1 00:00:00,551 --> 00:00:03,607 PROBLEM: "Express the rational number 19/27 2 00:00:03,607 --> 00:00:06,538 (or 19 27ths) as a terminating decimal 3 00:00:06,538 --> 00:00:08,666 or a decimal that eventually repeats. 4 00:00:08,666 --> 00:00:10,371 Include only the first six digits 5 00:00:10,371 --> 00:00:12,651 of the decimal in your answer." 6 00:00:12,651 --> 00:00:14,599 Let me give this a shot. 7 00:00:14,599 --> 00:00:18,791 So we want to express 19/27 – 8 00:00:18,791 --> 00:00:23,654 which is the same thing as 19 ÷ 27 – as a decimal. 9 00:00:23,654 --> 00:00:26,349 So let's divide 27 into 19. 10 00:00:26,349 --> 00:00:29,485 So 27 going into 19. 11 00:00:29,485 --> 00:00:31,264 And we know it's going to involve some decimals 12 00:00:31,264 --> 00:00:33,707 over here, because 27 is larger than 19, 13 00:00:33,707 --> 00:00:36,291 and it doesn't divide perfectly. 14 00:00:36,291 --> 00:00:38,131 So let's get into this. 15 00:00:38,131 --> 00:00:39,305 So 27 doesn't go into 1. 16 00:00:39,305 --> 00:00:40,923 It doesn't go into 19. 17 00:00:40,923 --> 00:00:43,994 It does go into 190. 18 00:00:43,994 --> 00:00:47,224 And it looks like 27 is roughly 30. 19 00:00:47,224 --> 00:00:48,794 It's a little less than 30. 20 00:00:48,794 --> 00:00:50,430 30 times 6 would be 180. 21 00:00:50,430 --> 00:00:53,132 So let's go with it going 6 times. 22 00:00:53,132 --> 00:00:54,045 Let's see if that works out. 23 00:00:54,045 --> 00:00:57,402 Well, 6 × 7 is 42. 24 00:00:57,402 --> 00:01:01,604 6 × 2 is 12, + 4 is 16. 25 00:01:01,604 --> 00:01:06,365 And when we subtract, 190 - 162 is going to get us – 26 00:01:06,365 --> 00:01:08,942 Actually, we could've had another 27 in there. 27 00:01:08,942 --> 00:01:10,716 Because when we subtract – 28 00:01:10,716 --> 00:01:12,510 So we get a 10 from the 10's place. 29 00:01:12,510 --> 00:01:13,966 So that becomes 8 10's. 30 00:01:13,966 --> 00:01:15,751 This became 28. 31 00:01:15,751 --> 00:01:17,522 So we could have put one more 27 in there. 32 00:01:17,522 --> 00:01:19,143 So let's do that. 33 00:01:19,143 --> 00:01:22,476 So let's put one more 27 in there. 34 00:01:22,476 --> 00:01:24,328 So 7 27's. 35 00:01:24,328 --> 00:01:26,923 7 × 7 is 49. 36 00:01:26,923 --> 00:01:31,190 7 × 2 is 14, + 4 is 18. 37 00:01:31,190 --> 00:01:33,459 And now our remainder is 1. 38 00:01:33,459 --> 00:01:37,859 We can bring down another 0. 39 00:01:37,859 --> 00:01:39,912 27 goes into 10 0 times. 40 00:01:39,912 --> 00:01:42,146 0 × 27 is 0. [Not "10," as Sal states by mistake.] 41 00:01:42,146 --> 00:01:44,102 Subtract – we have a remainder of 10. 42 00:01:44,102 --> 00:01:46,694 But now, we have to bring down another 0. 43 00:01:46,694 --> 00:01:51,014 So let's bring down this 0 right over here. 44 00:01:51,014 --> 00:01:56,494 So now, 27 goes into 100 3 times. 45 00:01:56,494 --> 00:02:05,944 3 × 27 is 60 + 21, is 81. 46 00:02:05,944 --> 00:02:09,154 And then we subtract: 100 - 81. 47 00:02:09,154 --> 00:02:10,436 Well, we could take 100 from 48 00:02:10,436 --> 00:02:13,331 the 100's place, and make it 10 10's. 49 00:02:13,331 --> 00:02:15,390 And then we could take 1 of those 10's from 50 00:02:15,390 --> 00:02:18,139 the 10's place and turn it into 10 1's. 51 00:02:18,139 --> 00:02:22,386 And so 9 10's minus 8 10's is equal to 1 10. 52 00:02:22,386 --> 00:02:24,719 And then 10 -1 is 9. 53 00:02:24,719 --> 00:02:25,616 So it's equal to 19. 