Answer :
Let's break down the information given in the survey and use the numerical results obtained to answer each part of the question.
### Part (a): Define \( T \) and \( C \) based on given ratios and percentages
1. The total population surveyed is 100 people (the problem statement refers to a community, so it is logical to assume 100 people for simplicity).
2. 10% of the people did not like both tea and coffee. Therefore:
[tex]\[ \text{People who disliked both} = 10 \% \times 100 = 10 \text{ people} \][/tex]
3. This implies that 90% of the people liked at least one of the items (tea or coffee or both):
[tex]\[ \text{People who liked at least one} = 90 \% \times 100 = 90 \text{ people} \][/tex]
4. 30% of the total people liked both tea and coffee:
[tex]\[ \text{People who liked both} = 30 \% \times 100 = 30 \text{ people} \][/tex]
5. The ratio of people who like tea only to those who like coffee only is 2:3. Let's denote the common ratio factor as \( x \). Hence:
[tex]\[ \text{People who like tea only} = 2x \][/tex]
[tex]\[ \text{People who like coffee only} = 3x \][/tex]
6. From the total number of people who like at least one of the items, we have:
[tex]\[ 90 = 2x + 3x + 30 \][/tex]
[tex]\[ 5x + 30 = 90 \][/tex]
[tex]\[ 5x = 60 \][/tex]
[tex]\[ x = 12 \][/tex]
7. Substituting \( x \) back to find the number of people who like tea only and coffee only:
[tex]\[ \text{People who like tea only} = 2 \times 12 = 24 \text{ people} \][/tex]
[tex]\[ \text{People who like coffee only} = 3 \times 12 = 36 \text{ people} \][/tex]
### Part (b): Venn Diagram
In a Venn diagram, we represent the two sets - people who like tea (T) and people who like coffee (C) - with overlapping circles.
- Number of people who like tea only: 24
- Number of people who like coffee only: 36
- Number of people who like both tea and coffee: 30
- Number of people who dislike both: 10
### Part (c): Number of people who like exactly one item
People who like exactly one item include those who like tea only and those who like coffee only:
[tex]\[ \text{Number of people who like exactly one item} = 24 + 36 = 60 \text{ people} \][/tex]
### Part (d): Ratio of people who like both items to those who do not
The ratio of people who like both items to those who do not is:
[tex]\[ \text{Number of people who like both} = 30 \text{ people} \][/tex]
[tex]\[ \text{Number of people who do not like both} = 10 \text{ people} \][/tex]
[tex]\[ \text{Ratio} = \frac{30}{10} = 3 \][/tex]
### Summary
(a)
- \( T \) (total number of people who like tea) = \( 24 + 30 = 54 \)
- \( C \) (total number of people who like coffee) = \( 36 + 30 = 66 \)
(b) Venn diagram should be drawn representing the sets
(c) Number of people who like exactly one item: \( 60 \)
(d) Ratio of people who like both items to those who do not: [tex]\( 3 \)[/tex]
### Part (a): Define \( T \) and \( C \) based on given ratios and percentages
1. The total population surveyed is 100 people (the problem statement refers to a community, so it is logical to assume 100 people for simplicity).
2. 10% of the people did not like both tea and coffee. Therefore:
[tex]\[ \text{People who disliked both} = 10 \% \times 100 = 10 \text{ people} \][/tex]
3. This implies that 90% of the people liked at least one of the items (tea or coffee or both):
[tex]\[ \text{People who liked at least one} = 90 \% \times 100 = 90 \text{ people} \][/tex]
4. 30% of the total people liked both tea and coffee:
[tex]\[ \text{People who liked both} = 30 \% \times 100 = 30 \text{ people} \][/tex]
5. The ratio of people who like tea only to those who like coffee only is 2:3. Let's denote the common ratio factor as \( x \). Hence:
[tex]\[ \text{People who like tea only} = 2x \][/tex]
[tex]\[ \text{People who like coffee only} = 3x \][/tex]
6. From the total number of people who like at least one of the items, we have:
[tex]\[ 90 = 2x + 3x + 30 \][/tex]
[tex]\[ 5x + 30 = 90 \][/tex]
[tex]\[ 5x = 60 \][/tex]
[tex]\[ x = 12 \][/tex]
7. Substituting \( x \) back to find the number of people who like tea only and coffee only:
[tex]\[ \text{People who like tea only} = 2 \times 12 = 24 \text{ people} \][/tex]
[tex]\[ \text{People who like coffee only} = 3 \times 12 = 36 \text{ people} \][/tex]
### Part (b): Venn Diagram
In a Venn diagram, we represent the two sets - people who like tea (T) and people who like coffee (C) - with overlapping circles.
- Number of people who like tea only: 24
- Number of people who like coffee only: 36
- Number of people who like both tea and coffee: 30
- Number of people who dislike both: 10
### Part (c): Number of people who like exactly one item
People who like exactly one item include those who like tea only and those who like coffee only:
[tex]\[ \text{Number of people who like exactly one item} = 24 + 36 = 60 \text{ people} \][/tex]
### Part (d): Ratio of people who like both items to those who do not
The ratio of people who like both items to those who do not is:
[tex]\[ \text{Number of people who like both} = 30 \text{ people} \][/tex]
[tex]\[ \text{Number of people who do not like both} = 10 \text{ people} \][/tex]
[tex]\[ \text{Ratio} = \frac{30}{10} = 3 \][/tex]
### Summary
(a)
- \( T \) (total number of people who like tea) = \( 24 + 30 = 54 \)
- \( C \) (total number of people who like coffee) = \( 36 + 30 = 66 \)
(b) Venn diagram should be drawn representing the sets
(c) Number of people who like exactly one item: \( 60 \)
(d) Ratio of people who like both items to those who do not: [tex]\( 3 \)[/tex]
