 So, here's a typical one mark exam style question. Check whether the decimal form of 27 by 48 terminates or not. So let me simplify the question first. We're given a fraction here and we want to talk about its decimal form and terminates in simple terms means ends. So we want to find whether the decimal form of this fraction looks something like This is the decimal form of half whether it terminates whether it ends or does it look something like this? point three three three three three goes on forever. So let's go ahead and solve it if I open my calculator and I punch in these numbers 27 by 48 I get point five six two five. So this means that this is actually Point five six two five and it clearly ends. So question solved the decimal form of this fraction actually terminates Now, I know what you're thinking in your exams calculators are not allowed So you're looking for a way to solve this question without a calculator And that's what we're going to do but instead of giving away that method What I would like us to do is together discover that method so by the end of this video will be able to look at any fraction not just this one and Talk about its decimal form whether it terminates or not and more importantly will have a very clear idea of why that method works So let's get started. Let's actually try to find the connection between fractions and decimals What is it about these fractions that tell us whether the decimal form will end or will never end to solve this mystery? We need a couple of more examples. Let me clean this board and Start writing a few decimals and fractions. Let me first write a few decimals that end 0.2 ends 0.25 ends and 0.175 all of these three end. I'm calling it decimals that end or They terminate Okay, let me write a few more decimals a few that don't end. I'll pick a few common ones 0.6 bar 0.6 repeating then there's this 0.83 bar and here's one of my favorite 0.142857 all of this repeating These decimals don't end. I'm going to call it don't end Don't end. They are non-terminating Non-terminating. Is there any other type of decimals? Hmm. Well, there's one type which doesn't end But it's also not repeating like 0.125798 0.8 So there's no clear pattern and it doesn't end But I don't think we need to consider these type of decimals for this video because we're looking for a relation between fractions and decimals and These decimals are not going to give us fractions. They're actually irrational numbers They're actually irrational numbers. So we don't get a fraction. We're not interested. Let's Get rid of them and only look at these decimals Okay, so now that we have these numbers in the decimal form. Let's also write their fractional form So 0.2 is actually 2 by 10 1.25 is 125 by 100 0.175 is 175 by thousand and I'm going to use a calculator here because I don't really remember what these fractions are 0.6 bar is actually two third 0.83 bar is where is it? It's five sixth point one four five one four two eight five seven that that's something that I remember That's one seventh. So yeah Here are a few fractions which give us terminating decimals and a few fractions which give us non-terminating decimals Can we see a pattern here? Do we do we have something special going on here? Something that's always giving us terminating decimals pause the video think about it So did you see it? Here's the pattern if you look at these fractions which give us decimals that end these denominators are somewhat special All of these denominators are multiples of 10. No wait powers of 10 It's the first power second power in the third power and you don't see that here So maybe Maybe for a fraction to give us a decimal that end. We need its denominator. I'm going to write dr here its denominator should be be a power of 10 Power of 10 now this does make sense, but let's be really sure about this. Is there no the fraction that does the job hmm Well, I can think of a few we started this video with a fraction half now half gives the decimal 0.5 which is terminating and this is not a power of 10 so clearly there's something else going on and Here's one more number that breaks the rule one by four, which is a quarter This gives us 0.25 even that's terminating and this is not a power of 10 Hmm. I think we jump to this conclusion a bit too fast because Even here I can see some non powers of 10 if I simplify these fractions. I'll get this is one by five This is five by four and this is if I cancel 25 out I'll get seven by 40. So I can see these numbers five four 40 and even two I can see these numbers popping up and and none of these numbers are powers of 10 So now what do we do? What's so special about these denominators that are giving us terminating decimals? What do these numbers have that these numbers don't three six and seven? Pause for a moment and see if you can figure it out yourself Okay, so here's what's going on. Let me clean this up first So here's what's going on We still need powers of 10 in the denominator to get these decimals that end But we don't always need to start with these powers of 10 We can start with some other