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Answer :
To find the unknown value [tex]\(x\)[/tex] in the sequence [tex]\(12.5, -10, -7.5, x\)[/tex] with a mean of 11.5, follow these steps:
1. Understand the Mean Formula: The mean (or average) of a set of numbers is calculated by adding up all the numbers and then dividing by the count of numbers. Mathematically, it is expressed as:
[tex]\[
\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of values}}
\][/tex]
2. Set up the Equation:
- We know the mean is 11.5.
- The numbers we have are [tex]\(12.5\)[/tex], [tex]\(-10\)[/tex], [tex]\(-7.5\)[/tex], and [tex]\(x\)[/tex].
- There are 4 numbers in total.
Plug these into the mean formula:
[tex]\[
\frac{12.5 + (-10) + (-7.5) + x}{4} = 11.5
\][/tex]
3. Calculate the Known Sum:
- Add together the known values:
[tex]\[ 12.5 + (-10) + (-7.5) = -5.0 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Substitute the known sum into the equation and solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-5.0 + x}{4} = 11.5
\][/tex]
- Multiply both sides by 4 to clear the fraction:
[tex]\[
-5.0 + x = 46
\][/tex]
- Add 5.0 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 46 + 5.0
\][/tex]
[tex]\[
x = 51.0
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(51.0\)[/tex].
1. Understand the Mean Formula: The mean (or average) of a set of numbers is calculated by adding up all the numbers and then dividing by the count of numbers. Mathematically, it is expressed as:
[tex]\[
\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Total number of values}}
\][/tex]
2. Set up the Equation:
- We know the mean is 11.5.
- The numbers we have are [tex]\(12.5\)[/tex], [tex]\(-10\)[/tex], [tex]\(-7.5\)[/tex], and [tex]\(x\)[/tex].
- There are 4 numbers in total.
Plug these into the mean formula:
[tex]\[
\frac{12.5 + (-10) + (-7.5) + x}{4} = 11.5
\][/tex]
3. Calculate the Known Sum:
- Add together the known values:
[tex]\[ 12.5 + (-10) + (-7.5) = -5.0 \][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Substitute the known sum into the equation and solve for [tex]\(x\)[/tex]:
[tex]\[
\frac{-5.0 + x}{4} = 11.5
\][/tex]
- Multiply both sides by 4 to clear the fraction:
[tex]\[
-5.0 + x = 46
\][/tex]
- Add 5.0 to both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x = 46 + 5.0
\][/tex]
[tex]\[
x = 51.0
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\(51.0\)[/tex].
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