Ratios with tape diagrams

We're told Kenzie makes quilts with some blue squares and some green squares. The ratio of blue squares to green squares is shown in the diagram. The table shows the number of blue squares and the number of green squares that Kenzie will make on two of her quilts.

All right, this is the table they're talking about. Based on the ratio, complete the missing values in the table. So why don't you pause this video and see if you can figure it out?

Well, first let's think about the ratio of blue to green squares. So for every three blue squares—let me do that same, a similar color—for every three blue squares, we are going to have one, two, three, four, five green squares. So the ratio of blue to green is three to five.

In quilt A, she has 21 blue squares. So she has 21 blue squares. How many green squares would she have? Well, to go from 3 to 21, you have to multiply by 7. And so, you would take 5 and then multiply that by 7, so you'd multiply 5 times 7 to get to 35. As long as you multiply both of these by the same number, or divide them by the same number, you're going to get an equivalent ratio. So 21 to 35 is the same thing as 3 to 5.

Now we have a situation in quilt B; they've given us the number of green squares, so that's 20. Well, how do we get 20 from 5? Well, we would multiply by 4. So if you multiply the number of green squares by 4, then you would do the same thing for the number of blue squares: 3 times 4, 3 times 4 is going to be equal to 12.

Twelve blue squares for every 20 green squares is the same ratio as three blue squares for every five green squares. Let's do another example here. We are told the following diagram describes the number of cups of blue and red paint in a mixture. What is the ratio of blue paint to red paint in the mixture? So try to work it out.

All right, so let's just see. We have one, two, three, four, five, six, seven, eight, nine, ten—ten cups of blue paint for every one, two, three, four, five, six cups of red paint. So this would be a legitimate ratio: a ratio of 10 cups of blue paint for every 6 cups of red paint.

But this isn't in, I guess you could say, lowest terms or most simplified terms because we can actually divide both of these numbers by two. So if you divide ten by two, you get five—do that blue color; and if you divide six by two, you get three. So one way to think about it is for every five blue squares you have three red squares in this diagram, in this tape diagram—that's sometimes called—or you could say for every five cups of blue paint you have three cups of red paint in our mixture.

And you can even see that here: three cups of red paint and one, two, three, four, five—five cups of blue paint, and you see that again right over here. Let's do another example here.

We're told Luna and Ginny each cast magic spells. The ratio of spells Luna casts to spells Ginny casts is represented in this tape diagram. All right, based on the ratio, what is the number of spells Ginny casts when Luna casts 20 spells? Pause this video and see if you can work it out.

All right, so let's just see the ratio here. For every one, two, three, four spells that Luna casts, Ginny casts one, two, three, four, five spells. So the ratio is four to five. But if Luna casts 20 spells—so if Luna casts 20 spells—well, to go from 4 to 20, we had to multiply by 5.

And so we would do the same thing with the number of spells Ginny casts: you'd multiply that by 5. So it's 25. So 4 Luna spells for every 5 Ginny spells is the same thing as 20 Luna spells for every 25 Ginny spells.

And so how many spells does Ginny cast when Luna casts 20 spells? She casts 25, and we're done.