1 00:00:00,000 --> 00:00:00,810 2 00:00:00,810 --> 00:00:03,109 Welcome to the presentation on dividing fractions. 3 00:00:03,109 --> 00:00:04,490 Let's get started. 4 00:00:04,490 --> 00:00:06,640 So before I give you the intuition-- actually, I might 5 00:00:06,639 --> 00:00:09,339 do that in a different module-- I'm just going to show you the 6 00:00:09,339 --> 00:00:11,740 mechanics of how you divide a fraction. 7 00:00:11,740 --> 00:00:13,740 And it turns out that it's actually not much 8 00:00:13,740 --> 00:00:16,030 more difficult than multiplying fractions. 9 00:00:16,030 --> 00:00:21,410 If I were to ask you, 1/2 divided by 1/2, whenever you 10 00:00:21,410 --> 00:00:25,109 divide by a fraction, or actually, when you divide by 11 00:00:25,109 --> 00:00:29,960 any number, it's the same thing as multiplying by its inverse. 12 00:00:29,960 --> 00:00:36,670 So 1/2 divided by 1/2 is equal to 1/2 times 2/1. 13 00:00:36,670 --> 00:00:44,990 We just inverted-- inverse-- the second 1/2. 14 00:00:44,990 --> 00:00:47,630 And we know from the multiplication module, 1/2 15 00:00:47,630 --> 00:00:51,109 times 2/1, well, that's just equal to 2/2, 16 00:00:51,109 --> 00:00:53,560 or it's equal to 1. 17 00:00:53,560 --> 00:00:56,020 And that makes sense because, actually, any number divided 18 00:00:56,020 --> 00:00:58,750 by itself is equal to 1. 19 00:00:58,750 --> 00:01:03,219 1/2 divided by 1/2 is 1, just like 5 divided by 5 is 1, just 20 00:01:03,219 --> 00:01:05,239 like 100 divided by 100 is 1. 21 00:01:05,239 --> 00:01:06,849 And this isn't a new principal. 22 00:01:06,849 --> 00:01:08,969 Actually, you were always doing it. 23 00:01:08,969 --> 00:01:15,000 Think about it this way: What is 2 divided by 2? 24 00:01:15,000 --> 00:01:16,290 Well, you know that's 1. 25 00:01:16,290 --> 00:01:20,560 But isn't this also the same thing as 2 times the 26 00:01:20,560 --> 00:01:24,210 inverse of 2, which is 1? 27 00:01:24,209 --> 00:01:24,949 I'll show it to you. 28 00:01:24,950 --> 00:01:26,990 Actually, let me give you a couple more examples to show 29 00:01:26,989 --> 00:01:31,339 that dividing fractions really isn't a new concept, this whole 30 00:01:31,340 --> 00:01:34,840 notion of multiplying by the inverse. 31 00:01:34,840 --> 00:01:40,540 If I were to tell you what is 12 divided by 4? 32 00:01:40,540 --> 00:01:42,650 Well, we know the answer to this, but I'm going to show 33 00:01:42,650 --> 00:01:50,640 you that this is the same thing as 12 times 1/4. 34 00:01:50,640 --> 00:01:56,230 12/1 times 1/4 4 is 12/4, which is 3. 35 00:01:56,230 --> 00:01:59,480 And 12/4 is really just another way of writing 12 divided by 4, 36 00:01:59,480 --> 00:02:02,535 so it's kind of a long way of getting to the same point. 37 00:02:02,534 --> 00:02:04,989 But I just wanted to show you that what we're doing in this 38 00:02:04,989 --> 00:02:07,969 module is nothing new than what we've always been doing 39 00:02:07,969 --> 00:02:09,319 when we divide by a number. 40 00:02:09,319 --> 00:02:11,359 Division is the same thing. 41 00:02:11,360 --> 00:02:14,310 Dividing by a number is the same thing as multiplying by 42 00:02:14,310 --> 00:02:15,960 the inverse of that number. 43 00:02:15,960 --> 00:02:19,879 And just as a review, an inverse, if I have a number 44 00:02:19,879 --> 00:02:28,069 A, the inverse-- inv, short for inverse-- is 1 over A. 45 00:02:28,069 --> 00:02:36,289 So the inverse of 2/3 is 3/2, or the inverse of 5, because 5 46 00:02:36,289 --> 00:02:39,669 is the same thing as 5/1, so the inverse is 1/5. 47 00:02:39,669 --> 00:02:41,000 We're just flipping it. 48 00:02:41,000 --> 00:02:43,319 We're switching the numerator and denominator. 