Multiply two-thirds by four-fifths using models.
We can see that the two models that we’re shown underneath this question represent the two fractions that are in the question. This shape has two out of three parts shaded. This is the same as two-thirds. And this shape has four out of five parts shaded. This is the same as four-fifths. And we can use these models to give us a clue as to how to solve this problem. The problem asks us to multiply these two fractions together, two-thirds multiplied by four-fifths.
Now, the idea of multiplying two fractions by each other can be tricky to understand. So another way we can think of this calculation is as trying to find two-thirds of four-fifths. In other words, we can start with four-fifths and then try to find what two-thirds of this is worth. And it’s this second idea that we’re going to go on to use. So to find two-thirds of four-fifths, we need to start with four-fifths. And we can use our model that shows four-fifths to help us.
Let’s begin by drawing a rectangle. Now, it doesn’t need to be the same size of the rectangle that we’re shown in the diagram. But it does need to be divided into five equal parts, in other words, divided into fifths. And to show four-fifths, we need to shade four out of our five parts. So this shaded area four-fifths is the part of the whole shape that we need to work with. And we need to multiply it by two-thirds. As we’ve said already, this is the same as finding two-thirds of this amount. And to show what two-thirds of this amount is, we need to use our second model to help us. It’s a reminder of what two-thirds is like. It’s two out of a possible three parts.
And so, to show two-thirds of our four-fifths, we’re going to firstly need to split four-fifths into three equal parts. And because we already have lines going across the shape, let’s do this from top to bottom, something like this. And as we’ve seen already in the diagram to show two-thirds, we need to shade two out of these three parts. Here’s one-third of four-fifths and here are two-thirds of four-fifths. We know that this new shaded area is worth two-thirds of four-fifths. But what fraction is it of the whole amount?
To find the answer to this, we need to divide the whole shape into equal parts. And at the moment, it’s not divided into equal parts. So we need to extend our two vertical lines downwards. And by doing this, we’ve split the whole shape into equal parts. We can see now that there are five rows and each row contains three equal parts. There must be 15 equal parts in the whole shape. We’ve divided the shape into 15ths. And so, our answer is going to be a number of 15ths.
In fact, if you look at our original fractions, can you see where the denominator 15th comes from? We’ve multiplied our two denominators together. One of our fractions was a number of thirds and the other was a number of fifths. And three multiplied by five equals 15. Let’s come back to our model. If we think about our shaded area that represented two-thirds of four-fifths, how many 15ths can we see? Let’s count in twos: two, four, six, eight. The fraction that shown in our model is eight 15ths.
If we look back at our original fractions, can you see where the numerator eight comes from? We had two-thirds and four-fifths. And if we multiply these two numerators together, two times four equals eight. Although we could have found the answer here just by multiplying the numerators and the denominators together, we were told in the question to use models. And so, that’s what we’ve done. First, we drew a rectangle that represented one whole. And then, we shaded four-fifths of it. And to find two-thirds of four-fifths, we needed to divide our four-fifths into three equal parts and only select two of them. We could see that the answer was going to be eight equal parts. And by extending our two lines downwards, we could see that each part was one 15th of the whole amount.
And so, we can say two-thirds multiplied by four-fifths equals eight 15ths.