# Lesson Video: Dividing Rational Numbers Mathematics • 7th Grade

In this video, we will learn how to divide rational numbers, including fractions and decimals.

17:17

### Video Transcript

In this lesson, what we’ll be looking at is dividing rational numbers. And this will include fractions and decimals. So by the end of the lesson, what we should be able to do is divide a rational decimal by a rational decimal, divide a fraction by a fraction, divide rational numbers in various different forms, and, finally, solve word problems involving the division of rational numbers.

But before we start to do any of that, what is a rational number? Well, in fact, a rational number is a real number that can be written as 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is not equal to zero. Another way of saying this is that it is any number that can be represented as the ratio between two integers. So what we have here are some examples. So first of all, we’ve got two-fifths which we can see is written in the form 𝑎 over 𝑏. Well, then we have 0.3 recurring. But we think, “hold on! This isn’t written as 𝑎 over 𝑏. So how come this is a rational number?” Well, 0.3 recurring can be written as one over three or one-third. So it is also worth noting at this point that, in fact, any recurring decimal is in fact a rational number. Then we have three.

But, again, we’re thinking, well, that’s not in the form 𝑎 over 𝑏. Well, in fact, it could be written as three over one or even nine over three; they would both give us three. Then we have another recurring decimal, which is 0.142857 recurring. This time, I’ve just put this in here just to show there’s a different way of showing recurring here. And that’s with a straight line, not just a dot. We could have a dot over the one and a dot over the seven to mean the same thing. And as we said before, all recurring decimals are rational numbers. And this one is the same as one over seven or one-seventh. Then, finally, for our examples, we have 0.125, which you might already know is an eighth. And it’s also worth pointing out here is that this is something called a terminating decimal. And any terminating decimal is also a rational number.

So therefore, what we can also surmise is that the converse of this, anything that doesn’t satisfy this rule, is not going to be a rational number. It’s going to be an irrational number. Okay, great. So we’ve looked at what rational numbers are. So now what we’re going to do is move on to our questions.

Evaluate 0.8 divided by 0.4.

So what we have here are two terminating decimals, and we’re going to divide them. And we have a couple of methods to do this. So what we’re gonna do is have a look at both of them. So for method one, what we’re gonna do is we’re gonna multiply both of our decimals. And we’re gonna multiply them both by 10. And that’s because what we’re gonna do is make it so, in fact, we’re not dividing decimals at all. We’re gonna be dividing units. So if we multiply 0.8 and 0.4 by 10, what’s gonna happen is that each of the numbers is going to move one place value to the left. So what we’re gonna get is eight divided by four. And we can do this because we’ve done it to both terms. So therefore, it’s going to give us the same result. Well, this is nice and straightforward. And that’s because eight divided by four is equal to two. So therefore, we can say that 0.8 divided by 0.4 is two.

So now we’re gonna take a look at method two. And for method two, what we’re going to do is convert both of our decimals to fractions. And we can do that because they’re both terminating decimals. And we know that terminating decimals are rational numbers, so therefore can be converted to a fraction with an integer as the numerator and an integer as the denominator. Well, if we start with 0.8, what this means is eight-tenths. Well, in turn, we can simplify eight-tenths by dividing the numerator and denominator by two, which will give us four-fifths. Then we have 0.4, which is gonna be equal to four-tenths, which again we can simplify by dividing the numerator and denominator by two to give us two-fifths.

Okay, great. We now got our two fractions. So we now have four-fifths divided by two-fifths. And we’ve got a method for dividing fractions. And what we can do is use our memory aid to remind us how to do that. And that is KCF, which is keep it, change it, flip it. And this means we keep the first fraction the same, we change the sign from a divide to a multiply, and we flip the second fraction. And it’s worth noting that if we flip a fraction, this is in fact the reciprocal of that fraction. And then if we multiply two fractions, all we do is multiply the numerators and multiply the denominators. So this is gonna be equal to 20 over 10, which once again gives us an answer of two.

Now this is quite straightforward because we’re multiplying two easy fractions. However, there is a quick tip, which can be useful. If you’re ever multiplying fractions, have a look at the numerators and denominators and see if there’s a common factor. So here we can see that five is a common factor. So if we divide both the numerator and denominator by five, we’re just left with four multiplied by one over one multiplied by two, which is four over two, which again would’ve given us two.

So that was our first example. So now what we’re gonna do is have a look at an example where we’re going to evaluate an expression using multiplication and division of rational numbers. And all of these are gonna be in fractional form.

Evaluate three-quarters multiplied by negative two over three divided by a fifth giving the answer in its simplest form.

