Rewriting decimals as fractions
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Rewriting tricky fractions to decimals
I'll now show you how to convert a fraction into a decimal. And if we have time, maybe we'll learn how to do a decimal into a fraction. So let's start with, what I would say, is a fairly straightforward example. Let's start with the fraction 1/2. And I want to convert that into a decimal. So the method I'm going to show you will always work. What you do is you take the denominator and you divide it into the numerator. Let's see how that works. So we take the denominator-- is 2-- and we're going to divide that into the numerator, 1. And you're probably saying, well, how do I divide 2 into 1? Well, if you remember from the dividing decimals module, we can just add a decimal point here and add some trailing 0's. We haven't actually changed the value of the number, but we're just getting some precision here. We put the decimal point here. Does 2 go into 1? No. 2 goes into 10, so we go 2 goes into 10 five times. 5 times 2 is 10. Remainder of 0. We're done. So 1/2 is equal to 0.5. Let's do a slightly harder one. Let's figure out 1/3. Well, once again, we take the denominator, 3, and we divide it into the numerator. And I'm just going to add a bunch of trailing 0's here. 3 goes into-- well, 3 doesn't go into 1. 3 goes into 10 three times. 3 times 3 is 9. Let's subtract, get a 1, bring down the 0. 3 goes into 10 three times. Actually, this decimal point is right here. 3 times 3 is 9. Do you see a pattern here? We keep getting the same thing. As you see it's actually 0.3333. It goes on forever. And a way to actually represent this, obviously you can't write an infinite number of 3's. Is you could just write 0.-- well, you could write 0.33 repeating, which means that the 0.33 will go on forever. Or you can actually even say 0.3 repeating. Although I tend to see this more often. Maybe I'm just mistaken. But in general, this line on top of the decimal means that this number pattern repeats indefinitely. So 1/3 is equal to 0.33333 and it goes on forever. Another way of writing that is 0.33 repeating. Let's do a couple of, maybe a little bit harder, but they all follow the same pattern. Let me pick some weird numbers. Let me actually do an improper fraction. Let me say 17/9. So here, it's interesting. The numerator is bigger than the denominator. So actually we're going to get a number larger than 1. But let's work it out. So we take 9 and we divide it into 17. And let's add some trailing 0's for the decimal point here. So 9 goes into 17 one time. 1 times 9 is 9. 17 minus 9 is 8. Bring down a 0. 9 goes into 80-- well, we know that 9 times 9 is 81, so it has to go into it only eight times because it can't go into it nine times. 8 times 9 is 72. 80 minus 72 is 8. Bring down another 0. I think we see a pattern forming again. 9 goes into 80 eight times. 8 times 9 is 72. And clearly, I could keep doing this forever and we'd keep getting 8's. So we see 17 divided by 9 is equal to 1.88 where the 0.88 actually repeats forever. Or, if we actually wanted to round this we could say that that is also equal to 1.-- depending where we wanted to round it, what place. We could say roughly 1.89. Or we could round in a different place. I rounded in the 100's place. But this is actually the exact answer. 17/9 is equal to 1.88. I actually might do a separate module, but how would we write this as a mixed number? Well actually, I'm going to do that in a separate. I don't want to confuse you for now. Let's do a couple more problems. Let me do a real weird one. Let me do 17/93. What does that equal as a decimal? Well, we do the same thing. 93 goes into-- I make a really long line up here because I don't know how many decimal places we'll do. And remember, it's always the denominator being divided into the numerator. This used to confuse me a lot of times because you're often dividing a larger number into a smaller number. So 93 goes into 17 zero times. There's a decimal. 93 goes into 170? Goes into it one time. 1 times 93 is 93. 170 minus 93 is 77. Bring down the 0. 93 goes into 770? Let's see. It will go into it, I think, roughly eight times. 8 times 3 is 24. 8 times 9 is 72. Plus 2 is 74. And then we subtract. 10 and 6. It's equal to 26. Then we bring down another 0. 93 goes into 26-- about two times. 2 times 3 is 6. 18. This is 74. 0. So we could keep going. We could keep figuring out the decimal points. You could do this indefinitely. But if you wanted to at least get an approximation, you would say 17 goes into 93 0.-- or 17/93 is equal to 0.182 and then the decimals will keep going. And you can keep doing it if you want. If you actually saw this on exam they'd probably tell you to stop at some point. You know, round it to the nearest hundredths or thousandths place. And just so you know, let's try to convert it the other way, from decimals to fractions. Actually, this is, I think, you'll find a much easier thing to do. If I were to ask you what 0.035 is as a fraction? Well, all you do is you say, well, 0.035, we could write it this way-- we could write that's the same thing as 03-- well, I shouldn't write 035. That's the same thing as 35/1,000. And you're probably saying, Sal, how did you know it's 35/1000? Well because we went to 3-- this is the 10's place. Tenths not 10's. This is hundreths. This is the thousandths place. So we went to 3 decimals of significance. So this is 35 thousandths. If the decimal was let's say, if it was 0.030. There's a couple of ways we could say this. Well, we could say, oh well we got to 3-- we went to the thousandths Place. So this is the same thing as 30/1,000. or. We could have also said, well, 0.030 is the same thing as 0.03 because this 0 really doesn't add any value. If we have 0.03 then we're only going to the hundredths place. So this is the same thing as 3/100. So let me ask you, are these two the same? Well, yeah. Sure they are. If we divide both the numerator and the denominator of both of these expressions by 10 we get 3/100. Let's go back to this case. Are we done with this? Is 35/1,000-- I mean, it's right. That is a fraction. 35/1,000. But if we wanted to simplify it even more looks like we could divide both the numerator and the denominator by 5. And then, just to get it into simplest form, that equals 7/200. And if we wanted to convert 7/200 into a decimal using the technique we just did, so we would do 200 goes into 7 and figure it out. We should get 0.035. I'll leave that up to you as an exercise. Hopefully now you get at least an initial understanding of how to convert a fraction into a decimal and maybe vice versa. And if you don't, just do some of the practices. And I will also try to record another module on this or another presentation. Have fun with the exercises.