Question 1199899: In a house of twenty people ,8 liked Ampesi,12 liked banku and 12 liked Fufu.5 liked both Ampesi
and Fufu,6 liked Banku and Fufu,2 only Ampesi and 3 liked all three types of food.
a). Illustrate the information on a Venn diagram
b). Find the number of in the house who liked ;
(i). only Fufu
(ii). Exactly two types
(iii).Only one type of food.
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
I'll get you started with part (a).
Draw a rectangle to represent the universal set.
Inside the rectangle will be 3 partially overlapped circles as shown below.
The 8 regions A,B,C,D,E,F,G,H represent the different scenarios.
For example, if someone is in region A, that person likes Ampesi only (they dont like Banku or Fufu).
This region is in the "Ampesi" circle but outside the other circles.
Another example: If you are in region E, then you like all three food types. This region is found in all three circles.
Here are the given facts:
| Number | Statement | | 1 | House of twenty people | | 2 | 8 liked Ampesi | | 3 | 12 liked Banku | | 4 | 12 liked Fufu | | 5 | 5 liked both Ampesi and Fufu | | 6 | 6 liked Banku and Fufu | | 7 | 2 [liked] only Ampesi | | 8 | 3 liked all three types of food. |
Fact 8 leads directly to E = 3
In other words, we replace the "E" in the diagram above with the value 3, since 3 people like all of the food types mentioned.
Fact 5 says "5 liked both Ampesi and Fufu"
Use this in conjunction with fact 8.
If 5 liked both Ampesi and Fufu, and 3 liked all the food types mentioned, then 5-3 = 2 people liked Ampesi and Fufu only (and not Banku).
In short, we write "2" in region D.
Now use fact 6 and fact 8 together to conclude that 6-3 = 3 people liked Banku and Fufu only (but not Ampesi). The value 3 goes in region F.
Let's update the diagram with what we know so far.
Now turn to fact 4
12 people like Fufu
The values in the "Fufu" circle must add to 12
G+2+3+3 = 12
G+8 = 12
G = 12-8
G = 4
Next we look at fact 7.
2 people like Ampesi only.
Update the value "A" to have it replaced with 2.
Update so far:
Fact 2 says "8 liked Ampesi"
Meaning the values in the "Ampesi" circle must add to 8
A+B+D+E = 8
2+B+2+3 = 8
B+7 = 8
B = 8-7
B = 1
One person likes Ampesi and Banku only, but does not like Fufu.
Another update:
Use fact 3 to find that...
B+C+E+F = set of twelve people who like Banku
1+C+3+3 = 12
C+7 = 12
C = 12-7
C = 5
Five people like Banku only. They do not like the other two food types.
Another update:
We're one step away from finishing.
Fact 1 says there are 20 people total.
The values in the circles add to:
A+B+C+D+E+F+G = 2+1+5+2+3+3+4 = 20
showing that there are 20-20 = 0 people outside of the circles. Meaning no one person disliked all of the food types mentioned. All 20 people are somewhere in at least one circle.
Here is the final Venn Diagram
This concludes part (a).
This step-by-step process might seem a bit lengthy. However, once you get the hang of the basic arithmetic that's going on, it should come second nature to you.
Use that Venn Diagram to answer the other questions. I'll let the student take over from here.
Here is another problem involving a Venn Diagram of 3 circles.
https://www.algebra.com/algebra/homework/Subset/Subset.faq.question.1196992.html
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