So to help us evaluate this expression, what we’re going to use is PEMDAS. And what PEMDAS is, is a way of remembering the order of operations. So here it says that the P — we’re gonna deal with the parentheses first, then exponents, multiplication, division, addition, then subtraction. So we can see that, in our expression, what we have are parentheses, so we’ll deal with these first. So what we have is three-quarters multiplied by negative two over three. So to multiply fractions, what we do is multiply numerators then multiply denominators. So this is gonna give us negative six over 12. Well, we can simplify this by dividing both the numerator and the denominator by six. So we’re gonna get negative one over two. So now we’ve dealt with our parentheses. So what we can do is put this value back into our expression.

Well, next we have a division. And that division is negative one over two, which is the result of the parentheses calculation, divided by one over five or one-fifth. Now, to help us remember what we’re gonna do with the division of fractions, we use our memory aid KCF, keep it, change it, flip it. So we keep the first fraction the same. We change our divide to a multiply. And we flip our second fraction. So we get negative one over two multiplied by five over one. So then what we do is multiply our numerators and denominators. So we can say that if we evaluate three-quarters multiplied by negative two over three divided by a fifth, we’re gonna get negative five over two.

So what we had a look at here was a problem involving fractions. What we’re gonna have a look at now is a problem that involves recurring decimals and the modulus or absolute value.

Evaluate 0.8 recurring divided by the modulus or absolute value of negative five over four giving the answer in its simplest form.

So in this question, the first thing we’re gonna have a look at is the recurring decimal. So we got 0.8 recurring. Now, this is, in fact, a rational number. And it’s a rational number because we can represent it as a fraction with an integer as the numerator and an integer as the denominator. And this is something we know about all recurring decimals. So to change this into a fraction, what we’re gonna do is let 𝑥 be equal to 0.8 recurring. And then what we’re gonna do is multiply 𝑥 by 10 to give us 10𝑥 and multiply 0.8 recurring by 10 to give us 8.8 recurring. So we now know that 10𝑥 is equal to 8.8 recurring.

So you might have thought, “Well, why have we just done that?” But it’s actually rather clever because what we’re gonna do now is eliminate the recurring part of our decimal. So let’s label them equation one and equation two. So what we can do is we can subtract equation one from equation two. So when we do that, what we’re gonna get on the left-hand side is nine 𝑥, cause we’ve got 10𝑥 minus 𝑥 which is nine 𝑥. And then on the right-hand side, we’re just gonna have eight, and that’s because if we have 8.8 recurring minus 0.8 recurring, the recurring parts cancel each other out and we’re left with just eight. So then what we’re gonna do is just divide through by nine. And what we get is 𝑥 is equal to eight over nine or eight-ninths. And as 𝑥 is equal to 0.8 recurring, we can say that 0.8 recurring is equal to eight over nine or eight-ninths. And this is a fraction with an integer numerator and an integer denominator.

Okay, great. So we converted that into a fraction. Well, now what about our absolute value or modulus of negative five over four? Well, these vertical lines mean the absolute value or modulus. And what this means is that we’re only interested in the positive value because what the absolute value or modulus means is the distance from zero or magnitude of a value. So therefore, we’re not interested in the negative part. So therefore, what we can do is just write our modulus or absolute value of negative five over four as five over four. So now what our calculation has become is eight over nine or eight-ninths divided by five over four. So now what we’re gonna do is divide our fractions.

And to remember how to do that, we can use our trusty memory aid, KCF: keep it, change it, flip it. So, to use this, we keep the first fraction the same. Then we change the sign from a divide to a multiply. And now we flip the second fraction. So we’ve now got eight over nine multiplied by four over five. So then if we multiply our numerators and denominators, we’re gonna get 32 over 45. And we can see this can’t be canceled down any further. So it is, in fact, in its simplest form. So then we can say the answer is 32 over 45.

Okay, great. So we’ve looked at a number of different skills so far, but what we’re gonna have look at now is to see if we can use these skills to solve problems. And what we’re gonna try and do is find our missing value.

The product of two rational numbers is negative 16 over nine. If one of the numbers is negative four over three, find the other number.

So the first thing we’re gonna do is look at a couple of key terms. So we’ve got product, which means multiply. So if we find the product of two numbers, that means we’re multiplying them together. And then we’re also looking at the term, rational. And what this means is a number that can be written as a fraction with an integer as the numerator and an integer as the denominator, which is gonna help us when we’re gonna try and find the number that we’re looking for. So taking the information we’ve got from the question, what we can do is write it down. And we’ve got negative four over three multiplied by 𝑎 over 𝑏 equals negative 16 over nine. And it’s this 𝑎 over 𝑏 that we’re trying to find.

