Let's 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.
### Step-by-Step Solution:
1. Understanding the Mean:
The mean (average) of a set of numbers is the sum of the numbers divided by the count of the numbers.
2. Given Information:
- Data set: [tex]\( x+10, 30, x+20, 4x+5, 40 \)[/tex]
- Mean of the data set: 33
- Count of the data points: 5
3. Setting Up the Equation for the Mean:
The formula for the mean is:
[tex]\[
\text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}}
\][/tex]
Therefore, we have:
[tex]\[
33 = \frac{(x+10) + 30 + (x+20) + (4x+5) + 40}{5}
\][/tex]
4. Calculate the Sum of the Data Points:
Combine all the data points:
[tex]\[
(x+10) + 30 + (x+20) + (4x+5) + 40
\][/tex]
Simplify inside the parentheses:
[tex]\[
x + 10 + 30 + x + 20 + 4x + 5 + 40
\][/tex]
Combine like terms:
[tex]\[
x + x + 4x + 10 + 30 + 20 + 5 + 40
\][/tex]
[tex]\[
6x + 105
\][/tex]
5. Set Up the Equation for the Mean:
Replace the sum in the mean formula:
[tex]\[
33 = \frac{6x + 105}{5}
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], first eliminate the fraction by multiplying both sides by 5:
[tex]\[
33 \times 5 = 6x + 105
\][/tex]
[tex]\[
165 = 6x + 105
\][/tex]
Subtract 105 from both sides:
[tex]\[
165 - 105 = 6x
\][/tex]
[tex]\[
60 = 6x
\][/tex]
Finally, divide both sides by 6:
[tex]\[
x = \frac{60}{6}
\][/tex]
[tex]\[
x = 10
\][/tex]
### Conclusion:
The value of [tex]\( x \)[/tex] that satisfies the condition where the mean of the given data set is 33 is [tex]\( x = 10 \)[/tex].
