WEBVTT 00:00:00.551 --> 00:00:03.607 PROBLEM: "Express the rational number 19/27 00:00:03.607 --> 00:00:06.538 (or 19 27ths) as a terminating decimal 00:00:06.538 --> 00:00:08.666 or a decimal that eventually repeats. 00:00:08.666 --> 00:00:10.371 Include only the first six digits 00:00:10.371 --> 00:00:12.651 of the decimal in your answer." 00:00:12.651 --> 00:00:14.599 Let me give this a shot. 00:00:14.599 --> 00:00:18.791 So we want to express 19/27 – 00:00:18.791 --> 00:00:23.654 which is the same thing as 19 ÷ 27 – as a decimal. 00:00:23.654 --> 00:00:26.349 So let's divide 27 into 19. 00:00:26.349 --> 00:00:29.485 So 27 going into 19. 00:00:29.485 --> 00:00:31.264 And we know it's going to involve some decimals 00:00:31.264 --> 00:00:33.707 over here, because 27 is larger than 19, 00:00:33.707 --> 00:00:36.291 and it doesn't divide perfectly. 00:00:36.291 --> 00:00:38.131 So let's get into this. 00:00:38.131 --> 00:00:39.305 So 27 doesn't go into 1. 00:00:39.305 --> 00:00:40.923 It doesn't go into 19. 00:00:40.923 --> 00:00:43.994 It does go into 190. 00:00:43.994 --> 00:00:47.224 And it looks like 27 is roughly 30. 00:00:47.224 --> 00:00:48.794 It's a little less than 30. 00:00:48.794 --> 00:00:50.430 30 times 6 would be 180. 00:00:50.430 --> 00:00:53.132 So let's go with it going 6 times. 00:00:53.132 --> 00:00:54.045 Let's see if that works out. 00:00:54.045 --> 00:00:57.402 Well, 6 × 7 is 42. 00:00:57.402 --> 00:01:01.604 6 × 2 is 12, + 4 is 16. 00:01:01.604 --> 00:01:06.365 And when we subtract, 190 - 162 is going to get us – 00:01:06.365 --> 00:01:08.942 Actually, we could've had another 27 in there. 00:01:08.942 --> 00:01:10.716 Because when we subtract – 00:01:10.716 --> 00:01:12.510 So we get a 10 from the 10's place. 00:01:12.510 --> 00:01:13.966 So that becomes 8 10's. 00:01:13.966 --> 00:01:15.751 This became 28. 00:01:15.751 --> 00:01:17.522 So we could have put one more 27 in there. 00:01:17.522 --> 00:01:19.143 So let's do that. 00:01:19.143 --> 00:01:22.476 So let's put one more 27 in there. 00:01:22.476 --> 00:01:24.328 So 7 27's. 00:01:24.328 --> 00:01:26.923 7 × 7 is 49. 00:01:26.923 --> 00:01:31.190 7 × 2 is 14, + 4 is 18. 00:01:31.190 --> 00:01:33.459 And now our remainder is 1. 00:01:33.459 --> 00:01:37.859 We can bring down another 0. 00:01:37.859 --> 00:01:39.912 27 goes into 10 0 times. 00:01:39.912 --> 00:01:42.146 0 × 27 is 0. [Not "10," as Sal states by mistake.] 00:01:42.146 --> 00:01:44.102 Subtract – we have a remainder of 10. 00:01:44.102 --> 00:01:46.694 But now, we have to bring down another 0. 00:01:46.694 --> 00:01:51.014 So let's bring down this 0 right over here. 00:01:51.014 --> 00:01:56.494 So now, 27 goes into 100 3 times. 00:01:56.494 --> 00:02:05.944 3 × 27 is 60 + 21, is 81. 00:02:05.944 --> 00:02:09.154 And then we subtract: 100 - 81. 00:02:09.154 --> 00:02:10.436 Well, we could take 100 from 00:02:10.436 --> 00:02:13.331 the 100's place, and make it 10 10's. 00:02:13.331 --> 00:02:15.390 And then we could take 1 of those 10's from 00:02:15.390 --> 00:02:18.139 the 10's place and turn it into 10 1's. 00:02:18.139 --> 00:02:22.386 And so 9 10's minus 8 10's is equal to 1 10. 00:02:22.386 --> 00:02:24.719 And then 10 -1 is 9. 