54 00:02:25,616 --> 00:02:26,291 You probably – 55 00:02:26,291 --> 00:02:28,343 You might have been able to do that in your head. 56 00:02:28,343 --> 00:02:29,254 And then we have – 57 00:02:29,254 --> 00:02:30,732 And I see something interesting here – 58 00:02:30,732 --> 00:02:33,593 because when we bring down our next 0, 59 00:02:33,593 --> 00:02:34,899 we see 190 again. 60 00:02:34,899 --> 00:02:36,814 We saw 190 up here. 61 00:02:36,814 --> 00:02:38,329 But let's just keep going. 62 00:02:38,329 --> 00:02:40,572 So 27 goes into 190 – 63 00:02:40,572 --> 00:02:42,654 And we already played this game. 64 00:02:42,654 --> 00:02:44,551 It goes into it 7 times. 65 00:02:44,551 --> 00:02:48,283 7 × 27 – we already figured out – was 189. 66 00:02:48,283 --> 00:02:49,146 We subtracted. 67 00:02:49,146 --> 00:02:50,591 We had a remainder of 1. 68 00:02:50,591 --> 00:02:54,013 Then we brought down another 0. 69 00:02:54,013 --> 00:02:57,307 We said 27 goes into 10 0 times. 70 00:02:57,307 --> 00:02:59,068 0 × 27 is 0. 71 00:02:59,068 --> 00:03:00,261 Subtract. 72 00:03:00,261 --> 00:03:01,641 Then you have – 73 00:03:01,641 --> 00:03:02,945 We still have the 10, 74 00:03:02,945 --> 00:03:07,298 but we've got to bring down another 0. 75 00:03:07,298 --> 00:03:09,083 So you have 27, which goes into 100 – 76 00:03:09,083 --> 00:03:10,438 (We've already done this.) 77 00:03:10,438 --> 00:03:11,599 –3 times. 78 00:03:11,599 --> 00:03:13,750 So you see something happening here. 79 00:03:13,750 --> 00:03:16,671 It's 0.703703. 80 00:03:16,671 --> 00:03:18,947 And we're just going to keep repeating 703. 81 00:03:18,947 --> 00:03:27,218 This is going to be equal to 0.703703703703 – 82 00:03:27,218 --> 00:03:29,718 on and on and on forever. 83 00:03:29,718 --> 00:03:32,267 So the notation for representing 84 00:03:32,267 --> 00:03:34,419 a repeating decimal like this 85 00:03:34,419 --> 00:03:36,405 is to write the numbers that repeat – 86 00:03:36,405 --> 00:03:38,547 in this case 7, 0, and 3 – 87 00:03:38,547 --> 00:03:40,057 and then you put a line over all of 88 00:03:40,057 --> 00:03:40,999 the repeating decimal numbers 89 00:03:40,999 --> 00:03:41,789 to indicate that they repeat. 90 00:03:41,789 --> 00:03:44,984 So you put a line over the 7, the 0, and the 3, 91 00:03:44,984 --> 00:03:47,130 which means that the 703 will keep 92 00:03:47,130 --> 00:03:49,423 repeating on and on and on. 93 00:03:49,423 --> 00:03:52,871 So let's actually input it into the answer box now. 94 00:03:52,871 --> 00:04:02,107 So it's 0.703703. 95 00:04:02,107 --> 00:04:03,341 And they tell us to include only 96 00:04:03,341 --> 00:04:05,838 the first six digits of the decimal in your answer. 97 00:04:05,838 --> 00:04:07,636 And they don't tell us to round or approximate – 98 00:04:07,636 --> 00:04:10,196 because, obviously, if they said to round 99 00:04:10,196 --> 00:04:13,957 that final sixth decimal place, 100 00:04:13,957 --> 00:04:15,017 then you would round it up from 6 to 7, 101 00:04:15,017 --> 00:04:16,107 because the next digit is a 7. 102 00:04:16,107 --> 00:04:17,485 But they don't ask us to round. 103 00:04:17,485 --> 00:04:19,337 They just say, "Include only the first six digits 104 00:04:19,337 --> 00:04:20,988 of the decimal in your answer." 105 00:04:20,988 --> 00:04:23,135 So that should do the trick. 106 00:04:23,135 --> 00:04:24,604 And it did.