Lesson Video: Multiplying and Dividing Fractions | Nagwa Lesson Video: Multiplying and Dividing Fractions | Nagwa

Lesson Video: Multiplying and Dividing Fractions Mathematics

In this video, we will learn how to multiply and divide fractions with like and unlike denominators, including proper, improper, and mixed fractions.

16:20

Video Transcript

In this video, we’ll learn how to multiply and divide fractions with like and unlike denominators, including proper, improper, and mixed fractions. It’s crucial that before accessing this video, you’re able to simplify fractions and convert between improper fractions and mixed numbers as these skills are crucial when it comes to confidently performing fraction arithmetic. Let’s begin by looking at how we multiply fractions.

What does it mean to multiply two fractions? Let’s take a pair of proper fractions. These are fractions that are between zero and one, say a third and two-fifths. If we think about the definition of multiplication, when we multiply these fractions, we’re asking ourself what is one-third lots of two-fifths or two-fifths lots of one-third. Now, clearly, this presents some issues beyond integer or whole number multiplication, when we can use a number line or similar. But we can represent this using a bar model. And we do so by reading this as one-third of two-fifths. We represent our first fraction using vertical lines.

We shade one out of the three bars to represent one-third. Then we model the second fraction by using horizontal lines. We shade two out of five of these rows to represent two-fifths. In doing so, we’re finding a third of two-fifths and vice versa. We now see that we have one, two squares shaded in our overlap and a total of three by five, which is 15 squares altogether. So a third of two-fifths which we said is the same as a third times two-fifths is two fifteenths.

But where do these numbers come from? Well, we saw that three times five gave us 15. This is the product of our denominators. It’s what we get if we multiply our denominators together. Similarly, if we multiply the numerators, we get two. In fact, we can generalize this. And we say that, to multiply two fractions, we simply multiply their numerators together and then separately multiply their denominators together. So one-third times two-fifths is one times two over three times five, which is two fifteenths. And we saw that using our slightly long-winded method earlier. And that’s it.

There are some steps we need to consider when working with mixed numbers, but we’ll look at that in a moment. For now, we’re just going to consider another simple example.

Find the result of one-half multiplied by one-third.

There are two ways we can answer this. One method involves using a bar model and recalling that a half times one-third is the same as finding a half of one-third. We can represent our fraction one-third by shading one bar out of three equally sized bars. Then in the opposite direction, in this case, going downwards, we represent a half of this same model by shading one bar out of two equally sized bars. We can now see we have one rectangle shaded out of a possible of three times two, which is six rectangles.

And so this means that a half of one-third which is the same as a half times one-third is simply one out of six; that’s one-sixth. Of course, there is a slightly quicker way we could’ve done this. We know that, to multiply a pair of fractions, we multiply their numerators together, then separately multiply their denominators together. So a half times one-third is one times one over two times three. One times one is one and two times three is six. So once again, we see that a half times one-third is equal to one-sixth.

We’re now going to move away from using a bar model and simply recall that when we multiply fractions, we multiply their numerators, then multiply their denominators. And in our next example, we’re going to see how this will work when multiplying a pair of mixed numbers.

Work out one and one-quarter multiplied by one and two-thirds.

Here, we have a pair of mixed numbers. Now, we know that, to multiply proper fractions, in other words, fractions between zero and one, we multiply the numerators together and then separately multiply the denominators. Now, in fact, this holds for improper fractions. And so to be able to find the product of a mixed number and any other fraction, we first begin by converting that mixed number into an improper fraction. So let’s begin by converting one and one-quarter into an improper fraction.

We begin by multiplying the integer part, that’s the whole number, by the denominator of the fraction. One times four is four. We then add this number to the numerator of our fraction. Four plus one is equal to five. This forms the numerator of the fraction; the denominator remains unchanged. So one and one-quarter is the same as five-quarters. Let’s do this again with one and two-thirds. Once again, we multiply the integer part by the denominator of the fraction; one times three is three. And then we take this number and we add it to the numerator of our fraction; three plus two is five.

