Lesson Video: Dividing Mixed Numbers Mathematics • 6th Grade

In this video, we will learn how to divide two mixed numbers and apply this in real-life situations.

16:34

Video Transcript

In this lesson, we’re going to learn how to divide using mixed numbers and apply this to real-life situations. Remember, a mixed number is a number that consists of an integer, a whole-number part, and a proper fraction. That’s a fraction where the numerator is smaller than the denominator. We’re also going to need to recall the technique for converting between mixed numbers and improper fractions. And that technique is going to be really important going forward. So make sure you feel confident with it.

To convert a mixed number into an improper fraction, we begin by multiplying the integer by the denominator of the proper fraction. We then take that number and we add it to the numerator of the proper fraction. This forms the numerator of our improper fraction, where the denominator remains unchanged. For instance, let’s take two and three-quarters. We multiply the integer, that’s two, by four. And two times four is equal to eight. We take this number and then we add that to the numerator. Eight add three is equal to 11. This forms the numerator of our improper fraction.

Then the denominator is the same as the denominator in the proper-fraction part of our mixed number. So two and three-quarters is equal to eleven quarters. Then, to convert an improper fraction to a mixed number, we sort of do the reverse. We begin by dividing the numerator by the denominator. The answer we get gives us the integer part of our mixed number. The remainder is the numerator to the proper-fraction part. And then once again the denominator is the same as the original denominator.

Let’s take, for instance, 15 over seven. We divide the numerator by the denominator. So we do 15 divided by seven. And that’s two with a remainder of course of one. The two is the integer part of our answer. And then one forms the numerator of the proper fraction. The denominator remains unchanged. And so it’s seven, meaning 15 over seven is equal, it’s equivalent, to two and one-seventh.

So now we’ve reminded ourselves how to convert between mixed numbers and improper fractions, we’re going to begin by looking at how we might divide a mixed number by an integer.

Evaluate three and three-fifths divided by six, giving the answer in its simplest form.

Before we do anything, we should always look to convert any mixed numbers in our calculation into improper fractions. We have a mixed number here. It’s three and three-fifths. So let’s convert three and three-fifths into an improper fraction. Remember, to do this, we take the integer part and we multiply it by the denominator of the proper fraction in our expression. Three times five is equal to 15. We then take that number and we add it to the numerator of the fraction. 15 plus three is 18. And this number forms the numerator of our improper fraction.

The denominator is the same as the denominator in the proper fraction in our expression. So three and three-fifths is equal to eighteen-fifths. And so an equivalent calculation to the one in our question is eighteen-fifths divided by six. And we can begin by thinking about this pictorially. Dividing by six is like saying, well, let’s share three and three-fifths or eighteen-fifths into six equal parts.

Now, we see we have 18 equally sized pieces. And of course we know how to divide 18 by six. 18 divided by six is equal to three. When we share 18 into six equal parts, each part consists of three equally sized pieces. And we can see then that eighteen-fifths shared into six equal parts will mean that each part consists of three-fifths. And so eighteen-fifths divided by six and therefore three and three-fifths divided by six is equal to three-fifths.

So one technique we do have for dividing a mixed number by an integer is to convert that mixed number into an improper fraction and then divide the numerator by the integer part. But what do we do if the integer isn’t a factor of the numerator? And what do we do if the divisor isn’t an integer at all? Let’s see if we can develop a technique that will work no matter what.

Calculate nine-tenths divided by seven and one-fifth. Give your answer in its simplest form.

Remember, when we’re performing calculations with mixed numbers, we always begin by converting those into improper fractions. We have seven and one-fifth here. So we begin by multiplying the integer part by the denominator of the fraction. Seven times five is 35. We then take this number and we add it to the numerator part of the fraction. 35 plus one is 36. This number forms the numerator part of our improper fraction. And then the denominator remains unchanged. So seven and one-fifth is equal to thirty-six fifths. And so we can rewrite our entire calculation as nine-tenths divided by thirty-six fifths. So how do we work this out?

