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Ratios and double number lines

Double number lines are a helpful tool for understanding and solving ratio problems. They visually represent the relationship between two quantities, making it easier to find the cost or amount of a single unit. Examples include calculating the price of avocados, building model trains, and making snowballs.

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Video transcript

- [Narrator] We're told that the double number line shows that five pounds of avocados cost $9. And so what is going on here with this double number line? This shows how as we increase the number of avocados, how the cost increases. So for example, when we have zero pounds of avocados, it costs us $0. When we have five pounds of avocados, it costs us $9. And so, if you look at any point over here, let's say you look right over here. This would be, let's see, this is one, right? This is one, two, three, four, five. If you look at one, this point on the cost number line would tell you how much one pound of avocado would cost. Two pounds of avocados, how much would that cost? You would look at this second number line right over there. So they ask us, based on the ratio shown in the double number line, what is the cost for one pound of avocados? So pause the video and think about it. Remember, one pound of avocados on this top number line, we look at this second number line, the cost would be right over here. What is this going to be? Well, we could just set it up as a ratio. The ratio of pounds of avocados to cost is going to be, let me do this in some colors. So if I have five pounds of avocados, it is going to cost me $9. So the ratio of pounds to dollars is five to nine. So if I were to have one pound of avocado, one pound of avocado, I have divided by five to get one pound of avocado. I would have to do the same thing for the cost. So if I divide nine by five, this is going to be nine fifths dollars. Nine fifths dollars would be the cost of one pound. Well, nine fifths isn't always the most natural way to write money, so you can view this nine fifths is equal to one and four fifths, which is equal to one and eight 10ths, which is equal to 1.8, or you could say this is $1.80. So if you were to go onto this double number line, the cost of one pound of avocado, this point right over here would be $1.80. If you said two pounds of avocados, well, now you would double it. So this would be $3.60, and you would go on and on and on all the way until you got to $9 here. Let's do another example. So here, we are told the double number line shows how many model trains Irene can build in a week. So we can see, in zero weeks she can't build any trains, but in one week she can build nine trains. And they asked us, which table represents the rate of Irene building model trains? So pause this video and see if you can figure it out. So once again, every week she can build nine trains. So one way to think about it is the ratio of week to trains would be one to nine. So let's see. If I look at this table, I just wanna see where the ratio between weeks to trains stays at one to nine. So five to 45, that is still one to nine. To go from one to five, I've multiplied by five. And then to go to nine to 45, I've also multiplied by five. So this one checks out. And another way to think about it, is 45 is nine times five. So nine times. That might be an easier way to think about it. Over here, 12 to 108, that's, once again, 12 times nine is 108. And then 26 to 234. Let's see, 26 times 10 would be 260, minus 26. Yeah, it would be 234. So this is nine times. So in all of these cases, the ratio of trains or the ratio of weeks to trains is one to nine. So this one is looking good, so I'll just circle that in. But let's just make sure that this one doesn't work. Over here, the ratio of weeks to trains is nine to one, not one to nine. The train should be nine times the weeks, while here the weeks is nine times the train. So just looking at that first one, we know that this is not going to work out. Let's do one last example. The double number line shows how many snowballs Jacob and his friends can make in one minute. No minutes, they can make zero snowballs. In one minute, they can make 12. Complete the table to show the same information as the double number line. So once again, pause this video and see if you can work this out. Well, we can think about it as a ratio. The ratio of minutes to snowballs is one to 12. So minutes to snowballs is one to 12. Or another way to think about it is, the snowballs is going to be 12 times the minutes. And so over here, if I have 12 snowballs, we already know that's going to be one minute. If I have 48 snowballs, let's just think about it. To go from 12 to 48 you have to multiply by four. So you have four times as many snowballs, it's gonna take you four times as many minutes. And then if you go to five, to go from one to five you multiply by five, so you're gonna have five times as many snowballs as you would be able to make in one minute. So five times 12 is 60, and we're done. It could be one to 12, four to 48, five to 60. For every five minutes, you can make 60 snowballs, or they make 60 snowballs.