High School

We appreciate your visit to Given the following information Universal set tex U tex such that tex n U 100 tex Subsets tex A tex and tex B tex of. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Given the following information:

- Universal set [tex] U [/tex] such that [tex] n(U) = 100 [/tex]
- Subsets [tex] A [/tex] and [tex] B [/tex] of [tex] U [/tex]
- [tex] n(A - B) = 32 + x [/tex]
- [tex] n(B - A) = 5x [/tex]
- [tex] n(A \cap B) = x [/tex]
- [tex] n(A \cup B) = [/tex] (value needed)

Tasks:

a) Represent the information above in a Venn diagram.

b) If [tex] n(A) = n(B) [/tex], find the value of [tex] n(A \cap B) [/tex].

c) Find the value of [tex] n(A \cup B) [/tex].

d) Calculate the ratio of [tex] n(A \cap B) [/tex] to [tex] n(A \cup B) [/tex] when [tex] n(U) = 50 [/tex].

e) By what percent is [tex] n(A \cap B) [/tex] more or less than [tex] n(A \cup B) [/tex]? Find the percentage.

Answer :

We are given the following information about two subsets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] of a universal set [tex]\( U \)[/tex] with [tex]\( n(U) = 100 \)[/tex]:

- [tex]\( n(A - B) = 32 + x \)[/tex]
- [tex]\( n(B - A) = 5x \)[/tex]
- [tex]\( n(A \cap B) = x \)[/tex]

We need to solve the following:

──────────────────────────────
Step (a): Venn Diagram

Draw two overlapping circles labeled [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Label the regions as follows:
- In the part of [tex]\( A \)[/tex] that is not in [tex]\( B \)[/tex]: [tex]\( n(A - B) = 32 + x \)[/tex]
- In the part of [tex]\( B \)[/tex] that is not in [tex]\( A \)[/tex]: [tex]\( n(B - A) = 5x \)[/tex]
- In the overlapping region (intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]): [tex]\( n(A \cap B) = x \)[/tex]

──────────────────────────────
Step (b): Finding [tex]\( n(A \cap B) \)[/tex] When [tex]\( n(A) = n(B) \)[/tex]

First, express the total number of elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in terms of [tex]\( x \)[/tex]:

[tex]\[
n(A) = n(A - B) + n(A \cap B) = (32 + x) + x = 32 + 2x
\][/tex]

[tex]\[
n(B) = n(B - A) + n(A \cap B) = 5x + x = 6x
\][/tex]

Since it is given that [tex]\( n(A) = n(B) \)[/tex], equate these two expressions:

[tex]\[
32 + 2x = 6x
\][/tex]

Solve for [tex]\( x \)[/tex]:

[tex]\[
32 = 6x - 2x \quad \Longrightarrow \quad 32 = 4x \quad \Longrightarrow \quad x = 8
\][/tex]

Thus,

[tex]\[
n(A \cap B) = x = 8
\][/tex]

──────────────────────────────
Step (c): Finding [tex]\( n(A \cup B) \)[/tex] and the Percent Difference

1. Calculate [tex]\( n(A) \)[/tex] and [tex]\( n(B) \)[/tex] with [tex]\( x = 8 \)[/tex]:

[tex]\[
n(A) = 32 + 2(8) = 32 + 16 = 48
\][/tex]
[tex]\[
n(B) = 6(8) = 48
\][/tex]

2. Use the formula for the union of two sets:

[tex]\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\][/tex]

Substitute the values we have:

[tex]\[
n(A \cup B) = 48 + 48 - 8 = 96 - 8 = 88
\][/tex]

3. Determine by what percent [tex]\( n(A \cap B) \)[/tex] is less than [tex]\( n(A \cup B) \)[/tex]:

The difference between [tex]\( n(A \cup B) \)[/tex] and [tex]\( n(A \cap B) \)[/tex] is:

[tex]\[
n(A \cup B) - n(A \cap B) = 88 - 8 = 80
\][/tex]

The percent difference is:

[tex]\[
\text{Percent Difference} = \frac{80}{88} \times 100\% \approx 90.91\%
\][/tex]

──────────────────────────────
Final Answers:

- [tex]\( n(A \cap B) = 8 \)[/tex]
- [tex]\( n(A \cup B) = 88 \)[/tex]
- [tex]\( n(A \cap B) \)[/tex] is approximately [tex]\( 90.91\% \)[/tex] less than [tex]\( n(A \cup B) \)[/tex].

This completes the detailed step-by-step solution.

Thanks for taking the time to read Given the following information Universal set tex U tex such that tex n U 100 tex Subsets tex A tex and tex B tex of. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada