Sure, let's work through each part of the problem step-by-step.
Part (a):
We are given the following information:
- The mean, [tex]\(\bar{x}\)[/tex], is 9.
- The sum of the values, [tex]\(\Sigma x\)[/tex], is [tex]\(80 + p\)[/tex].
- The number of values, [tex]\(N\)[/tex], is 10.
To find [tex]\(p\)[/tex], we use the formula for the mean of a dataset:
[tex]\[
\bar{x} = \frac{\Sigma x}{N}
\][/tex]
Substituting the given values into the formula, we get:
[tex]\[
9 = \frac{80 + p}{10}
\][/tex]
To isolate [tex]\(p\)[/tex], we multiply both sides of the equation by 10:
[tex]\[
9 \times 10 = 80 + p
\][/tex]
Simplifying further:
[tex]\[
90 = 80 + p
\][/tex]
To solve for [tex]\(p\)[/tex], we subtract 80 from both sides:
[tex]\[
90 - 80 = p
\][/tex]
[tex]\[
p = 10
\][/tex]
So, the value of [tex]\(p\)[/tex] is [tex]\(10\)[/tex].
Part (b):
We are given the following information:
- The sum of the values, [tex]\(\Sigma x\)[/tex], is [tex]\(60 + a\)[/tex].
- The number of values, [tex]\(N\)[/tex], is [tex]\(a - 4\)[/tex].
- The mean, [tex]\(\bar{x}\)[/tex], is 5.
To find [tex]\(a\)[/tex], we use the formula for the mean of a dataset:
[tex]\[
\bar{x} = \frac{\Sigma x}{N}
\][/tex]
Substituting the given values into the formula, we get:
[tex]\[
5 = \frac{60 + a}{a - 4}
\][/tex]
To isolate [tex]\(a\)[/tex], we multiply both sides of the equation by [tex]\((a - 4)\)[/tex]:
[tex]\[
5(a - 4) = 60 + a
\][/tex]
Expanding the left-hand side:
[tex]\[
5a - 20 = 60 + a
\][/tex]
To solve for [tex]\(a\)[/tex], we first subtract [tex]\(a\)[/tex] from both sides:
[tex]\[
5a - a - 20 = 60
\][/tex]
[tex]\[
4a - 20 = 60
\][/tex]
Next, we add 20 to both sides:
[tex]\[
4a - 20 + 20 = 60 + 20
\][/tex]
[tex]\[
4a = 80
\][/tex]
Finally, we divide by 4:
[tex]\[
a = \frac{80}{4}
\][/tex]
[tex]\[
a = 20
\][/tex]
So, the value of [tex]\(a\)[/tex] is [tex]\(20\)[/tex].
To summarize:
- The value of [tex]\(p\)[/tex] is [tex]\(10\)[/tex].
- The value of [tex]\(a\)[/tex] is [tex]\(20\)[/tex].