49 00:02:43,319 --> 00:02:46,474 So let's do some fraction division problems. 50 00:02:46,474 --> 00:02:49,269 51 00:02:49,270 --> 00:02:56,340 What is 2/3 divided by 5/6? 52 00:02:56,340 --> 00:03:05,969 Well, we know that this is the same thing as 2/3 times 6/5, 53 00:03:05,969 --> 00:03:09,229 and that's equal to 12/15. 54 00:03:09,229 --> 00:03:14,569 We can divide the numerator and denominator by 3, that's 4/5. 55 00:03:14,569 --> 00:03:22,900 What is 7/8 divided by 1/4? 56 00:03:22,900 --> 00:03:30,520 Well, that's the same thing as 7/8 times 4/1. 57 00:03:30,520 --> 00:03:32,820 Remember, I just flipped this 1/4. 58 00:03:32,819 --> 00:03:36,840 Divide by 1/4 is the same thing as multiplying by 4/1. 59 00:03:36,840 --> 00:03:38,229 That's all you've got to do. 60 00:03:38,229 --> 00:03:39,989 And then we could use a little shortcut we learned in the 61 00:03:39,990 --> 00:03:41,480 multiplication module. 62 00:03:41,479 --> 00:03:42,949 8 divided by 4 is 2. 63 00:03:42,949 --> 00:03:44,799 4 divided by 4 is 1. 64 00:03:44,800 --> 00:03:47,450 So that equals 7/2. 65 00:03:47,449 --> 00:03:49,899 Or if you wanted to write that as a mixed number, this is, of 66 00:03:49,900 --> 00:03:51,200 course, an improper fraction. 67 00:03:51,199 --> 00:03:53,439 Improper fractions have a numerator larger 68 00:03:53,439 --> 00:03:54,829 than the denominator. 69 00:03:54,830 --> 00:03:58,670 If you wanted to write that as a mixed number, 2 goes into 7 70 00:03:58,669 --> 00:04:03,679 three times with a remainder of 1, so that's 3 and a half. 71 00:04:03,680 --> 00:04:04,439 You can write it either way. 72 00:04:04,439 --> 00:04:05,990 I tend to keep it this way because it's 73 00:04:05,990 --> 00:04:07,800 easier to deal with. 74 00:04:07,800 --> 00:04:10,130 Let's do a ton of more problems, or at least as many 75 00:04:10,129 --> 00:04:13,829 more as we can do in the next four or five minutes. 76 00:04:13,830 --> 00:04:23,850 What's negative 2/3 divided by 5/2? 77 00:04:23,850 --> 00:04:29,110 Once again, that's the same thing as minus 2/3-- whoops-- 78 00:04:29,110 --> 00:04:34,850 as minus 2/3 times what? 79 00:04:34,850 --> 00:04:40,110 It's times the inverse of 5/2, which is 2/5, and 80 00:04:40,110 --> 00:04:45,629 that equals minus 4/15. 81 00:04:45,629 --> 00:04:52,300 What is 3/2 divided by 1/6? 82 00:04:52,300 --> 00:04:59,850 Well, that's just the same thing as 3/2 times 6/1, 83 00:04:59,850 --> 00:05:03,000 which equals 3 and 1. 84 00:05:03,000 --> 00:05:09,610 We just divided the 6 by 2 and the 2 by 2, so that equals 9. 85 00:05:09,610 --> 00:05:11,280 I think you might be getting it now. 86 00:05:11,279 --> 00:05:12,949 Let's see, let's do a couple more. 87 00:05:12,949 --> 00:05:16,289 And, of course, you can always pause, and look at this whole 88 00:05:16,290 --> 00:05:19,420 presentation again, so you can get confused all over again. 89 00:05:19,420 --> 00:05:27,240 Let's see, let's do minus 5/7 divided by 10/3. 90 00:05:27,240 --> 00:05:33,879 Well, this is the same thing as minus 5/7 times 3/10. 91 00:05:33,879 --> 00:05:35,420 I just multiplied by the inverse. 92 00:05:35,420 --> 00:05:38,120 That's all I keep doing over and over again. 93 00:05:38,120 --> 00:05:40,180 Minus 5 times 3. 94 00:05:40,180 --> 00:05:42,610 Minus 15. 95 00:05:42,610 --> 00:05:47,350 7 times 10 is 70. 96 00:05:47,350 --> 00:05:49,900 If we divide the numerator and the denominator by 97 00:05:49,899 --> 00:05:56,049 5, we get minus 3/14. 98 00:05:56,050 --> 00:05:57,500 We could have also just done it here. 99 00:05:57,500 --> 00:05:59,889 We could have done 5, 2, and we would have gotten 100 00:05:59,889 --> 00:06:02,509 minus 3/14 as well. 101 00:06:02,509 --> 00:06:05,420 Let's do one or two more problems. 