Well, we do have a couple of techniques for dividing fractions. Let’s remind ourselves what those are. Method one is to create a common denominator. So if we look at nine-tenths and thirty-six fifths, we know that the common denominator will actually be equal to 10. And to achieve this with our second fraction, we’re going to need to multiply both the numerator and denominator by two. So nine-tenths divided by thirty-six fifths is equal to nine-tenths divided by seventy-two tenths.

Once the denominators are the same, we can simply divide the numerators. Nine-tenths then divided by seventy-two tenths is the same as nine divided by 72 or nine over 72. And of course we can simplify this by dividing both the numerator and denominator of the fraction by nine. And that gives us an answer of one-eighth. And this is a really lovely method for dividing fractions as it really shows us what’s going on. But you may have heard of an alternative method.

In that alternative method, to divide by a fraction, you multiply by the reciprocal of that fraction, where the reciprocal of a number is one over that number. Now, what this looks like when you’re working with a fraction is that the numerator and denominator switch. And so nine-tenths divided by thirty-six fifths is the same as nine-tenths times five over 36.

Now we might look to multiply both the numerators together and then both the denominators together. But we can also cross cancel to make this calculation a little bit easier. We can divide through by five. And then we can divide through by nine. And so our calculation becomes a half times a quarter. One times one is one, and two times four is eight. And that shows us once again that nine-tenths divided by thirty-six fifths is equal to an eighth.

Now, of course, either of these methods is perfectly valid. It’s very much a matter of personal preference. But the big takeaway here is that when doing any calculation with mixed numbers, it’s always sensible to convert them into improper fractions first.

In general, we say that to divide with mixed numbers, we begin by converting all mixed numbers into improper fraction form. And if necessary, if we’re working with any integers, we write them with a denominator of one. Then we use whatever method we prefer for dividing fractions. And we divide as normal.

Note that if the question asks us to give our answer in its simplest form, and we end up with an improper fraction after performing the division, we do then need to convert that back into a mixed number. Let’s see what that might look like.

Calculate fourteen fifteenths plus ten-fifths divided by one and two-fifths, giving your answer in its simplest form.

Remember, when we’re performing calculations involving mixed numbers, we always begin by turning those mixed numbers into improper fractions. Here we have one and two-fifths. And we know that to convert a mixed number into an improper fraction, we begin by multiplying the integer part by the denominator. Here that’s one times five, which is five. We then take that number and we add it to the numerator of the proper-fraction part. That gives us five plus two, which is equal to seven. This number forms the numerator part of our improper fraction. And the denominator is the same as the denominator in the proper fraction. So one and two-fifths is equal to seven-fifths.

Now, we’re also going to apply the order of operations. And we’re going to begin by performing the calculation inside the pair of parentheses. That’s fourteen fifteenths plus ten-fifths. Now we might also even notice that ten-fifths or 10 divided by five is equal to two. And then we might look to create a mixed number by adding two and fourteen fifteenths. But of course then we will need to convert that back into an improper fraction.

So let’s recall how we actually add fractions. We create a common denominator. So what we’re going to do is multiply both the numerator and denominator of our second fraction by three to give us a denominator of 15. When we do, we find that ten-fifths is equivalent to thirty fifteenths. And so we get fourteen fifteenths plus thirty fifteenths. And of course now we have that common denominator; we just add the numerators. And we get forty-four fifteenths. And so our calculation now becomes forty-four fifteenths divided by seven-fifths.

And we know that there are a couple of ways that we can divide fractions. Let’s look at the first method. That involves creating a common denominator. Once again, that denominator is actually going to be 15. And so we’re going to multiply the numerator and denominator of our second fraction by three. And so we get forty-four fifteenths divided by twenty-one fifteenths.

Now that the denominators are equal, we simply divide the numerators. We can write 44 divided by 21 as 44 over 21. And since we’re asked to give our answer in its simplest form, we’re going to finally turn this back into a mixed number. 44 divided by 21 is two with a remainder of two. So two forms the integer part, and then another two forms the numerator of the proper-fraction part. The denominator remains unchanged, so 44 over 21 is two and two twenty-oneths.

