Question 834691: Hi there, I think this goes here.
I have this word problem and I can't seem to wrap my head around it.
300 people attend an interview
100 had mobile phones (x)
70 had Passports (y)
140 had drivers licenses (z)
40 had x+y
30 had y+z
60 had x+z
10 had x+y+z
How many had none?
any help with this would be awesome! Thanks!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! A venn diagram really helps you visualize what is going on. So start off by drawing 3 overlapping circles inside a rectangle.
10 had x+y+z (aka 10 had all 3). Keep this in mind for the next parts. This value goes in the very center of the 3 overlapping circles.
60 had x+z. So this means that 60-10 = 50 had only x and z but not y. Notice how we took out the people who had all 3 to make sure of this. This number 50 goes in the region between X and Z (but NOT in any region where Y is overlapping).
Similarly, 30 had y+z, so 30-10 = 20 had only y and z but not x. The number 30 goes in the region between Y and Z (but NOT x). Also, 40 had x+y which means 40-10 = 30 had only x and y but not z. Write the number 40 in the region between X and Y (but not z)
Now we must fill out the rest of the circular regions of the venn diagram. In the x circle for instance, all of the numbers must add to 100 (since there are 100 people who are in region x), so
x + 50 + 10 + 30 = 100
x + 90 = 100
x = 100 - 90
x = 10
The other values are found the same way
y + 30 + 10 + 20 = 70
y + 60 = 70
y = 70 - 60
y = 10
z + 50 + 10 + 20 = 140
z + 80 = 140
z = 140 - 80
z = 60
Finally, we add up all the values in the circular regions (but not outside the 3 circles) to get
10+30+10+50+10+20+60 = 190
This means that there are 190 people who...
a) have just one thing x, y or z
b) have 2 things (x+y, x+z, y+z)
OR
c) have all 3 things (x+y+z)
So there are 300 - 190 = 110 people who have none of these 3 things.
Here is what the venn diagram would look like:
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