Again, five forms the numerator of our improper fraction, and the denominator remains unchanged. So one and two-thirds is five-thirds. And so to multiply one and a quarter by one and two-thirds, we’re going to multiply five-quarters by five-thirds. We know, of course, that to multiply fractions, we simply multiply their numerators together and then multiply their denominators together. So here, that’s five times five over four times three, which gives us 25 over 12. Now, of course, we were given a question in mixed numbers, so we’re not quite finished.

We’re going to need to convert twenty-five twelfths back into a mixed number. And to do so, we essentially perform a reverse process to changing a number from a mixed number into an improper fraction. We divide our numerator by our denominator. We ask ourselves how many 12s make 25? And this is one of the very few times we use remainders. We say that 25 divided by 12 is two remainder one. Two forms the integer part of our answer; it’s the whole number. And one is the numerator of the fraction. The denominator remains unchanged, so it’s 12. And so we see that one and a quarter times one and two-thirds is two and one twelfth.

Let’s now consider how we divide fractions.

Work out twelve-fifths divided by two-fifths.

Let’s begin by thinking about what it actually means to divide twelve-fifths by two-fifths. Essentially, when we divide, we’re sharing. We want to share twelve-fifths into two-fifths sized pieces. So let’s draw a diagram to represent this. Each of these bars are split into five equally sized squares, so each square must represent one-fifth. We shade 12 of these to represent twelve-fifths. And we’re going to share these into pieces sized two-fifths each. Two-fifths must be two squares. So we can take two squares here. We’ll take two more. And we’ll continue in this manner, until we’ve used up all of our squares.

So how many two-fifths size pieces did we share our twelve-fifths into? Let’s count them. We have one, two, three, four, five, six two-fifths size pieces. And so this tells us that twelve-fifths divided by two-fifths must simply be equal to six. But this is quite a long way to perform division. So how else can we consider this problem? We don’t want to draw these diagrams out each time. Well, we can say that to divide twelve-fifths by two-fifths, we simply divide their numerators. And this works because their pieces are the same size, and that’s because their denominators are equal.

So one technique that we have to divide a pair of fractions is to create a common denominator and equivalent fractions and then divide their numerators. In fact, there is another method, but we’ll consider this method first.

Calculate four-fifths divided by three-quarters.

One method we have to divide fractions is to create equivalent fractions with a common denominator and then simply divide the numerators. Now, whilst we don’t need to find the least common denominator, it does ensure that there’ll be minimal simplifying at the other end. So what is the common denominator of our two fractions? Well, the least common multiple of five and four is 20. So that’s the denominator we’re going to use. So how do we convert from fifths into twentieths? Well, to get from five to 20, we multiply by four. To create an equivalent fraction, we must do the same to the numerator. So four times four is 16, meaning four-fifths is equivalent to sixteen twentieths.

We repeat this process for three-quarters. To turn quarters into twentieths, we multiply by five. And so we have to do the same to our numerator. We’re going to multiply three by five, which is 15, meaning three-quarters is equivalent. It’s the same as fifteen twentieths. So we’re dividing sixteen twentieths by fifteen twentieths. Once their denominators are the same, we’re able simply to divide the numerators. So we’re going to divide 16 by 15. Of course, the fraction line actually means divide. So 16 divided by 15 can be written as 16 over 15.

And since this is an improper fraction — in other words, the numerator is greater than the denominator — we’re going to convert it into a mixed number. We ask ourselves how many 15s make 16? Well, it’s one remainder one. This means the integer part is one and the numerator of our fraction is also one. The denominator remains unchanged, so 16 over 15 is equivalent to one and one fifteenth. And so four-fifths divided by three-quarters is one and one fifteenth. But this isn’t the only method we have to divide fractions.

