Ratio and proportion
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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-- when the numerator is 24, the denominator is 40. So in that way, you could think of 24/40. But then they want us to write equivalent ratios where we have to fill in different blanks over here-- 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 in 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 doubled 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 four equivalent ratios. The ratio 3 to 5 or 3/5 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 27/75. So then they wrote all of the different equivalent fractions. This table shows ratios equivalent to 18/55. Fair enough. All right, so these are all equivalent to 27/75. These are all equivalent to 18/55, so all of these. Which fraction is greater, 27/75 or 18/55? So this is an interesting thing. What we want to do-- because 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 this situation where we see 27/75 is 54/150. And over here, we see that 18/55 is 54-- and this 54 jumped out at me because it's the same numerator-- over 165. And that makes the comparison much easier. What is smaller? 54/150 or 54/165? Well, if you have the same numerator, having a larger denominator will make the number smaller. So 54/165 is smaller than 54/150, which tells us that 18/55 is smaller than 27/75. So let's see, which of these? So this is saying that 27/75 is greater than 18/55, and that is absolutely right. And 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 0. So table 1-- so distance run in meters. So they're running at a constant speed. So 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 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 right over here is not a legitimate table. Table 3-- so 1 to 1. Then when you double the distance, we double the time. When you triple the distance from 1, you didn't triple the time. So table 3 doesn't seem to make sense, either. Table 4-- so 14 to 10. So that's the same thing as-- let's see, that's the same ratio as, if we were to divide by 2, as 7 to 5 ratio. If we divide both of these by 3, 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 4 seems like a completely reasonable scenario. And we can check our answer, and it is.