Converting a fraction to a repeating decimal

PROBLEM: "Express the rational number 19/27 (or 19 27ths) as a terminating decimal or a decimal that eventually repeats. Include only the first six digits of the decimal in your answer." Let me give this a shot. So we want to express 19/27 – which is the same thing as 19 ÷ 27 – as a decimal. So let's divide 27 into 19. So 27 going into 19. And we know it's going to involve some decimals over here, because 27 is larger than 19, and it doesn't divide perfectly. So let's get into this. So 27 doesn't go into 1. It doesn't go into 19. It does go into 190. And it looks like 27 is roughly 30. It's a little less than 30. 30 times 6 would be 180. So let's go with it going 6 times. Let's see if that works out. Well, 6 × 7 is 42. 6 × 2 is 12, + 4 is 16. And when we subtract, 190 - 162 is going to get us – Actually, we could've had another 27 in there. Because when we subtract – So we get a 10 from the 10's place. So that becomes 8 10's. This became 28. So we could have put one more 27 in there. So let's do that. So let's put one more 27 in there. So 7 27's. 7 × 7 is 49. 7 × 2 is 14, + 4 is 18. And now our remainder is 1. We can bring down another 0. 27 goes into 10 0 times. 0 × 27 is 0. [Not "10," as Sal states by mistake.] Subtract – we have a remainder of 10. But now, we have to bring down another 0. So let's bring down this 0 right over here. So now, 27 goes into 100 3 times. 3 × 27 is 60 + 21, is 81. And then we subtract: 100 - 81. Well, we could take 100 from the 100's place, and make it 10 10's. And then we could take 1 of those 10's from the 10's place and turn it into 10 1's. And so 9 10's minus 8 10's is equal to 1 10. And then 10 -1 is 9. So it's equal to 19. You probably – You might have been able to do that in your head. And then we have – And I see something interesting here – because when we bring down our next 0, we see 190 again. We saw 190 up here. But let's just keep going. So 27 goes into 190 – And we already played this game. It goes into it 7 times. 7 × 27 – we already figured out – was 189. We subtracted. We had a remainder of 1. Then we brought down another 0. We said 27 goes into 10 0 times. 0 × 27 is 0. Subtract. Then you have – We still have the 10, but we've got to bring down another 0. So you have 27, which goes into 100 – (We've already done this.) –3 times. So you see something happening here. It's 0.703703. And we're just going to keep repeating 703. This is going to be equal to 0.703703703703 – on and on and on forever. So the notation for representing a repeating decimal like this is to write the numbers that repeat – in this case 7, 0, and 3 – and then you put a line over all of the repeating decimal numbers to indicate that they repeat. So you put a line over the 7, the 0, and the 3, which means that the 703 will keep repeating on and on and on. So let's actually input it into the answer box now. So it's 0.703703. And they tell us to include only the first six digits of the decimal in your answer. And they don't tell us to round or approximate – because, obviously, if they said to round to that smallest, sixth decimal place, then you would round up because the next digit is a 7. But they don't ask us to round. They just say, "Include only the first six digits of the decimal in your answer." So that should do the trick. And it did.