To find the value of [tex]\( x \)[/tex] given that the mean of the data set [tex]\( x+10, 30, x+20, 4x+5, 40 \)[/tex] is 33, follow these steps:
1. List the data elements: The data elements are [tex]\( x + 10 \)[/tex], [tex]\( 30 \)[/tex], [tex]\( x + 20 \)[/tex], [tex]\( 4x + 5 \)[/tex], and [tex]\( 40 \)[/tex].
2. Set up the mean equation: The mean (average) of a set of data is found by summing all the data elements and then dividing by the number of elements. Given that the mean is 33, we start with the equation for the sum of the elements divided by 5 (since there are 5 elements) being equal to 33:
[tex]\[
\frac{(x + 10) + 30 + (x + 20) + (4x + 5) + 40}{5} = 33
\][/tex]
3. Combine like terms within the numerator: Sum all the [tex]\( x \)[/tex] terms and constant terms inside the fraction:
[tex]\[
\frac{(x + x + 4x) + (10 + 30 + 20 + 5 + 40)}{5} = 33
\][/tex]
Simplify the terms:
[tex]\[
\frac{6x + 105}{5} = 33
\][/tex]
4. Clear the fraction by multiplying both sides by 5:
[tex]\[
6x + 105 = 165
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
[tex]\[
6x = 165 - 105
\][/tex]
[tex]\[
6x = 60
\][/tex]
[tex]\[
x = 10
\][/tex]
6. Verify the solution by checking the mean: Substitute [tex]\( x = 10 \)[/tex] back into the original data set:
[tex]\[
x + 10 = 10 + 10 = 20
\][/tex]
[tex]\[
30 \, \text{(unchanged)}
\][/tex]
[tex]\[
x + 20 = 10 + 20 = 30
\][/tex]
[tex]\[
4x + 5 = 4(10) + 5 = 40 + 5 = 45
\][/tex]
[tex]\[
40 \, \text{(unchanged)}
\][/tex]
Calculate the mean of the new data set [tex]\( 20, 30, 30, 45, 40 \)[/tex]:
[tex]\[
\frac{20 + 30 + 30 + 45 + 40}{5} = \frac{165}{5} = 33
\][/tex]
Since the calculated mean matches the given mean of 33, the solution is verified to be correct.
Therefore, the value of [tex]\( x \)[/tex] is [tex]\( 10 \)[/tex].
