Visually dividing unit fraction by a whole number

We're asked to figure out what is one-seventh divided by four. They help us out with this diagram. We have a whole divided into seven equal sections; each of those is a seventh. We have one of those sevenths filled in, so this is one-seventh right over here. Then, they divide it into four equal sections. In fact, they divide all of the sevenths into four equal sections.

So, one-seventh, which is this whole green bar, divided by four, well that would be this fraction of the whole that is in a question mark. So, can you pause this video and figure out what fraction of the whole is this question mark?

Well, when we divided the first seventh into four equal sections, we also divided all of the sevenths into four equal sections. Now, the entire whole is 28 equal sections because you have a four by seven grid. You have one, two, three, four rows, and you still have your seven columns. You could count them: 7, 14, 21, 28.

So, one-seventh divided by four is going to be one of these 28 sections. So this right over here is 1 over 28. So, this is 1 twenty-eighth.

Let's do another example. We're told to use the number line below to help visualize one-fifth being divided by three. As we go from zero to one on the number line, you could divide into five equal sections, where that's one-fifth, two-fifths, three-fifths, four-fifths, and of course five-fifths is equal to one.

But we want one-fifth divided by three, so we took the section from zero to one-fifth and divided it into three equal sections. The first of those sections, this one right over here, would be one-fifth divided by three.

So, what is this going to be equal to? Pause this video again and see if you can figure that out.

Well, the key realization is that when we divided each of the fifths into three more equal sections, we can now think of each of these steps as a fifteenth because now we have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 equal sections between 0 and 1.

Where did that 15 come from? Well, we had 5 equal sections and then we split each of those 5 into 3 more equal sections, so 5 times 3 is 15.

So this right over here is 1 fifteenth, this is 2 fifteenths, this is 3 fifteenths, which is equivalent to one-fifth. We could keep going on and on and on, but the key realization here is if I take that first one-fifth and if I divide it into three equal sections and I go only as far as that first of the three equal sections, that is going to be one fifteenth. One fifteenth, and we are done.