Certainly! Let's determine the value of [tex]\( x \)[/tex] given that the numbers in the list are [tex]\( 12.5, -10, -7.5, x \)[/tex] and their mean is [tex]\( 11.5 \)[/tex].
1. Calculate the sum of the given numbers:
[tex]\[
12.5 + (-10) + (-7.5) = 12.5 - 10 - 7.5 = -5.0
\][/tex]
So, the sum of the three given numbers is [tex]\( -5.0 \)[/tex].
2. Determine the total number of elements:
We have three numbers given and one unknown number [tex]\( x \)[/tex]. Thus, there are four numbers in total.
[tex]\[
\text{Number of elements} = 4
\][/tex]
3. Use the mean to find the total sum:
The mean of a set of numbers is defined as the total sum of the numbers divided by the number of elements. Given the mean (11.5) and the number of elements (4), we set up the equation:
[tex]\[
\frac{\text{Total sum}}{\text{Number of elements}} = 11.5
\][/tex]
Therefore:
[tex]\[
\text{Total sum} = 11.5 \times 4 = 46.0
\][/tex]
4. Determine the value of [tex]\( x \)[/tex]:
We know the sum of the three given numbers ([tex]\( -5.0 \)[/tex]) and need to combine this with [tex]\( x \)[/tex] to get the total sum (46.0). The equation to solve for [tex]\( x \)[/tex] is:
[tex]\[
-5.0 + x = 46.0
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by adding 5.0 to both sides of the equation:
[tex]\[
x = 46.0 + 5.0 = 51.0
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( 51.0 \)[/tex].
