Answered

1) In a survey of 100 people in the community, it was found that the ratio of people who like tea only to those who like coffee only is [tex]$2:3[tex]$[/tex]. Additionally, [tex]$[/tex]30\%[tex]$[/tex] of people liked both and [tex]$[/tex]10\%$[/tex] of people did not like either.

a) If [tex]$T[tex]$[/tex] and [tex]$[/tex]C$[/tex] denote the number of people who like tea and coffee respectively, what does this imply?

b) Represent the above information in a Venn diagram.

c) Find the number of people who like exactly one item.

d) Find the ratio of people who like both items to those who do not like either.



Answer :

GinnyAnswer
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]