Let's start with a detailed step-by-step solution to the problem.
Given:
- Total people surveyed (T) = 80
- People who like only mango (M) = 20
- People who like only apple (A) = 25
- People who like neither fruit (N) = 5
We need to find:
i. The number of people who like both fruits.
ii. The total number of people who like mango.
iii. The total number of people who like apple.
iv. Represent the above information in a Venn diagram.
### Step-by-Step Solution:
1. Find the number of people who like both fruits (B):
First, calculate the number of people who like either mango or apple or both. This is found by subtracting the people who like neither fruit from the total number of people surveyed.
[tex]\[
\text{People who like either or both} = 80 - 5 = 75
\][/tex]
Next, find the total number of people who like only one fruit (either mango only or apple only).
[tex]\[
\text{People who like only mango or only apple} = 20 + 25 = 45
\][/tex]
Now, subtract the number of people who like only one fruit from those who like either or both to get the number of people who like both fruits.
[tex]\[
\text{People who like both fruits} (B) = 75 - 45 = 30
\][/tex]
2. Find the total number of people who like mango (M_total):
The total number of people who like mango is the sum of those who like only mango and those who like both fruits.
[tex]\[
M_{\text{total}} = 20 + 30 = 50
\][/tex]
3. Find the total number of people who like apple (A_total):
Similarly, the total number of people who like apple is the sum of those who like only apple and those who like both fruits.
[tex]\[
A_{\text{total}} = 25 + 30 = 55
\][/tex]
4. Represent the above information in a Venn diagram:
Draw two overlapping circles:
- One circle represents people who like mango.
- The other circle represents people who like apple.
- The overlapping part represents people who like both mango and apple.
Place the values in the Venn diagram:
- The part of the Mango circle that does not overlap with the Apple circle will have 20 (people who like only mango).
- The part of the Apple circle that does not overlap with the Mango circle will have 25 (people who like only apple).
- The overlapping part will have 30 (people who like both fruits).
- Outside both circles, place 5 for people who like neither fruit.
#### Venn Diagram Representation:
```
________
( )
/ 30 \
____/_______\________
( 20 ) ( 25 )
(_______) (_______)
5 (outside both circles)
```
In summary:
i. The number of people who like both fruits is 30.
ii. The total number of people who like mango is 50.
iii. The total number of people who like apple is 55.
iv. The Venn diagram represents the distribution described above.
