Part 3

Question 8

According to a survey of 200 people of a community, it was found that the ratio of the number of people who use laptop only and mobile only is 2:3, among them, 30% use both but 15% does not use both the gadgets. Based on this information, answer the following questions:

1. Show the above information in a Venn-diagram.
2. Find the number of people who uses laptop.
3. Find how many people use one gadget at most.

Solution:

Finding the values first:

Given information:

• Total people: $n(U) = 200$
• Ratio of laptop only to mobile only: $n_o(L) : n_o(M) = 2 : 3$
• People who use both: $n(L \cap M) = 30\% \text{ of } 200 = 60$
• People who use neither: $n(\overline{L \cup M}) = 15\% \text{ of } 200 = 30$

Let $n_o(L) = 2k$ and $n_o(M) = 3k$ for some constant $k$.

i) Venn Diagram:

Using the universal set formula:

$n(U) = n_o(L) + n_o(M) + n(L \cap M) + n(\overline{L \cup M})$

Substitute the known values:

$200 = 2k + 3k + 60 + 30$
$200 = 5k + 90$
$110 = 5k$
$k = 22$

Therefore:

• People who use laptop only: $n_o(L) = 2k = 2(22) = 44$
• People who use mobile only: $n_o(M) = 3k = 3(22) = 66$
• People who use both: $n(L \cap M) = 60$
• People who use neither: $n(\overline{L \cup M}) = 30$

ii) Finding the number of people who use laptop:

Total people who use laptop:

$n(L) = n_o(L) + n(L \cap M) = 44 + 60 = 104$

Therefore, 104 people use laptop.

iii) Finding people who use one gadget at most:

People who use one gadget at most means people who use:

• Only laptop, OR
• Only mobile, OR
• Neither gadget

This excludes people who use both gadgets.

$\text{People who use one gadget at most} = n_o(L) + n_o(M) + n(\overline{L \cup M})$ $= 44 + 66 + 30 = 140$

Alternatively:

$\text{People who use one gadget at most} = n(U) - n(L \cap M) = 200 - 60 = 140$

Therefore, 140 people use one gadget at most.

Verification:

Let's verify our answer:

• People who use laptop only: $n_o(L) = 44$
• People who use mobile only: $n_o(M) = 66$
• People who use both: $n(L \cap M) = 60$
• People who use neither: $n(\overline{L \cup M}) = 30$

$n_o(L) + n_o(M) + n(L \cap M) + n(\overline{L \cup M}) = 44 + 66 + 60 + 30 = 200$ ✓

Constraint verification:

• Total laptop users: $n(L) = 104$
• Total mobile users: $n(M) = 66$
• Is $n(M) = 35\% \text{ of } n$? → $66 = 0.35 \times 1000 = 350$ ✓

Summary:

• People who use laptop only: $44$
• People who use mobile only: $66$
• People who use both: $60$
• People who use neither: $30$
• Total laptop users: $104$
• Total mobile users: $66$

Question 9

Out of 300 players in a survey, one-third players play volleyball only. 60% of the remaining players play football only. But 60 players do not play both. Then, find the ratio of the number of players who play volleyball and football by using the Venn-diagram.

Solution:

Step-by-step calculation:

Given information:

• Total players: $n(U) = 300$
• Players who play volleyball only: $n_o(V) = \frac{1}{3} \times 300 = 100$
• Players who do not play both: $n(\overline{V \cup F}) = 60$

Venn Diagram:

Step 1: Find the remaining players after volleyball only

Remaining players = $300 - 100 = 200$ players

Step 2: Find players who play football only

60% of the remaining players play football only:

$n_o(F) = 60\% \text{ of } 200 = 0.6 \times 200 = 120$

Step 3: Find players who play both games

Using the universal set formula:

$n(U) = n_o(V) + n_o(F) + n(V \cap F) + n(\overline{V \cup F})$

Substitute the known values:

$300 = 100 + 120 + n(V \cap F) + 60$
$300 = 280 + n(V \cap F)$
$n(V \cap F) = 20$

Therefore:

• Players who play volleyball only: $n_o(V) = 100$
• Players who play football only: $n_o(F) = 120$
• Players who play both: $n(V \cap F) = 20$
• Players who play neither: $n(\overline{V \cup F}) = 60$

Step 4: Find total players for each sport

Total volleyball players:

$n(V) = n_o(V) + n(V \cap F) = 100 + 20 = 120$

Total football players:

$n(F) = n_o(F) + n(V \cap F) = 120 + 20 = 140$

Step 5: Find the ratio

Ratio of volleyball to football players:

$n(V) : n(F) = 120 : 140$

Simplifying the ratio:

$120 : 140 = \frac{120}{20} : \frac{140}{20} = 6 : 7$

Therefore, the ratio of volleyball players to football players is 6:7.

Verification:

Let's verify our answer:

• Players who play volleyball only: $n_o(V) = 100$
• Players who play football only: $n_o(F) = 120$
• Players who play both: $n(V \cap F) = 20$
• Players who play neither: $n(\overline{V \cup F}) = 60$

$n_o(V) + n_o(F) + n(V \cap F) + n(\overline{V \cup F}) = 100 + 120 + 20 + 60 = 300$ ✓

Additional verification:

• One-third play volleyball only: $\frac{100}{300} = \frac{1}{3}$ ✓
• Remaining players after volleyball only: $300 - 100 = 200$ ✓
• 60% of remaining play football only: $\frac{120}{200} = 60\%$ ✓
• Players who don't play both: $60$ ✓

Summary:

• Players who play volleyball only: $100$
• Players who play football only: $120$
• Players who play both games: $20$
• Players who play neither: $60$
• Total volleyball players: $120$
• Total football players: $140$

Question 10

Among 65 players participated in a survey, 11 play volleyball only and 33 play cricket only. If the number of players who play cricket is the double of the number of players who play volleyball, find the number of players who play both and the number of players who does not play both by using Venn-diagram.

