5.4

## National STEM Learning Centre

Skip to 0 minutes and 8 secondsPAULA KELLY: So let's have a look at how we divide a decimal by another decimal.

Skip to 0 minutes and 13 secondsMICHAEL ANDERSON: So for this example, we're going to look at 1.2 and we're going to divide that by 0.3. Now we can think of a division much in the same way as we think of fractions. Now, it might look a little bit unusual, but I could write this as a fraction as 1.2 over 0.3. Now, that might not help us too much at the moment, but what we can do is instead of simplifying these down, we can multiply up instead. So I'm going to take the top, the numerator, and the bottom, the denominator, and I'm going to multiply them both by 10. So if I multiply the top by 10, 1.2 multiplied by 10 gives us 12.

Skip to 0 minutes and 56 secondsAnd if I multiply the bottom as well-- so I'm going to times that by 10. It's going to give me 3. So I've changed the fraction from 1.2 over 0.3, to 12 divided by 3. And I know my 3 times tables. How many 3s go into 12? Well, that's going to be 4.

Skip to 1 minute and 17 secondsPAULA KELLY: So written as a fraction, that makes perfect sense. As you were saying, dividing by whole numbers is so much easier. We've kept both of our numbers in proportion by doing the same to the top and bottom. We've multiplied both top and bottom by 10. So if we could see it visually and see what we've actually done as well. So if we're looking here, we've got our 1.2. Because we're dividing by not 0.3, we're seeing how many not 0.3s will fit into 1.2.

Skip to 1 minute and 46 secondsMICHAEL ANDERSON: And do you know your not 0.3 times tables? [LAUGHTER]

Skip to 1 minute and 49 secondsPAULA KELLY: Well, we can count up on here in not 0.3s, but simpler is to use our 3 times table and see how many 3s go into 12.

Skip to 1 minute and 59 secondsMICHAEL ANDERSON: And we've got hopefully the answer to both is 4. 0, then 3, 6, 9, 12. And because we've scaled our numbers up here and we've scaled them both by a factor of 10, that means that the answer to this question 4 is also the answer to this question. So 1.2 divided by 0.3 is 4. So let's have a look at another example. So say we're doing 12 divided by 0.5. So what we're asking here is how many 0.5s can fit into 12? So we can do that same method with the fractions. So our numerator is going to be 12 and our denominator is 0.5. Now, this looks a bit weird. We're not used to having decimals in fractions.

Skip to 2 minutes and 47 secondsWe normally like to have fractions where both the top and the bottom are integers. So what I can do to change that is to multiply that 0.5 by 10. So if I just extend this line a little bit, and if I multiply that by 10, then I'm going to get in the bottom of my fraction a 5. And I feel a lot better about that. There's no decimals involved now. But I also have to do the same to the 12. So I'm going to multiply this by 10 as well. 12 multiplied by 10 is 120.

Skip to 3 minutes and 17 secondsSo I've changed the question a little bit, but the answers should still be the same as we've multiplied both the top and the bottom by the same amount. Now, it might not help us too much. My 5 times tables up to 120 are a little bit rusty. So I can always play around with this fraction to make it an easier question. So let's try doubling both sides. I'm going to times this by 2 and I'm going to times this by 2, because that will give me 10 on the bottom. And if I double that, that'll give 240. And I can divide by 10 quite easily. 240 divided by 10. The answer is going to be 24.

Skip to 3 minutes and 58 secondsSPEAKER 1: So really clear with our fractions. We want to make our denominator into a whole number, into an integer. And if we can divide by 10, that's much easier than dividing by 5.

Skip to 4 minutes and 8 secondsSPEAKER 2: Yeah, you can play around with these as much as you like. You can multiply by whatever number you think is going to help, so long as you multiply the numerator and the denominator at the same time by the same amount.

Skip to 4 minutes and 18 secondsSPEAKER 1: So if we have a look at our diagram to see visually what we've done, we started with our question of 12 divided by 0.5.

Skip to 4 minutes and 27 secondsPAULA KELLY: So we want to know how many 0.5s fit into our 12.

Skip to 4 minutes and 30 secondsMICHAEL ANDERSON: Yeah. And one strategy could be just to count them up. So we've got the little markers here, which are all at 0.5 intervals. So we could just count them. And start at 0, 0.5, 1, 1.5, 2, et cetera, et cetera, et cetera, keeping track of what we've got to up until 12. And then how many not 0.5s we're counted on in, that will be our answer. But it's quite time-consuming. So a better way would be something that takes less time.

Skip to 4 minutes and 56 secondsPAULA KELLY: Absolutely. So we know our 0.5s-- we have 24 of them here. By the same scale, rather than 12, if we multiplied all of our numbers by 10 till 120--

Skip to 5 minutes and 9 secondsPAULA KELLY: --so this time, like with our fraction, we're looking at how many 5s go into 120, rather than how many 0.5s go into 12. We can see it's exactly the same amount. There's still going to be 24 lots of 5 in 120, as there were 24 lots of 0.5 in 12.

Skip to 5 minutes and 30 secondsMICHAEL ANDERSON: Yeah, they're the same questions. 12 divided by not 0.5 is the same question as 120 divided by 5. We've just scaled one up. So let's look at a more difficult example involving smaller numbers. So say we're trying to divide 0.24 and we're going to divide that by 0.03.

Skip to 5 minutes and 52 secondsPAULA KELLY: So looking at this, it doesn't look very nice.

Skip to 5 minutes and 55 secondsPAULA KELLY: How could we make it a bit nicer?

Skip to 5 minutes and 56 secondsMICHAEL ANDERSON: Well, the same kind of idea of scaling it up to a different question that we can solve a lot more easily. So if we think about this as a fraction again, we've got 0.24 over 0.03. Now, ideally, I'd like to deal with integers, with whole numbers. So I'm going to multiply both the top and the bottom by 100 this time, because that will shift our digits two places to the left. So let's times them both by 100 on the top and 100 on the bottom. And notice that 100 over 100 is just 1, so essentially we're multiplying by 1. So we're not changing the actual value of this calculation. We're going to end up with 24 over 3.

Skip to 6 minutes and 41 secondsMICHAEL ANDERSON: Yeah. How many 3s go into 24? Well, I know my 3 times tables. It's 8.

Skip to 6 minutes and 48 secondsPAULA KELLY: Fantastic. And once more we have a look visually. We have our 0.24 and we're looking at how many 0.03s fit into here.

Skip to 6 minutes and 58 secondsPAULA KELLY: If we counted, we'd have eight of these. So eight lots of 0.03 will give us 0.24. Same way we've scaled our numbers up.

Skip to 7 minutes and 8 secondsPAULA KELLY: Rather than 0.24, we've got 24.

Skip to 7 minutes and 12 secondsPAULA KELLY: Fantastic. And then 3 is also 100 times larger than 0.03. But still we have eight 3s to give us 24. Eight 0.03s give us 0.24.

# Dividing decimals

Dividing a decimal number by a decimal number can be difficult for some students as they attempt to use tried and tested algorithms with more challenging values. It is important that students are able to manipulate the question efficiently, using basic skills, in order to create a less challenging problem.

## True or false?

True or false: When dividing one number by another the answer will always be smaller than the number you started with?