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we're told this table shows equivalent ratios to 24 to 40 fill in the missing values and they write the ratio 24 to 40 right over here 24 24 when the numerator is 24 the denominator is 40 so in that way you could think of 24 over 40 but then they want us to write equivalent ratios where we have to fill in different blanks over here in the denominator and here in the numerator and there's a bunch of ways that we could actually tackle this but maybe the easiest is to start with the ratio that they gave us where they gave us both the numerator and the denominator and then move from there so for example if we look at this one right over here the numerator is 12 it is half of the 24 so the denominator is also going to be half of the denominator here it's going to be half of 40 so we could stick a 20 right over there and then we could go up here if you compare the 3 to the 12 to go from 12 to 3 you have to divide by 4 so in the numerator you're dividing by 4 so the denominator you also want to divide by 4 so 20 divided by 4 is 5 and then we have one more to fill in this numerator right over here and we see from the denominator we double the denominator we went from 40 to 80 so we would double the numerator as well and so you would get 48 and what we just did here is we wrote 4 equivalent ratios the ratio 3 to 5 or 3 over 5 is the same thing as 12 to 20 is the same thing as 24 to 40 is the same thing as 48 to 80 let's make sure we got the right answer let's do a couple more of these the following table shows equivalent fractions to 27 70 fifths so then they wrote all of the different equivalent fractions this table shows ratios equivalent to 1850 fifths fair enough all right so these are all equivalent to 27 70 fifths these are all equivalent to 1850 fifths so all of these which fraction is greater 27 70 fifths or 1850 fifths so this is an interesting thing what we want to do because you know you look at these two things and you're like well I don't know their denominators are different how do I compare them and the best way that I can think of comparing them is look at a point where you're getting an equivalent fraction and either the numerators are going to be the same or the denominators are going to be the same so let's see if there's any situation here so you have the situation where we see 27 over 75 is 54 over 150 and over here we see that 1855 is 54 and and I this 54 jumped out at me because it's the same numerator over 165 and that makes the comparison much easier what is smaller 54 over 150 or 54 over 165 well if you have the same numerator having a larger denominator will make the number smaller so 54 over 165 is smaller than 54 over 150 which tells us that 18 over 55 is smaller than 27 over 75 so let's see which of these so this is saying that 27 over 75 is greater than 1850 fifths and that is absolutely right let's do one more of these lunara's friends are running a race each of them runs at a constant speed starting at time 0 which of these tables might show the distances one of lunara's friends traveled over time so they're running a race each of them runs at a constant speed starting at time zero so table so table 1 so distance run in meters so they're running at a constant speed so the really the ratio between distance and time should be constant throughout all of these possible tables so here you have a ratio of 3 to 2 if you if you triple the distance we're tripling the time if you multiply the distance by 5 we're multiplying the time by 5 so table 1 seems completely reasonable let's keep going table 2 11 to 4 and then 12 to 5 here it's just incrementing by 1 but the ratios are not the same 11 to 4 is not the same thing as 12 to 5 so we're not going to be able to this is not a legitimate this right over here is not a legitimate of of table Table three so one-to-one then when you double the distance we double the time when you triple the distance when you triple to the distance from one you didn't triple the time so Table three doesn't seem to make sense either Table four so 14 to ten so that's the same thing as let's see that's the same ratio as if we were divided by two as 7 to 5 ratio if we divide both of these by three this is also a 7 to 5 ratio and if you divide both of these by 7 this is also a 7 to 5 ratio so Table four seems like a completely reasonable a completely reasonable scenario and we can check our answer and it is