e) Calculate the average of the first five prime numbers.

f) Calculate the average of the first five multiples of 5.

5. a) If the average of [tex]7, 12, 6, 10, [/tex] and [tex] p [/tex] is 8, find the value of [tex] p [/tex].



Answer :

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].