Worked example: Solving proportions

whereas to solve the proportion and we have 836 is equal to 10 over what or 8 to 36 or the ratio of 8 over 36 is equal to the ratio of 10 to what and there's a bunch of different ways to solve this and I'll explore really all of them or not all of them are a good selection of them so one way to think about is these two need to be equivalent ratios or really equivalent fractions so whatever happened to the numerator also has to happen to the denominator so what do we have to multiply 8 by to get 10 well you could multiply 8 times 10 over 8 will definitely give you 10 so we're multiplying by 10 over 8 over here or another way to write 10 over 8 10 over 8 is the same thing as 5 over 4 so we're multiplying by 5 over 4 to get to 10 from 8 to 10 well if we did that to the numerator in order to have an equivalent fraction you have to do the same thing to the denominator you have to multiply it you have to multiply it times 5 over 4 and so we could say this n this thing that we just solve for this n is going to be equal to 36 times 5 divided by 4 or you could say that this is going to be equal to 36 times 5 divided by 4 and now 36 divided by 4 we know what that is we could divide both the numerator and the denominator by 4 you divide the numerator by 4 you get 9 divide the denominator by 4 you get 1 you get 45 so that's one way to think about it 8 over 36 8 over 36 is equal to 10 over 45 another way to think about it is how much what do we have to multiply by 8 by to get its denominator how much larger is the denominator 36 then 8 well let's just divide 36 over 8 so 36 over 8 is the same thing as so we can simplify dividing the numerator in the denominator by 4 that's there that's the greatest common divisor that's the same thing as 9 halves so when you if you multiply the numerator by 9 halves you get the denominator so we're multiplying by 9 halves to get the denominator over here well then we have to do the same thing over here if this if 36 is 9 halves tying xate let me write this 36 or let me write 8 times 9 halves is equal to is equal to 36 all right that's how we go from the numerator denominator then to figure out what the denominator here is if we want the same fraction we have to multiply by 9 halves again so then we'll get 10 times 9 halves is going to be equal to n is going to be equal to this denominator and so this is the same thing as saying 10 times 9 over 2 divide the numerator in the denominator by 2 you get 5 over 1 which is 45 so 45 is equal to n once again we got the same way completely legitimate way to solve it now sometimes when you see a proportion like this sometimes people say oh you could cross multiply and you can cross multiply and I'll teach you how to do that and that's sometimes a quick way to do it but I don't like teaching it the first time you look at proportions because it's really just something mechanical you really don't understand what you're doing and it really comes out of a little bit of algebra and I'll show you the algebra as well but if you don't understand it if it doesn't make as much sense to you at this point don't worry too much about it so we have 8 over 36 is equal to 10 over N when you cross multiply you're saying that the numerator here times the denominator over here is going to be equal to so 8 times n is going to be equal to the new the denominator over here the denominator over here let me do this in a different color the denominator over here times the numerator over here this is what it means to cross multiply so this is going to be equal to 36 times 10 36 times 10 or you could say you could say let me do this in a neutral color now you could say that 8 n is equal to 360 and so you're saying 8 times what is equal to 360 or to figure out what that times what is you divide 360 divided by 8 so we could divide and this is a little bit of algebra here we're dividing both sides of the equation by 8 and we're getting n is equal to 360 divided by 8 and you know you can do that without thinking in kind of strict algebraic terms you say 8 times what is 360 well 8 times 3 sixty over eight if I write eight times question mark is equal to 360 well question mark could definitely be 360 over eight if I multiply these out this guy and that guy cancel out and it's definitely 360 and that's why it's 360 over eight but now we want to actually divide this to actually get a right answer or a simplified answer eight goes into 360 eight goes into 36 4 times 4 times 8 is 32 you have a remainder of 4 bring down the 0 8 goes into 40 five times five times eight is 40 and then you have no remainder and then you have no remainder and you're done once again we got n is equal to n is equal to 45 now the last way I'm going to show you involves a little bit of algebra if any of the ways before this work that's fine and where this is sitting in the playlist you're not expected to know the algebra but I want to show you the algebra just because I wanted to show you that this cross multiplication isn't some magic that using algebra we will get this exact same thing but you can stop watching this if you'll find in this part confusing so let's let's rewrite our proportion 8 over 36 is equal to 10 over N and we want to solve we want to solve for n well the easiest way to solve for n is maybe multiply both this thing on the left is equal to this thing on the right so we can multiply them both by the same thing and the Equality will still hold so we could multiply both of them both of them by n on the right hand side the ends cancel out on the left hand side we have 8 36 times n is equal to 10 now if we want to solve for n we can literally multiply if we want if we want just an N here we could we would want to multiply this side times 36 do that in a different color we want to multiply this times this side times 36 times 8 because if you multiply these guys out you get 1 and you just have an N but since we're doing it to the left hand side we also have to do it to the right hand side so times 36 over 8 these guys cancel out and we're left with n is equal to 10 times 36 is 360 over 8 and no we're getting the exact we're getting the exact same value that we got with cross multiplying and with cross multiplying you're actually doing you're actually doing two steps actually doing an extra step here your multiplying both sides by n so that you get your a 10 and then you're multiplying both sides by 36 so that you get your 3 your your 36 on both sides and you get this value here but at the end when you simplify it you'll get the exact same answer so those are all different ways to solve this proportion probably the most obvious way or the easiest way to do it in your head was well either just looking at what you have to multiply the numerator by and then doing the denominator doing the same thing of the denominator or maybe by cross multiplication