Question 834691
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 = <font color="red">110 people</font> who have none of these 3 things.


Here is what the venn diagram would look like:


<img src = "http://i150.photobucket.com/albums/s91/jim_thompson5910/untitled8_zpsd74be0f7.png">