Well, there are, in fact, a couple of ways we could solve this. So we’re gonna have a look at both of those. So first of all, what we could do is divide both sides by negative four over three. So when we do that, we’ll have 𝑎 over 𝑏 equals negative 16 over nine divided by negative four over three. So then what we can do is divide our fractions. And to do that, we can use our memory aid, KCF — keep it, change it, flip it — which is gonna give us 𝑎 over 𝑏 is equal to negative 16 over nine multiplied by negative three over four. So now, before we multiply, what we can do is divide through by any common factors. Well, first of all, we can divide numerators and denominators by four and then by three.

So now what we’ve got is negative four over three multiplied by negative one over one. Well, a negative multiplied by a negative is a positive. So therefore, what we’re gonna get is 𝑎 over 𝑏 is equal to four over three. So therefore, we’ve found our missing number. And what we can do is check this by using the alternate method. And the alternate method is equating the numerators and denominators. Well, as we know, we’ve got negative four over three in the left-hand side and the result is negative 16 over nine. We know that a negative has to be multiplied by a positive to give us a negative result. So therefore, we know that 𝑎 over 𝑏 will be positive. So we could ignore the signs when we’re gonna equate the numerators and denominators.

Well, if you equate the numerators, we’ve got four 𝑎 cause four multiplied by 𝑎 is equal to 16. So therefore, 𝑎 will be equal to four. So then if we equate the denominators, we’re gonna get three 𝑏 is equal to nine. So 𝑏 is equal to three. So therefore, 𝑎 over 𝑏 is gonna be equal to four over three, which is what we’ve got with the first method.

Okay, great. So we’ve just looked at a problem where we had to find a missing value. So for our final example, we’re gonna take a look at a worded problem, so a problem where we’re gonna use these skills in context.

Noah constructs three-quarters of a wall in one and two-thirds days. How many days will he need to construct the wall?

So what we’re gonna do is use a diagram to help us visualize the problem. So what we can see is it takes one and two-thirds days to build three-quarters of the wall. So therefore, if we want to find out how long it takes to build one-quarter of the wall, what we can do is divide one and two-thirds by three. Well, to complete this calculation, what we want to do is convert one and two-thirds, a mixed number, into a top heavy or improper fraction. So to do that, what we do is we see that there are three-thirds in a whole one add two gives us five over three. So it’s one multiplied by three add two over three. So we’ve got five-thirds divided by three.

Well, if we want to divide five-thirds by three, we can think of this as five-thirds divided by three over one. Well then what we can use is our memory aid for dividing by a fraction, which is keep it, change it, flip it, KCF, which will give us five-thirds multiplied by one over three, cause we keep the first fraction, change the sign, flip the second fraction. So therefore, what we’ve got is the time taken for one-quarter of the wall to be built is five over nine or five-ninths because five multiplied by one is five and three multiplied by three is nine.

So now what’s the next stage? Now what do we need to do? Well, we can see that the whole wall is four-quarters. So therefore, we need to multiply the time taken for a quarter of the wall to be built by four. So this means five-ninths multiplied by four. Well, to help us think what we’ll do with this calculation, we can think of four as four over one. So then we’re gonna do five multiplied by four over nine, which is equal to 20 over nine. Well, this is an improper fraction. So what we could do now is convert this back into a mixed number. Well, to do that, what we do is we see how many nines go into 20, which is two with a remainder of two. So therefore, we know that it takes two and two-ninths days to construct the wall.

In fact, it is worth noting that we could’ve completed this question with one calculation. And that would’ve been one and two-thirds multiplied by four over three because if we divide by three and multiply by four, it’s the same as multiplying by four over three. And it could also have been solved by the calculation one and two-thirds divided by three over four because if we go backwards from keep it, change it, flip it, we could see that one and two-thirds divided by three over four is the same as one and two-thirds multiplied by four over three.

So we’ve taken a look at a number of examples and covered the key objectives for the lesson, so now let’s take a look at the key points. So the first key point is that a rational number is in fact a real number that can be written as 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers and 𝑏 is not equal to zero. So therefore, it can be written as a fraction. Another way we can think of that is that a rational number is a number that can be represented as the ratio of two integers. And we also know that both recurring decimals and terminating decimals are also rational numbers because these can be written as fractions.

For example, on the left, we have our recurring decimal, which can be written as a seventh. And on the right, we have a terminating decimal which could be written as an eighth. And for our final key point, if we’re gonna divide two fractions, so 𝑎 over 𝑏 divided by 𝑐 over 𝑑, then this is equal to 𝑎 over 𝑏 multiplied by 𝑑 over 𝑐. So you multiply it by the reciprocal with the second fraction. And we have a memory aid to help us remember this. And that is KCF: keep it, change it, flip it. Keep the first fraction the same, change the sign from a divide to a multiply, and flip the second fraction.