numbers like these and here's why they work if you look at 10 That's made up of a 2 and a 5 and all higher powers of 10 are also made up of 2s and 5s 100 is made up of 2 2s and 2 5s Similarly, 1000 will be made up of 3 2s and 3 5s So you can start with 10, but you can also start with a 5 If you have a 5 with the denominator simply multiply numerator and denominator by 2 to get 2 by 10 And there you have it a 10 in the denominator That will give you a terminating decimal and this is what's happening for all of these numbers If you look at 4, that's actually 2 times 2 To make it a power of 10 you multiply this by 25 Now let's look at 40 40 is made up of if I factorize 40 That's made up of a 2 and a 2 and a 2 and a 5 8 times 5 is 40 to make this a power of 10 We'll multiply this by 5 times 5 which is 25 This will give us 1000 and that's exactly what's going on here And this also explains why the other fractions were not working We now know why 2 by 3 was not working and And 5 by 6 was not working and 1 by 7 was not working For all three denominators 3 6 and 7 we have numbers that will never give us 10 There is no integer that you can multiply 3 with to get 10 or a power of 10 And that's the problem with these two as well Okay, I think we've figured it out now. Let's summarize If we want a terminating decimal if we want a decimal that ends We need to start with a fraction that has a power of 10 in the denominator But if you're not that lucky if we don't have a power of 10 Then at least we should start with a number that has only 2s and 5s as their factors If we start with anything else Then we're not going to get a terminating decimal. Isn't that amazing just by looking at these denominators We can figure out whether the decimal is going to end or not end I'm excited to try a few more problems. Let's get this out And this time we're not going to use a calculator So let me write down a few fractions 13 by 12 and let's let's add one more 29 by 35 and one more 21 by 30 pause the video and check whether these fractions are going to give us Terminating decimals or non-terminating decimals Okay, so let's do this together If you look at the denominator, we have 12 And 12 is made up of a 2 and a 2 and a 3 2s are fine, but this 3 is going to be a problem So this is not going to give us a terminating decimal. This is going to give us a non-terminating decimal What about 29 by 35? Well 35 is made up of a 5 and a 7 5 is fine, but 7 is a problem 7 is a problem. So a non-terminating decimal And 20 by 30 if you look at 30 It's made up of a 2 and a 5 and a 3 2 is fine, 5 is fine, but 3 is a problem So this is also going to give us a non-terminating decimal Now let's check our answers using a calculator 30 by 12 is actually It's actually 1.083 and 3 is repeating. So yeah, that's going to give us a non-terminating decimal. So we were right 29 by 35 is actually 0.828571 for okay, that's that's a really long one, but It's also repeating and this is not going to end. It's a non-terminating decimal. So we were right And the third one is 21 by 30. That's actually 0.7 Okay, hold on something's wrong. This was supposed to be a non-terminating decimal, but Why did it end? Try to pause the video and see if you can figure out what's going on here Okay, are you ready? Did you see it 21 by 30? We had a 2 and a 5 and a 3 which was a problem But all this while we were only looking at the denominator If I also look at the numerator 21 that's made up of a 3 and a 7 Surprisingly this 3 which was a problem in the denominator actually gets cancelled out So My fraction becomes I'm going to get rid of NT my fraction becomes 7 upon 2 times 5 which is 10 and 10 will always give us a terminating decimal So We should still check for these 2s and 5s in the denominator But only after we have the fraction written in the simplified form We should make sure that everything that's a problem actually gets cancelled out first Okay, now let's really summarize and let's do that using the first question that we started with We had 27 by 48 and we know that this is going to be 0.5625 But suppose we don't have the calculator And we want to know whether this fraction terminates or not in the decimal form. What do we do? So the first thing that we do is we prime factorize We see that the denominator is 48 That's made up of a 2 and a 2 and a 2 and a 2 that's 16 and one more 3 That's 3 and The 2s are fine. I think We know where this is going If this 3 cancels out, we're going to get a terminate decimal. If it doesn't we're going to get a non-terminate decimal So let's look at the numerator as well 27. Well, that's a bunch of 3s 3 times 3 times 3 and Thankfully we have a 3 in the numerator as well, which we can cancel so What we left with is 9 by 16 And we don't even need to write that we can simply say that it's the denominator is made up of a bunch of 2s And that's why it's going to give us a terminating decimal A terminating decimal And what a beautiful technique without using calculator We can simply look at a fraction and talk about its decimal form And I hope you got the technique, but also why it works