Averages

welcome to the presentation on averages average is probably a concept that you've already used before maybe not in a mathematical way but you know people will talk in terms of the average voter wants the politician to do this or the average student in the class wants to get out early so you're probably already familiar with the concept of an average and you probably already intuitively knew that an average is just a number that represents the the different the different values that a group could have but it can represent that as one number as opposed to giving all of the different values and let's give a couple of examples of how to compute an average and you might already know how to do this so let's say I had the numbers 1 3 5 and 20 and I asked you what is the average of these four numbers well what we do is we literally just add up the numbers and then divide by the number of numbers we have so we say 1 plus 3 is 4 so let me write that 1 plus 3 plus 5 plus 20 equals let's see 1 plus 3 is 4 4 plus 5 is 9 9 plus 20 is 29 right and we had 4 numbers 1 2 3 4 so 4 goes into 29 it was 7 7 28 and then we have 10 I didn't have to do that decimal there go to 825 so 4 goes into 29 7.2 five times so the average of these four numbers is equal to seven point two five and that might make sense to you because seven point two five is is is someplace in between these numbers and we can kind of view this seven point two five as one way to represent these four numbers without having to list these four numbers there are other representations you'll learn later on like the mode you'll also see that the mean which we'll talk about later is actually the same thing the average but the average is just one number that you can use to represent a set of numbers so let's do some problems which I think are going to be close to your heart let's say on the first four tests of an exam I got a let's see I got an 80 80 and 81 and 87 and an 88 what's my average in the class so far well all I have to do is add up these four numbers so I say 80 plus 81 plus 87 plus 88 well 0 plus 1 is 1 1 plus 7 is 7 set 1 plus 7 is 8 8 plus 8 is 16 I just ran 8 miles so I'm a bit tired and then you have 4 eighths so that's 32 plus 1 is 33 and now we divide this number by 4 4 goes into 336 it goes into 33 8 times 8 times 4 is 32 33 minus 32 is 116 4 so the average is equal to 84 so depending on what school you go to that's either a B or C so so far my average after the first four exams is an 84 now let's make this a little bit more difficult we know that the average after 4 exams hit for exams is equal to 84 now if I were to ask you what do I have to get on the next test to average an 88 to average an 88 in the class right so let's say that X is what I get on the next test so now what we can say is is that the first four exams I could either list out the first four exams that I took or I already know what the average is so I know the sum of the first four exams is going to be four times 84 and now I want to add the what I get on the fifth exam X and I'm going to divide that by all five exams so notice this number is what is the average of my first five exams we just figured out the average of the first four exams but now we sum up the first four exams here we add what I get on the fifth exam and then we divide it by five because now we're averaging five exams and I said that I need to get an 88 in the class and now we solve for X so let's make some space here so 5 times 88 is let's see five times a D is 400 so it's four forty four forty equals four times eighty four we just saw that is 320 plus 16 is 336 336 plus X is equal to 440 it turns out if you subtract three 36 from both sides you get x is equal to 104 so unless you have a exam that has some bonus problems on it it's probably impossible for you to get an 88 average in the class after just the next exam you'd have to get 104 on that next exam let's just look at what we just did we said after four exams we had an 84 what do I have to get on that next exam to average an 88 in the class after five exams and that's what we solved for when we got X now let's ask another question I said after four exams after four exams I had an 84 average if I said that there are six exams in the class and the highest score I can get on an exam is 100 what is the highest average I can finish in the class if I were to really study hard and get 100 on the next two exams well once again what we'll want to do is sit assume we get 100 on the next two exams and then take the average so we'll have to solve up all six exams so we're going to have the average of 6 so we're in the denominator we're going to have 6 the first four exams the sum is we already learned is is a for exams times the 84 average and this dot is just x + there's going to be two more exams right because there's six exams in the class and I'm get 100 in each so that's two hundred and what's this average well four times 84 we already said is 336 plus 200 over six so that's five thirty six over six six goes into five thirty six I don't know if I give myself enough space but 6 goes into 53 eight times 48 56 goes into it 9 times 9 times 6 is 54 6 minus 26 goes into so we'll see it's actually 89.3 33333 it goes on forever so 89.3 repeating so no matter how hard i try in this class the best I can do because I only have two exams left even if I were to get 100 on the next two exams I can finish the class with an 89 0.333 average hopefully I think some of this might have been a little bit of review for you you already kind of had a sense of what an average is and hopefully these last two problems not only taught you how to do some algebra problems involving average but they'll also help you figure out how well you have to do on your exams to get an A in your math class I think you're now ready for the average module I have fun