in this video we're going to focus on solving problems associated with venn diagrams so let's start with this one all the students in a class were surveyed regarding what type of pet they own at home 20 students said they own cats and 25 said they own dogs eight said they own both and 12 students said they own neither draw a venn diagram so i already have that on the board an event diagram is basically two overlapping circles with data inside of it so the first circle we're going to say represent the number of cats or rather the number of students who own cats and for the second circle this is going to represent the number of students who own dogs now in this overlapping region that represents the students who own both cats and dogs and that was given to us in a problem eight students said they own both so what that means is we can put eight in this region here now what about part b how many students own cats but no dogs so how many students own cats only we have a total of 20 students who own cats so this includes the eight who own both so to get those who own cats only we need to subtract 20 by eight and that's going to give us 12. so we have 12 students who own cats 8 own cats and dogs so we have a total of 20 students who own cats now what about part c how many students own dogs but no cats so we have a total of 25 students who own dogs eight own both cats and dogs so the difference is 17 17 students own dogs only or dogs and no cats now we have 12 students who own neither cats nor dogs so we can put that on the outside of the venn diagram so now how many students are in the class how can we get the answer for part d there's two ways we can do this one is when we have the venn diagram all filled out and completed the other way involves getting the answer without using the venn diagram with the information that we had in the beginning so let's start with the first way all we need to do is add up the four numbers that we have here so we have 12 students who own cats only 17 only owned dogs eight on both 12 own neither 12 plus 17 is 29 8 plus 12 is 20. 29 plus 20 is 49 so we have a total of 49 students who were surveyed in this class and since all the students in the class were surveyed there are 49 students in this class now to do it the second way without the venn diagram using the information that we started with we're going to add up the number of students who own cats and dogs so we have 20 students who own cats 25 own dogs but now both of these numbers include the students who own cats and dogs that is the students who own both and since we've counted it twice we need to subtract it once because we only want to include this number once so since we already included twice in both these numbers we're going to subtract this by eight once so that we've counted this quantity only one time and then finally we need to add the number of students who own neither which is 12. so 20 plus 25 is 45 and then negative 8 plus 12 that's positive 4 and this will give us the same number of 49 students who are in this class so those are two ways in which you can get the final answer if you have to get the total sum of everyone involved in a survey now let's work on a slightly different problem in this problem we're given the total number of students in the class we didn't have the information in the last problem however in this problem we don't know how many students are studying neither spanish or french in the last problem we had the number of students who own neither dogs or cats that's the difference between these two problems so go ahead and feel free to pause the video and try this problem so there are 40 students in a math class of these 16 students are studying spanish and 19 are studying french five students from this class are studying both spanish and french part a how many students are studying spanish but not french so let's put s for spanish f for french now the first thing i like to do just like in the last problem is i like to put the number of students who are studying in both categories and that's five so let's put that in the middle so now how many students are studying spanish but not french so we have a total of 16 students who are studying spanish five of them are studying both spanish and french though the difference will give us those who are studying only spanish but not french and that's going to be 11. part b how many students are studying french but not spanish so we have a total of 19 who are studying french five are studying both so only 14 are studying french but not spanish now how do we find out in part c how many students in this math class are studying neither french or spanish so basically how can we find the number that belongs outside of the venn diagram now remember how we got our total in the last problem the total number of students this is the first method using the venn diagram is basically the sum of all of these numbers so it's the 11 students studying spanish only plus the 14 students studying french but not spanish the five were studying both and the number who are studying neither which is what we're looking for so that's x and we have the total the total is we got 40 students in the class so we have 40 is equal to 11 plus 14 plus 5 plus x so now all we got to do is solve for x so let's go ahead and combine like terms 11 plus 14 that's going to be 25 and then 25 plus 5 that's 30 so i'm going to write that here so we have 40 is equal to 30 plus x now we need to subtract both sides by 30. so we're gonna have 40 minus 30 is equal to x and 40 minus 30 is 10. so we have 10 students who belong here who are studying neither french nor spanish so that's it for this problem now let's move on to the third problem out of a group of 100 college students 40 said they own a car and 25 said they own an suv 47 students send they own neither a car or suv so how many students own both a car and suv now let's recap what we've done so far so in problem one we needed to calculate the total number of students who own cats and dogs in problem two we didn't know the neither part we didn't know the number of students who took neither spanish or french so that was the missing element in problem three the missing element is both we don't know how many students own both a car and the suv and so those are the three types of variations that we can have with this two category venn diagram problem so how can we calculate the both part it helps to write a formula so let's put c for cars s for suv b for both so that's the number that goes in here and n for neither now using method two to get the total it would be all the students who own cars which would be this entire circle plus all the students who own suvs which would be this entire circle but doing it this way we counted this part twice so we would have to subtract by b and then add n so that would be method one so it's c plus s minus b plus n method two to calculate the total would be the students who own cars only that would be c minus b c represents all the students who own cars so if you subtract those who own both cars and suvs you get those who own cars only now to get those who own suvs only it's s minus b so that would represent this part here now we need to add the middle part so we would have plus b and then those who own neither plus n but notice that the equation on the right simplifies to the equation on the left if you cancel negative b and b you get c plus s minus b plus n so both equations lead to the same result so let's go ahead and use this equation to calculate b so the total number of students is a hundred college students now we know that 40 instead they own a car 25 said they own an suv our goal is to calculate b and 47 said they own neither so let's put this here n is 47. so now let's do some algebra let's combine like terms 40 plus 25 that's 65 and 65 plus 45 i mean 65 plus 47 rather that is a hundred and twelve so a hundred and twelve minus what number is a hundred b has to be 12. if you want to show your work subtract both sides by 112 and you're going to get negative 12 is equal to negative b multiply both sides by negative 1 you get b is 12. so we have 12 students who own both a car and an suv so that's it for part a part b how many students own a car but not an suv so this is going to be c minus b we have 40 students who own a car minus both that will give us 28 who own a car only now part c how many students own an suv but not a car that's going to be s minus b we have 25 who own an suv minus the 12 who own both so only 13 own an suv only now to check your work all four of these numbers must add up to the total so if you add 28 plus 12 which is 40 plus 13 that's 53 plus 47 you get 100. and so that's it for this video so now you know how to solve word problems that is associated with venn diagrams involve in two categories thanks for watching