We read your photo as:

A group of 70 females play Volleyball, Basketball, or Table-tennis. 20 play only Volleyball, 10 play only Basketball, and 6 play only Table-tennis. 4 play all three games. An equal number play exactly

Mathematics

ScanSolve AI Answer

Answers

4 parts
b(i)

44\text{44}

b(ii)

34\text{34}

b(iii)

30\text{30}

c

30\text{30}

Step-by-step solution

2 steps

Step 1: Determine the number of females who play exactly two games.
Let V, B, and T represent the sets of females who play Volleyball, Basketball, and Table-tennis, respectively.
Given information:

  • Total females = 70
  • V only = 20
  • B only = 10
  • T only = 6
  • V ∩\cap B ∩\cap T (all three games) = 4
  • An equal number play exactly two games only. Let this number be xx. So, (V ∩\cap B) only = xx, (V ∩\cap T) only = xx, (B ∩\cap T) only = xx.

The sum of all disjoint regions in the Venn diagram must equal the total number of females:

(V only)+(B only)+(T only)+(V∩B only)+(V∩T only)+(B∩T only)+(V∩B∩T)=70(\text{V only}) + (\text{B only}) + (\text{T only}) + (\text{V} \cap \text{B only}) + (\text{V} \cap \text{T only}) + (\text{B} \cap \text{T only}) + (\text{V} \cap \text{B} \cap \text{T}) = 70 20+10+6+x+x+x+4=7020 + 10 + 6 + x + x + x + 4 = 70

Combine the constant terms:

40+3x=7040 + 3x = 70

Subtract 40 from both sides:

3x=70−403x = 70 - 40 3x=303x = 30

Divide by 3:

x=10x = 10

So, 10 females play exactly two games for each pair of sports.

Step 2: Illustrate the information on a Venn diagram.

Part a· 1 step

The Venn diagram should have three overlapping circles (V, B, T) within a universal set U.

Label the regions as follows:

  • V only: 20
  • B only: 10
  • T only: 6
  • V ∩\cap B only: 10
  • V ∩\cap T only: 10
  • B ∩\cap T only: 10
  • V ∩\cap B ∩\cap T: 4
  • The region outside the circles (females playing no games) is 0, as all females play at least one game. The universal set U = 70.

  1. Find the number of females who play each game.

Part b(i)

44\text{44}

Volleyball

The number of females who play Volleyball includes those who play V only, V and B only, V and T only, and all three.

∣V∣=(V only)+(V∩B only)+(V∩T only)+(V∩B∩T)|\text{V}| = (\text{V only}) + (\text{V} \cap \text{B only}) + (\text{V} \cap \text{T only}) + (\text{V} \cap \text{B} \cap \text{T}) ∣V∣=20+10+10+4=44|\text{V}| = 20 + 10 + 10 + 4 = 44

The number of females who play Volleyball is 44\boxed{\text{44}}.

Part b(ii)

34\text{34}

Basketball

The number of females who play Basketball includes those who play B only, V and B only, B and T only, and all three.

∣B∣=(B only)+(V∩B only)+(B∩T only)+(V∩B∩T)|\text{B}| = (\text{B only}) + (\text{V} \cap \text{B only}) + (\text{B} \cap \text{T only}) + (\text{V} \cap \text{B} \cap \text{T}) ∣B∣=10+10+10+4=34|\text{B}| = 10 + 10 + 10 + 4 = 34

The number of females who play Basketball is 34\boxed{\text{34}}.

Part b(iii)

30\text{30}

Table-tennis

The number of females who play Table-tennis includes those who play T only, V and T only, B and T only, and all three.

∣T∣=(T only)+(V∩T only)+(B∩T only)+(V∩B∩T)|\text{T}| = (\text{T only}) + (\text{V} \cap \text{T only}) + (\text{B} \cap \text{T only}) + (\text{V} \cap \text{B} \cap \text{T}) ∣T∣=6+10+10+4=30|\text{T}| = 6 + 10 + 10 + 4 = 30

The number of females who play Table-tennis is 30\boxed{\text{30}}.

  1. Find the number of females who play exactly two games.

Part c

30\text{30}

The number of females who play exactly two games is the sum of those who play (V ∩\cap B) only, (V ∩\cap T) only, and (B ∩\cap T) only.

(V∩B only)+(V∩T only)+(B∩T only)=x+x+x=3x(\text{V} \cap \text{B only}) + (\text{V} \cap \text{T only}) + (\text{B} \cap \text{T only}) = x + x + x = 3x

Substitute the value of xx found in Step 1:

3x=3×10=303x = 3 \times 10 = 30

The number of females who play exactly two games is 30\boxed{\text{30}}.

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question

More Mathematics Questions

QA group of 70 females participates in volleyball, basketball, or table-tennis. If 20 play all three…QA set of 70 females plays Volleyball, Basketball, and Table-tennis. Given the number of females…QIn a group of 100 students, 60 play football, 50 play volleyball, and 40 play basketball. 25 play…QIn a group of students, 189 play Soccer, 129 play Basketball, 116 play Volleyball. 70 play Soccer…QA group of 87 students play hockey, football, or volleyball. Given the number of students who play…
See all Mathematics questions →
Got a syllabus? Photograph it and I'll tell you what's due and what's worth the most.
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp