Ratios and double number lines

We're told the double number line shows that five pounds of avocados cost nine dollars. 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.

For example, when we have zero pounds of avocados, it costs us zero dollars. When we have five pounds of avocados, it costs us nine dollars. 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. 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 the 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. If I have five pounds of avocados, it is going to cost me nine dollars. So, the ratio of pounds to dollars is five to nine.

If I were to have 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 could view this nine-fifths as equal to one and four-fifths, which is equal to one and eight-tenths, which is equal to one point eight.

Or you could say this is one dollar and eighty cents. So, if you were to go on to this double number line, the cost of one pound of avocado, this point right over here, would be one dollar and eighty cents. If you said two pounds of avocados, well now you would double it. So, this would be three dollars and sixty cents, and you would go on and on and on all the way until you got to nine dollars.

Here, let's do another example. We are told the double number line shows how many model trains Irene can build in a week. We can see in zero weeks she can't build any trains, but in one week, she can build nine trains. They ask us which table represents the rate of Irene building model trains. So, pause this video and see if you can figure it out.

Once again, every week she can build nine trains. So, one way to think about it is the ratio of weeks to trains would be one to nine. If I look at this table, I just want to see where the ratio between weeks and trains stays at one to nine. Five to forty-five, that is still one to nine. To go from one to five, I've multiplied by five, and then to go to nine to forty-five, I've also multiplied by five.

So, this one checks out. Another way to think about it is forty-five is nine times five. That might be an easy way to think about it. Over here, twelve to one hundred eight, well that's once again, twelve times nine is one hundred eight. Then twenty-six to two hundred thirty-four, let's see, twenty-six times ten would be two hundred sixty, minus twenty-six, yeah it would be two hundred thirty-four.

So, this is nine times. 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. Well, 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.

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. In zero minutes, they can make zero snowballs, and in one minute, they can make twelve. 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 twelve. Minutes to snowballs is one to twelve, or another way to think about it is the snowballs are going to be twelve times the minutes.

So, over here, if I have twelve snowballs, we already know that's going to be one minute. If I have forty-eight snowballs, well, let's just think about it. To go from twelve to forty-eight, you have to multiply by four. So, you have four times as many snowballs; it is going to take you four times as many minutes.

Then, if you go to five, well to go from one to five, you multiply by five. So, you're gonna have five times as many snowballs than you would be able to make in one minute. So, five times twelve is sixty, and we're done. It could be one to twelve, four to forty-eight, five to sixty. For every five minutes, you can make sixty snowballs, or they make sixty snowballs.