Now, of course, we do have one second method, so we’ll briefly consider that. In the second method, we simply multiply by the reciprocal of the second fraction, by the divisor. So forty-four fifteenths divided by seven-fifths is equal to forty-four fifteenths times five-sevenths. Then we could multiply the numerators and separately multiply the denominators. But we might notice that we can divide both five and 15 by five. And so now we do 44 times one to get 44 and three times seven to get 21. And once again we find that forty-four fifteenths divided by seven-fifths is 44 over 21, which we’ve seen is equal to two and two over 21.

We’re now going to consider a contextual question.

Sophia and Liam have walked 13 and one-quarter kilometers in two and one-quarter hours. As they want to walk for three hours, what fraction of their walk have they completed so far? Then the second part says if they keep up the same pace, what distance will they have walked in three hours?

Sophia and Liam have walked for two and a quarter hours. And they want to walk for three. To find the fraction of the walk that they’ve completed so far, we’re going to divide the amount they’ve completed by the total. Our instinct might be to write this as two and one-quarter over three. But actually, we know that fraction line means divide. So we’re going to write it as two and one-quarter divided by three, as shown.

Next, we know that to divide when we’re working with mixed numbers, we need to make sure any mixed numbers are written in improper fraction form. Similarly, any integers we write with a denominator of one. Now two times four is eight. And when we add the one, we get nine. So two and one-quarter is equivalent to nine-quarters. We also know that three, since it’s an integer, can be written as three over one. And so the calculation we’re doing is nine over four divided by three over one.

Now, in fact, since three is a factor of nine, we could actually simply divide nine by three. But let’s prove to ourselves that our methods that we have for dividing fractions work when we’re working with integers two. One of those involves multiplying the first fraction by the reciprocal of the second. And so we can say that nine-quarters divided by three over one is actually the same as nine-quarters times one-third. Then we cross cancel. Nine divided by three is three, and three divided by three is one. So we get three-quarters times one over one. And that of course is simply equal to three-quarters. Sophia and Liam have completed three-quarters of the walk so far.

Then the second part says that they’re going to keep up the same pace. So they’re going to have the same average speed. What distance will they have walked in three hours? And so one thing that we could do is use the speed–distance–time formula. Speed is equal to distance divided by time. So we could work out the average speed for the first part of their journey by dividing 13 and a quarter by two and a quarter. Then we can work out the total distance that they’ll walk in three hours by multiplying this speed by the time taken, by three.

But there is in fact another method. We know that they’re going to be walking at the same pace. So the total distance they travel will be directly proportional to the amount of time taken. And so we’re going to divide the distance that they walked in the first part of the journey by the fraction of the walk that they’d completed so far. So that’s 13 and one-quarter divided by three-quarters.

To perform this calculation, we convert 13 and one-quarter into a mixed number. 13 and one-quarter is the same as 53 over four. And that’s because 13 times four is 52. And then we add the numerator one to get 53. So we’re doing fifty-three quarters divided by three-quarters.

Now, of course, since the denominators of these fractions are equal, we simply divide the numerators. And so we get 53 over three. We do of course need to change this back into a mixed number. 53 divided by three is 17 with a remainder of two. Since we’re converting an improper fraction into a mixed number, we know that the denominator remains unchanged. It’s still three. And so we can say 53 over three is equivalent to 17 and two-thirds. And therefore, the total distance they will have walked in three hours, assuming that they maintain the same pace, will be 17 and two-thirds of a kilometer.

In this video, we’ve learned that whenever we’re performing a calculation involving a mixed number, we begin by converting that mixed number or even more than one mixed number into improper fractions. We then saw that if we’re dividing a mixed number by an integer, as long as the integer is a factor of the numerator of the improper fraction, we can simply divide those.

But actually, alternatively, we can write integers with a denominator of one. And then we can use the standard methods that we have for dividing any fraction, that is, creating a common denominator and then simply dividing the numerators or multiplying by the reciprocal of the divisor. Finally, we saw that, when necessary, we should always look to try and convert our answers back into mixed number form.

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