Sure, let's solve this step-by-step together.
We are given the following values in a continuous series:
- The mean [tex]\(\bar{X} = 20\)[/tex]
- [tex]\(\Sigma fm = 800 + 15p\)[/tex]
- [tex]\(N = 10 + p\)[/tex]
We need to find the value of [tex]\(p\)[/tex] and the exact value of [tex]\(N\)[/tex].
The formula for the mean [tex]\(\bar{X}\)[/tex] in a continuous series is given by:
[tex]\[
\bar{X} = \frac{\Sigma fm}{N}
\][/tex]
Substituting the known values into the formula, we get:
[tex]\[
20 = \frac{800 + 15p}{10 + p}
\][/tex]
To solve for [tex]\(p\)[/tex], we will first eliminate the fraction by multiplying both sides of the equation by [tex]\((10 + p)\)[/tex]:
[tex]\[
20 (10 + p) = 800 + 15p
\][/tex]
Next, distribute the 20 on the left side:
[tex]\[
200 + 20p = 800 + 15p
\][/tex]
Now, we will isolate [tex]\(p\)[/tex] by moving all [tex]\(p\)[/tex] terms to one side and the constants to the other side:
[tex]\[
20p - 15p = 800 - 200
\][/tex]
[tex]\[
5p = 600
\][/tex]
Now, divide both sides by 5 to solve for [tex]\(p\)[/tex]:
[tex]\[
p = \frac{600}{5}
\][/tex]
[tex]\[
p = 120
\][/tex]
So, the value of [tex]\(p\)[/tex] is 120.
To find the exact value of [tex]\(N\)[/tex], substitute [tex]\(p\)[/tex] back into the equation for [tex]\(N\)[/tex]:
[tex]\[
N = 10 + p
\][/tex]
[tex]\[
N = 10 + 120
\][/tex]
[tex]\[
N = 130
\][/tex]
Therefore, the value of [tex]\(p\)[/tex] is [tex]\(120\)[/tex], and the exact value of [tex]\(N\)[/tex] is [tex]\(130\)[/tex].
