To find the value of [tex]\( y \)[/tex] given that the mean (average) of the numbers [tex]\( 11, 15, y, 13, \)[/tex] and 18 is 12, follow these steps:
1. Understand the Mean Formula:
The mean of a set of numbers is given by the sum of the numbers divided by the count of the numbers.
[tex]\[
\text{Mean} = \frac{\sum \text{(numbers)}}{\text{Count of numbers}}
\][/tex]
2. Define the Known Values:
- The numbers are [tex]\( 11, 15, 13, 18 \)[/tex], and [tex]\( y \)[/tex].
- The mean is given as 12.
- Let's denote the sum of the known numbers as [tex]\( \text{Sum\_known} \)[/tex].
3. Calculate the Sum of the Known Numbers:
[tex]\[
\text{Sum\_known} = 11 + 15 + 13 + 18 = 57
\][/tex]
4. Total Count of Numbers:
Since there are 4 known numbers and 1 unknown number [tex]\( y \)[/tex], the total count of numbers is 5.
5. Set Up the Mean Equation:
According to the mean formula:
[tex]\[
\text{Mean} = \frac{\text{Sum\_total}}{\text{Count of numbers}}
\][/tex]
Where [tex]\(\text{Sum\_total}\)[/tex] is the total sum of the numbers including [tex]\( y \)[/tex].
[tex]\[
12 = \frac{57 + y}{5}
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
Multiply both sides of the equation by 5 to remove the denominator:
[tex]\[
12 \times 5 = 57 + y
\][/tex]
[tex]\[
60 = 57 + y
\][/tex]
Subtract 57 from both sides of the equation to isolate [tex]\( y \)[/tex]:
[tex]\[
y = 60 - 57
\][/tex]
[tex]\[
y = 3
\][/tex]
Conclusion:
The value of [tex]\( y \)[/tex] that makes the mean of the set [tex]\( 11, 15, y, 13, 18 \)[/tex] equal to 12 is [tex]\( 3 \)[/tex].
