Equivalent fractions on number lines

So they're telling us that r fifths is equal to eight tenths, and we need to figure out what r is going to be equal to. They help us out with this number line where they've put eight tenths on the number line. That makes sense because to go from zero to one, they've split it into one, two, three, four, five, six, seven, eight, nine, ten equal jumps. At this point, we have gone eight of those ten equal jumps between zero and one, so that is eight tenths.

They've also labeled one-fifth for us, and one way to think about it is if we look at these bold lines: zero, one, two, three, four, five. If you just look at the purple, we have five equal jumps, so each of those jumps is a fifth. It makes sense that our first jump right over here gets us to one-fifth, and you can see that this is equivalent to two of the tenths. I'll just write that up here so we can see that equivalence: one-fifth is equal to two-tenths.

But how many fifths is equal to eight tenths? Pause this video and try to figure it out.

All right, well this is one-fifth. If we do one more jump of a fifth, that would be two-fifths. Then if we go another fifth, that will get us to three-fifths. And then if we go another fifth, that will get us to four-fifths. We see that four-fifths is exactly equivalent to eight tenths. That makes sense because we also saw that every fifth is equivalent to two tenths. So four-fifths is going to be equivalent to eight of those tenths. We see that very clearly right over here, and so r is equal to four. Four-fifths is equal to eight tenths, so r is equal to four.

Let's do another example. What fraction is equivalent to point a? Pause this video and see if you can figure that out.

All right, so let's figure out where point a is. To go from zero to one, we have one, two, three, four, five, six equal jumps, so each of these jumps is a sixth. Going from zero, one jump will get us to one-sixth, then two-sixths, then three-sixths, then four-sixths, then five-sixths. Can we see four-sixths in the choices? No, I do not see four-sixths, so we have to find an equivalent fraction to four-sixths.

So we could go choice by choice. The first choice has five-sixths. Well, we very clearly see that five-sixths would be here on the number line, which is clearly a different place than four-sixths, so we could rule out this first choice. But what about these other ones? Let’s see how we can think about it.

Four-fifths versus four-sixths: could those be equivalent? If I have four out of five versus four out of six, that's not feeling too good. So I’m going to put a curly line to it; that's not feeling right. If I could have four out of five equal jumps or five equal sections, that would be the same as four out of six equal sections. If I divide it into six equal sections, each of those sections is going to be a little bit smaller than if I divide it into five equal sections. So if I have four of each, they're going to be a different value. Actually, when I talk it out like that, I feel even more confident that I could rule this one out.

Now what about six fourths? Well, one way to think about it is that four-fourths would be equal to one. So six-fourths is going to be beyond one, so it's definitely not going to be where a is. I could rule that one out, and we could say, “Oh well, maybe it's just going to be d.” But let's make sure that this makes sense: what does two-thirds look like?

Well, let me try to divide this part of the number line from zero to one into thirds—into three equal sections. So I have zero there, and then that could be one-third, two-thirds, and then three-thirds. That looks like three equal sections. So this is one-third, this is two-thirds. I'm making another jump of a third, and then when I get to one, of course, that is three-thirds—or we could have said six-sixths.

Point a, which is right over here, I’m writing over it, that is indeed equal to two-thirds. You can see each jump of a third is equal to two-sixths, so it makes sense that four-sixths is equal to two-thirds or that two-thirds is equal to four-sixths. So I like this one.