00:02:24.719 --> 00:02:25.616 So it's equal to 19. 00:02:25.616 --> 00:02:26.291 You probably – 00:02:26.291 --> 00:02:28.343 You might have been able to do that in your head. 00:02:28.343 --> 00:02:29.254 And then we have – 00:02:29.254 --> 00:02:30.732 And I see something interesting here – 00:02:30.732 --> 00:02:33.593 because when we bring down our next 0, 00:02:33.593 --> 00:02:34.899 we see 190 again. 00:02:34.899 --> 00:02:36.814 We saw 190 up here. 00:02:36.814 --> 00:02:38.329 But let's just keep going. 00:02:38.329 --> 00:02:40.572 So 27 goes into 190 – 00:02:40.572 --> 00:02:42.654 And we already played this game. 00:02:42.654 --> 00:02:44.551 It goes into it 7 times. 00:02:44.551 --> 00:02:48.283 7 × 27 – we already figured out – was 189. 00:02:48.283 --> 00:02:49.146 We subtracted. 00:02:49.146 --> 00:02:50.591 We had a remainder of 1. 00:02:50.591 --> 00:02:54.013 Then we brought down another 0. 00:02:54.013 --> 00:02:57.307 We said 27 goes into 10 0 times. 00:02:57.307 --> 00:02:59.068 0 × 27 is 0. 00:02:59.068 --> 00:03:00.261 Subtract. 00:03:00.261 --> 00:03:01.641 Then you have – 00:03:01.641 --> 00:03:02.945 We still have the 10, 00:03:02.945 --> 00:03:07.298 but we've got to bring down another 0. 00:03:07.298 --> 00:03:09.083 So you have 27, which goes into 100 – 00:03:09.083 --> 00:03:10.438 (We've already done this.) 00:03:10.438 --> 00:03:11.599 –3 times. 00:03:11.599 --> 00:03:13.750 So you see something happening here. 00:03:13.750 --> 00:03:16.671 It's 0.703703. 00:03:16.671 --> 00:03:18.947 And we're just going to keep repeating 703. 00:03:18.947 --> 00:03:27.218 This is going to be equal to 0.703703703703 – 00:03:27.218 --> 00:03:29.718 on and on and on forever. 00:03:29.718 --> 00:03:32.267 So the notation for representing 00:03:32.267 --> 00:03:34.419 a repeating decimal like this 00:03:34.419 --> 00:03:36.405 is to write the numbers that repeat – 00:03:36.405 --> 00:03:38.547 in this case 7, 0, and 3 – 00:03:38.547 --> 00:03:40.057 and then you put a line over all of 00:03:40.057 --> 00:03:40.999 the repeating decimal numbers 00:03:40.999 --> 00:03:41.789 to indicate that they repeat. 00:03:41.789 --> 00:03:44.984 So you put a line over the 7, the 0, and the 3, 00:03:44.984 --> 00:03:47.130 which means that the 703 will keep 00:03:47.130 --> 00:03:49.423 repeating on and on and on. 00:03:49.423 --> 00:03:52.871 So let's actually input it into the answer box now. 00:03:52.871 --> 00:04:02.107 So it's 0.703703. 00:04:02.107 --> 00:04:03.341 And they tell us to include only 00:04:03.341 --> 00:04:05.838 the first six digits of the decimal in your answer. 00:04:05.838 --> 00:04:07.636 And they don't tell us to round or approximate – 00:04:07.636 --> 00:04:10.196 because, obviously, if they said to round 00:04:10.196 --> 00:04:13.957 that final sixth decimal place, 00:04:13.957 --> 00:04:15.017 then you would round it up from 6 to 7, 00:04:15.017 --> 00:04:16.107 because the next digit is a 7. 00:04:16.107 --> 00:04:17.485 But they don't ask us to round. 00:04:17.485 --> 00:04:19.337 They just say, "Include only the first six digits 00:04:19.337 --> 00:04:20.988 of the decimal in your answer." 00:04:20.988 --> 00:04:23.135 So that should do the trick. 00:04:23.135 --> 00:04:24.604 And it did.