102 00:06:05,420 --> 00:06:06,629 I think you kind of get it, though. 103 00:06:06,629 --> 00:06:09,600 104 00:06:09,600 --> 00:06:14,500 Let's say 1/2 divided by minus 3. 105 00:06:14,500 --> 00:06:14,964 Ah-ha! 106 00:06:14,964 --> 00:06:17,939 So what happens when you take a fraction and you divide it by 107 00:06:17,939 --> 00:06:19,730 a whole number or an integer? 108 00:06:19,730 --> 00:06:22,970 Well, we know any whole number can be written as a fraction. 109 00:06:22,970 --> 00:06:29,010 This is the same thing as 1/2 divided by minus 3/1. 110 00:06:29,009 --> 00:06:33,870 And dividing by a fraction is the same thing as multiplying 111 00:06:33,870 --> 00:06:37,430 by it's inverse. 112 00:06:37,430 --> 00:06:42,150 So the inverse of negative 3/1 is negative 1/3, and this 113 00:06:42,149 --> 00:06:45,199 equals negative 1/6. 114 00:06:45,199 --> 00:06:46,039 Let's do it the other way. 115 00:06:46,040 --> 00:06:51,879 What if I had negative 3 divided by 1/2? 116 00:06:51,879 --> 00:06:52,500 Same thing. 117 00:06:52,500 --> 00:07:00,370 Negative 3 is the same thing as minus 3/1 divided by 1/2, which 118 00:07:00,370 --> 00:07:07,939 is the same thing as minus 3/1 times 2/1, which is equal to 119 00:07:07,939 --> 00:07:12,009 minus 6/1, which equals minus 6. 120 00:07:12,009 --> 00:07:15,810 121 00:07:15,810 --> 00:07:17,350 Now, let me give you a little bit of intuition 122 00:07:17,350 --> 00:07:19,730 of why this works. 123 00:07:19,730 --> 00:07:24,240 Let's say I said 2 divided by 1/3. 124 00:07:24,240 --> 00:07:27,650 Well, we know that this is equal to 2/1 times 125 00:07:27,649 --> 00:07:30,120 3/1, which equals 6. 126 00:07:30,120 --> 00:07:32,699 So how does 2, 1/3, and 6 relate? 127 00:07:32,699 --> 00:07:33,689 Well, let's look at it this way. 128 00:07:33,689 --> 00:07:36,930 If I had two pieces of pizza. 129 00:07:36,930 --> 00:07:38,660 I have two pieces of pizza. 130 00:07:38,660 --> 00:07:41,520 Here's my two pieces of pizza right. 131 00:07:41,519 --> 00:07:42,529 Two right here. 132 00:07:42,529 --> 00:07:45,049 So I have two pieces of pizza, and I'm going to divide 133 00:07:45,050 --> 00:07:48,079 them into thirds. 134 00:07:48,079 --> 00:07:50,599 So I'm going to divide each pizza into a third. 135 00:07:50,600 --> 00:07:52,860 I'll draw the little Mercedes sign. 136 00:07:52,860 --> 00:07:57,050 So I'm dividing each pizza into a third, right? 137 00:07:57,050 --> 00:07:58,210 How many pieces do I have? 138 00:07:58,209 --> 00:08:02,924 Let's see, 1, 2, 3, 4, 5, 6. 139 00:08:02,925 --> 00:08:04,800 I have 6 pieces. 140 00:08:04,800 --> 00:08:08,139 So you might want to sit and ponder that for a little bit, 141 00:08:08,139 --> 00:08:12,849 but I think it might make a little bit of sense to you. 142 00:08:12,850 --> 00:08:17,189 Let's do one more just to tire your brain. 143 00:08:17,189 --> 00:08:25,750 If I had negative 7/2 divided by 4/9-- let's pick a negative 144 00:08:25,750 --> 00:08:30,579 4/9-- well, that's the same thing as minus 7/2 times 145 00:08:30,579 --> 00:08:33,720 minus 9/4, right? 146 00:08:33,720 --> 00:08:37,950 I just multiplied by the inverse of negative 4/9. 147 00:08:37,950 --> 00:08:41,220 9 times 7 is equal to-- negative 7 times negative 148 00:08:41,220 --> 00:08:47,800 9 is positive 63, and 2 times 4 is 8. 149 00:08:47,799 --> 00:08:51,459 Hopefully, I think you have a good idea of how to divide by 150 00:08:51,460 --> 00:08:55,960 a fraction now, and you can try out the dividing 151 00:08:55,960 --> 00:08:57,310 fractions modules. 152 00:08:57,309 --> 00:08:58,889 Have fun!