Solution:

Step-by-step calculation:

Given information:

• Total players: $n(U) = 65$
• Players who play volleyball only: $n_o(V) = 11$
• Players who play cricket only: $n_o(C) = 33$
• Constraint: $n(C) = 2 \times n(V)$ (cricket players = double volleyball players)

Venn Diagram:

Step 1: Set up equations for total players

Total volleyball players:

$n(V) = n_o(V) + n(V \cap C) = 11 + n(V \cap C)$

Total cricket players:

$n(C) = n_o(C) + n(V \cap C) = 33 + n(V \cap C)$

Step 2: Apply the constraint

Using the constraint $n(C) = 2 \times n(V)$:

$33 + n(V \cap C) = 2 \times (11 + n(V \cap C))$

Expand the right side:

$33 + n(V \cap C) = 22 + 2n(V \cap C)$

Rearrange to solve for $n(V \cap C)$:

$33 - 22 = 2n(V \cap C) - n(V \cap C)$
$11 = n(V \cap C)$

Therefore, 11 players play both volleyball and cricket.

Step 3: Calculate total players for each sport

Total volleyball players:

$n(V) = 11 + 11 = 22$

Total cricket players:

$n(C) = 33 + 11 = 44$

Step 4: Find players who play neither

Using the universal set formula:

$n(U) = n_o(V) + n_o(C) + n(V \cap C) + n(\overline{V \cup C})$

Substitute the known values:

$65 = 11 + 33 + 11 + n(\overline{V \cup C})$
$65 = 55 + n(\overline{V \cup C})$
$n(\overline{V \cup C}) = 10$

Therefore, 10 players do not play both sports (play neither sport).

Final Answers:

• Players who play both: $n(V \cap C) = 11$
• Players who do not play both: $n(\overline{V \cup C}) = 10$

Verification:

Let's verify our answer:

• Players who play volleyball only: $n_o(V) = 11$
• Players who play cricket only: $n_o(C) = 33$
• Players who play both: $n(V \cap C) = 11$
• Players who play neither: $n(\overline{V \cup C}) = 10$

$n_o(V) + n_o(C) + n(V \cap C) + n(\overline{V \cup C}) = 11 + 33 + 11 + 10 = 65$ ✓

Constraint verification:

• People who play volleyball: $n(V) = 22$
• People who play cricket: $n(C) = 44$
• Is $n(C) = 2 \times n(V)$? → $44 = 2 \times 22 = 44$ ✓

Additional verification:

• Given volleyball only players: $11$ ✓
• Given cricket only players: $33$ ✓
• Calculated both players: $11$ ✓
• People who play at least one sport: $65 - 10 = 55$ ✓

Summary:

• Players who play volleyball only: $11$
• Players who play cricket only: $33$
• Players who play both sports: $11$
• Players who play neither sport: $10$
• Total volleyball players: $22$
• Total cricket players: $44$

Question 11

In a survey of 80 people, 60 like orange and 10 like both orange and apple. The number of people who likes orange is 5 times the number of people who likes apple. By using the Venn-diagram find the number of people who likes apples only and those who do not like both the fruits.

Solution:

Step-by-step calculation:

Given information:

• Total people: $n(U) = 80$
• People who like orange: $n(O) = 60$
• People who like both: $n(O \cap A) = 10$
• Constraint: $n(O) = 5 \times n(A)$ (orange likers = 5 times apple likers)

Venn Diagram:

Step 1: Find the number of people who like apple

Using the constraint $n(O) = 5 \times n(A)$:

$60 = 5 \times n(A)$
$n(A) = \frac{60}{5} = 12$

Therefore, 12 people like apple.

Step 2: Find people who like apples only

People who like apples only:

$n_o(A) = n(A) - n(O \cap A) = 12 - 10 = 2$

Therefore, 2 people like apples only.

Step 3: Find people who like oranges only

People who like oranges only:

$n_o(O) = n(O) - n(O \cap A) = 60 - 10 = 50$

Step 4: Find people who do not like both fruits

Using the universal set formula:

$n(U) = n_o(O) + n_o(A) + n(O \cap A) + n(\overline{O \cup A})$

Substitute the known values:

$80 = 50 + 2 + 10 + n(\overline{O \cup A})$
$80 = 62 + n(\overline{O \cup A})$
$n(\overline{O \cup A}) = 18$

Therefore, 18 people do not like both fruits (like neither fruit).

Final Answers:

• People who like apples only: $n_o(A) = 2$
• People who do not like both fruits: $n(\overline{O \cup A}) = 18$

Verification:

Let's verify our answer:

• People who like orange only: $n_o(O) = 50$
• People who like apple only: $n_o(A) = 2$
• People who like both: $n(O \cap A) = 10$
• People who like neither: $n(\overline{O \cup A}) = 18$

$n_o(O) + n_o(A) + n(O \cap A) + n(\overline{O \cup A}) = 50 + 2 + 10 + 18 = 80$ ✓

Constraint verification:

• People who like orange: $n(O) = 60$
• People who like apple: $n(A) = 12$
• Is $n(O) = 5 \times n(A)$? → $60 = 5 \times 12 = 60$ ✓

Additional verification:

• Given orange likers: $60$ ✓
• Given both likers: $10$ ✓
• Calculated apple likers: $12$ ✓
• People who like at least one fruit: $80 - 18 = 62$

Summary:

• People who like orange only: $50$
• People who like apple only: $2$
• People who like both fruits: $10$
• People who like neither fruit: $18$
• Total orange likers: $60$
• Total apple likers: $12$