To divide by a fraction, we can multiply by the reciprocal of that fraction. So the first fraction remains unchanged. And instead of dividing, we’re timesing. Reciprocal means one over. But if we already have it in fraction form, we simply invert it. We switch the numerators and denominators. So the reciprocal of three-quarters is four-thirds. And the sum becomes four-fifths times four-thirds.

But why does this work? Well, what we’re doing is dividing by the numerator of our second fraction to get a unit fraction and then multiplying by the denominator to get the whole. And the beauty of this method is we already know how to multiply fractions. We simply multiply their numerators and then separately multiply their denominators. So this becomes four times four over five times three, which is sixteen fifteenths. Once again, we know that this becomes one and one fifteenth. So we have our alternative method for dividing four-fifths by three-quarters. Now, both of these methods are equally valid.

In our next example, we’ll consider how each method works when we divide mixed numbers.

Work out three and one-third divided by two and one-half.

We have two methods for dividing fractions. But before we perform either of these, we need to spot that we’ve been given a problem involving mixed numbers. And so before we can perform the division, we need to convert the mixed numbers into improper fractions. Let’s begin with three and one-third. We multiply the integer part by the denominator of the fraction. Three times three is nine. We then add this value to the numerator of the fraction. So nine plus one is 10. This forms the numerator part of our improper fraction, and the denominator remains unchanged. So three and one-thirds is equal to ten-thirds.

We’ll repeat this process for two and a half. This time, the integer part multiplied by the denominator is two times two, which is four. And adding this to our numerator gives us five. So two and a half is equal to five over two. We’ll now consider one method we have for dividing fractions. And that is to find a common denominator. Once we’ve created equivalent fractions with that denominator, we simply divide the numerator. So what’s the least common denominator for our two fractions? Well, the least common multiple of three and two is six. So we create equivalent fractions by making a denominator of six.

With our first fraction to achieve this, we multiply by two. So we do the same to our numerator. This means our first fraction, ten-thirds, becomes 20 over six. With our second fraction, we need to multiply the numerator and denominator by three. And so five over two is equivalent to 15 over six. Now that our denominators are equal, to divide, we simply divide the numerators. 20 divided by 15 we can write as 20 over 15. We’ll simplify this fraction by dividing through by a common factor of five to give us four-thirds.

Now, of course, since the question was given to us as a pair of mixed numbers, we should convert this back. Four divided by three is one remainder one. The denominator remains unchanged. So we see that three and one-third divided by two and a half is one and one-third. Let’s now consider the alternative method. To divide by a fraction, we multiply by the reciprocal of that fraction. Informally, this has the name keep, change, flip, though this name can be quite unpopular. To divide by five over two, we multiply by the reciprocal of five over two. The reciprocal of a fraction involves inverting the two numbers by switching the numerator with the denominator.

So we’re going to multiply by two over five. We keep the first fraction the same, we change the divides to a multiply, and we flip the second fraction. That’s a nice way of saying, “find the reciprocal.” Then what we could do is multiply the two numerators and multiply the two denominators, 10 times two over three times five. Alternatively, we can perform a little time-saver. And notice that we have a common factor of five diagonally across our sum. And so let’s divide through by this factor. And we get two-thirds multiplied by two over one. Now, when we multiply the numerators and then the denominators, we get two times two over three times one which, once again, gives us four-thirds. Three and one-third divided by two and a half is one and one-third.

In this video, we’ve learned that, to multiply fractions, we multiply their numerators and then separately multiply their denominators. We also saw that we have two methods to divide fractions. We can either create a common denominator and a pair of equivalent fractions and then divide the numerators. Alternatively, we say that to divide by a fraction, we multiply it by the reciprocal of the second fraction. And this is sometimes called informally keep, change, flip. We also saw that if we want to perform these sorts of calculations with mixed numbers, we must first convert them into improper fractions. From that stage, the same rules hold.

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