Two-thirds of a cake is divided equally into four pieces. The diagram shows this situation. The cake was divided first into three parts to identify two-thirds of the cake and then divided into four to get four pieces from the two-thirds. How many fractional units are there after these two divisions? How many units is one piece of cake? And what fraction, in its simplest form, of the cake is that?
The first sentence of this problem tells us very clearly what calculation we’re doing. Two-thirds of a cake is divided equally into four pieces. The calculation is two-thirds divided by four. Now, one of the strategies that we can use to help us when we divide a fraction by a whole number is to use a diagram to help. And in this question, it talks us through how this diagram can help us find the answer. Now, as part of the question, we’re given the diagram to look at. But perhaps, it might be helpful if we draw the diagram as it’s described too so we can see how it came about.
So to begin with, we’ve got a cake. Let’s use this rectangle to represent our cake. Now we know that the cake isn’t a whole cake. We’ve only got two-thirds of it. And that two-thirds is divided equally into four pieces. So the first thing we need to do is to find two-thirds of this cake. And the third sentence of our problem tells us how to do this. It says the cake was divided first into three parts to identify two-thirds of the cake. So let’s do this to begin with. We’ll divide our cake into thirds. Here we go. And we’ve done this to help us identify two-thirds of the cake. So in our diagram, we could shade in two-thirds of the cake.
Now we know that the problem is asking us to divide this two-thirds of the cake equally into four pieces. And we’re also told how to do this in the question. It says the cake was then divided into four to get four pieces from the two-thirds. We can see the orange part of the diagram. We can imagine sitting there with this big orange part of the cake. And we need to now divide it into four slices. So we’re going to cut here and here and here. So one of these four pieces that we’ve cut our two-thirds into is where we’ve drawn the pink diagonal lines here. This is one piece of the cake. And this is where our diagram comes from. And we can now answer the questions.
The questions in this problem are designed to help us to find the final answer. Which is what fraction, in its simplest form, of the cake have we got. Let’s look at our first question. “How many fractional units are there after these two divisions?” We had a whole cake. We divided it into three equal parts so that we could find two-thirds of it. And then, we divided it again into four. We divided by three and then by four. How many fractional units do we have now? Let’s count them. One, two, three, four, five, six, seven, eight, nine, 10, 11. There are 12 fractional units.
The next question asks us, “How many units is one piece of cake?” We could see as we worked through the problem that two-thirds of the cake was the orange part of the diagram. And one piece of that two-thirds when we divided into four is where we’ve drawn the diagonal pink lines. And we can see that these are two out of those 12 units. We’ve colored in two of the squares. So what fraction of the cake have we got if we have two out of 12? Well, the answer is two twelfths. But this isn’t the answer to the final question, because we’re asked to find the fraction in its simplest form. And we can make two twelfths simpler. What can we divide both two and 12 by to simplify this fraction? Well, they’re both even numbers. So we can divide them both by two. Two divided by two equals one. And 12 divided by two equals six. So we can say that two-thirds divided by four equals one-sixth.
We wanted to divide two-thirds of the cake equally into four pieces. And we used a diagram to do so. First, we divided the cake into three parts to find two-thirds of the cake. And then we divided it again into four to get four pieces from the two-thirds. The number of fractional units that there were after dividing by three and then by four is 12. And by using the diagram to help, we could see that the number of units that were the same as one piece of cake was two. So we knew that each piece of cake would be worth two twelfths of the whole cake. But we were asked to simplify this. And we could write the answer as one-sixth. Two-thirds divided by four equals one-sixth.