Certainly! Let's solve the problem step by step.
### Problem:
If the average of the numbers [tex]\(7, 12, 6, 10\)[/tex], and [tex]\(p\)[/tex] is 8, find the value of [tex]\(p\)[/tex].
### Step-by-Step Solution:
1. Understanding the Average Formula:
The average of a set of numbers is given by the sum of the numbers divided by the number of elements in the set.
Mathematically:
[tex]\[
\text{Average} = \frac{\text{Sum of elements}}{\text{Number of elements}}
\][/tex]
2. Given Data:
- The numbers are [tex]\(7, 12, 6, 10\)[/tex], and [tex]\(p\)[/tex].
- The target average is 8.
3. Setting Up the Equation:
Let’s denote the sum of these numbers by [tex]\(S\)[/tex].
The sum [tex]\(S\)[/tex] for the numbers [tex]\(7, 12, 6, 10\)[/tex], and [tex]\(p\)[/tex] can be written as:
[tex]\[
S = 7 + 12 + 6 + 10 + p
\][/tex]
4. Number of Elements:
There are 5 numbers in total.
5. Using the Average Formula:
Given that the average of these numbers is 8, we can set up the equation:
[tex]\[
8 = \frac{7 + 12 + 6 + 10 + p}{5}
\][/tex]
6. Solving the Equation:
To find [tex]\(p\)[/tex], we first solve for the sum:
[tex]\[
8 \times 5 = 7 + 12 + 6 + 10 + p
\][/tex]
[tex]\[
40 = 7 + 12 + 6 + 10 + p
\][/tex]
7. Calculating the Current Sum:
Calculate the sum of the given numbers:
[tex]\[
7 + 12 + 6 + 10 = 35
\][/tex]
8. Finding [tex]\(\mathbf{p}\)[/tex]:
Now substitute the known sum into the equation:
[tex]\[
40 = 35 + p
\][/tex]
9. Isolating [tex]\(p\)[/tex]:
Solve for [tex]\(p\)[/tex] by subtracting 35 from both sides:
[tex]\[
p = 40 - 35
\][/tex]
[tex]\[
p = 5
\][/tex]
### Conclusion:
Therefore, the value of [tex]\(p\)[/tex] is [tex]\(\boxed{5}\)[/tex].
