex]
2. The result when the mean of the numbers is subtracted from the median of the numbers is [tex]\( 0 \)[/tex].
