Question 191874This question is from textbook
: I am taking my first online algebra course and it scares the heck out of me. Can someone please help me with a few problems I missed on a quiz. I am lost! Here is the first one:
In a survey of 100 consumers, 33 indicated that they were going to buy a new car, 18 said they were going to buy a new refrigerator, and 34 said they were going to buy a new washer. Of these, 7 were going to buy both a car and a refrigerator, 15 were going to buy a car and a washer, and 9 were going to buy a washer and a refrigerator. Three consumers indicated that they were going to buy all three items. And she wants a diagram too! Help me please.
Construct a Venn diagram, label your diagram clearly.
Use your diagram to answer the following questions:
(a) How many were going to buy only a car?
(b) How many were going to buy only a washer?
(c) How many were going to buy only a refrigerator
(d) How many were going to buy a car and a washer but not a refrigerator?
(e) How many were going to buy none of these items?
This question is from textbook
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
To start things off, let's draw three blank overlapping circles and label them "car", "refrigerator", and "washer". Now draw a large rectangle that surrounds the three circles and label this rectangle U (for the universal set).
Since "Three consumers indicated that they were going to buy all three items", this means that the value "3" goes in the direct center of the overlapping circles.
Now because "9 were going to buy a washer and a refrigerator", this means that some of those 9 people might have bought all three. So to find the ones who ONLY bought a washer and a refrigerator (not all three), we need to subtract 3 from 9 to get:  . So this tells us that 6 people bought a washer and a refrigerator (and not all three). So place this value in the overlapping region between the circles labeled "refrigerator" and "washer".
Also, since "15 were going to buy a car and a washer" and we just want to find those who ONLY bought a a car and a washer (not all three), we just subtract 3 from 15 to get  . So 12 people bought a car and a washer (not all three). Now place this value in the region between the "car" and "washer" circles.
Since "7 were going t o buy both a car and a refrigerator", and we're only interested in those who ONLY bought the two (not all three), we subtract 3 from 7 to get  . So there are 4 people who bought a car and a refrigerator (not all three). Stick this value in the region between the "car" and "refrigerator" circles.
So we should have the following so far:
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Now because "34 said they were going to buy a new washer" and we're only interested in the number who ONLY bought a washer (not a car or refrigerator), we need to subtract the number who bought all three (3 people), the number who bought a car and washer (12 people), and the number of people who bought a refrigerator and washer (6 people) from 34 to get  . So there are 13 people who ONLY bought a washer. Place this value in the "washer" circle (and not in any overlapping regions)
If the method above is a bit cumbersome, then let's try another method in finding the number of people who ONLY bought a car:
Since 12 people bought a car and a washer, 4 people bought a car and a refrigerator, and 3 people bought all three, this means that  people bought AT LEAST 2 items (where one of the items is a car). Since we ONLY want the car, we need to subtract this from 33 (the number people who bought a car and maybe something else) to get:  . So 15 people ONLY bought a car. Place this value in the "car" circle (and not in any overlapping regions)
Because 4 people bought a refrigerator and a car, 6 people bought a refrigerator and a washer, and 3 bought all three, this means that  people bought AT LEAST two items in which one item is a refrigerator. Subtract this value from 18 (the number who bought a refrigerator and maybe something more) to get  . So 5 people bought ONLY a refrigerator. Place this value in the "refrigerator" circle (and not in any overlapping regions)
Finally, add up EVERY element that is in every region to get:  . So 58 people bought AT LEAST one item (either a car, refrigerator, or washer). Subtract this value from the number who participated (100 people) to get  . So 42 people did NOT buy any of the items listed. Place this value outside the three circles but inside the set U (the rectangle).
So we should now have the completed Venn diagram:
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Now let's answer the questions:
a)
How many were going to buy only a car?
Looking at the diagram (and going over our previous work), we get 15 people only bought a car.
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b)
How many were going to buy only a washer?
Similarly, we see from the Venn diagram that 13 people only bought a washer.
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c)
How many were going to buy only a refrigerator?
From the diagram, we see that 5 people only bought a refrigerator.
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d) How many were going to buy a car and a washer but not a refrigerator?
From the diagram, we see that the number that lies in the region between the "car" and "washer" is 12. So 12 people bought a car and a washer but not a refrigerator.
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e) How many were going to buy none of these items?
Since the value 42 is outside all of the circles, this means that 42 people did not buy any of these items.
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