1 00:00:00,000 --> 00:00:01,610 2 00:00:01,610 --> 00:00:03,710 Welcome to the presentation on adding and 3 00:00:03,710 --> 00:00:05,299 subtracting fractions. 4 00:00:05,299 --> 00:00:08,449 Let's get started. 5 00:00:08,449 --> 00:00:12,119 Let's start with what I hope shouldn't confuse you too much. 6 00:00:12,119 --> 00:00:15,129 This should hopefully be a relatively easy question. 7 00:00:15,130 --> 00:00:23,940 If I were to ask you what 1/4 plus 1/4 is. 8 00:00:23,940 --> 00:00:25,270 Let's think about what that means. 9 00:00:25,269 --> 00:00:32,420 Let's say we had a pie and it was divided into four pieces. 10 00:00:32,420 --> 00:00:35,179 So this is like saying this first 1/4 right here -- let me 11 00:00:35,179 --> 00:00:37,729 do it in a different color. 12 00:00:37,729 --> 00:00:39,529 This 1/4 right here, let's say it's this 13 00:00:39,530 --> 00:00:42,399 1/4 of the pie, right? 14 00:00:42,399 --> 00:00:45,570 And we're going to add it to another 1/4 of the pie. 15 00:00:45,570 --> 00:00:51,609 Let's make it this one -- let me change the color -- pink. 16 00:00:51,609 --> 00:00:57,070 This 1/4, this pink 1/4 is this 1/4 of the pie. 17 00:00:57,070 --> 00:01:02,270 So if I were to eat both 1/4s or 1/4 and then I eat another 18 00:01:02,270 --> 00:01:04,570 1/4, how much have I eaten? 19 00:01:04,569 --> 00:01:08,259 Well, you could look from just the picture, I have now eaten 2 20 00:01:08,260 --> 00:01:10,280 out of the 4 pieces of the pie. 21 00:01:10,280 --> 00:01:15,480 So if I eat 1/4 of a piece of pie or 1/4 of a pie, and then I 22 00:01:15,480 --> 00:01:21,550 eat another 1/4 of a pie, I will have eaten 2/4 of the pie. 23 00:01:21,549 --> 00:01:23,939 And we know from the equivalent fractions module that this is 24 00:01:23,939 --> 00:01:27,489 the same thing as I've eaten 1/2 of the pie, 25 00:01:27,489 --> 00:01:28,329 which makes sense. 26 00:01:28,329 --> 00:01:32,149 If I eat 2 out of 4 pieces of a pie, then I've eaten 1/2 of it. 27 00:01:32,150 --> 00:01:33,700 And if we look at it mathematically, what 28 00:01:33,700 --> 00:01:34,960 happened here? 29 00:01:34,959 --> 00:01:39,229 Well the denominators or the bottom numbers, the bottom 30 00:01:39,230 --> 00:01:41,270 numbers in the fraction stayed the same. 31 00:01:41,269 --> 00:01:42,729 Because that's just the total number of pieces 32 00:01:42,730 --> 00:01:44,359 I have in this example. 33 00:01:44,359 --> 00:01:47,929 Well, I added the numerators, which makes sense. 34 00:01:47,930 --> 00:01:51,490 I had 1 out of the 4 pieces of pie, then I ate another 1 out 35 00:01:51,489 --> 00:01:54,599 of the 4 pieces of pie, so I ate 2 out of the 4 pieces 36 00:01:54,599 --> 00:01:56,509 of pie, which is 1/2. 37 00:01:56,510 --> 00:01:57,840 Let me do a couple more examples. 38 00:01:57,840 --> 00:02:01,820 39 00:02:01,819 --> 00:02:09,239 What is 2/5 plus 1/5? 40 00:02:09,240 --> 00:02:11,760 Well we do the same thing here. 41 00:02:11,759 --> 00:02:14,229 We first check to make sure the denominators are the same -- 42 00:02:14,229 --> 00:02:16,129 we'll learn in a second what we do when the denominators 43 00:02:16,129 --> 00:02:16,900 are different. 44 00:02:16,900 --> 00:02:19,340 If the denominators are the same, the denominator of the 45 00:02:19,340 --> 00:02:21,030 answer will be the same. 46 00:02:21,030 --> 00:02:22,479 And we just add the numerators. 47 00:02:22,479 --> 00:02:27,579 2/5 plus 1/5 is just 2 plus 1 over 5, which 48 00:02:27,580 --> 00:02:31,100 is equal to 3 over 5. 49 00:02:31,099 --> 00:02:33,389 And it works the same way with subtraction. 50 00:02:33,389 --> 00:02:42,429 If I had 3 over 7 minus 2 over 7, that just equals 1 over 7. 51 00:02:42,430 --> 00:02:45,480 I just subtracted the 3, I subtracted the 2 from the 3 52 00:02:45,479 --> 00:02:48,149 to get 1 and I kept the denominator the same. 53 00:02:48,150 --> 00:02:48,860 Which makes sense. 54 00:02:48,860 --> 00:02:53,040 If I have 3 out of the 7 pieces of a pie and I were to give 55 00:02:53,039 --> 00:02:57,799 away 2 out of the 7 pieces of a pie, I'd be left with 1 of 56 00:02:57,800 --> 00:03:00,180 the 7 pieces of a pie. 57 00:03:00,180 --> 00:03:02,370 So now let's tackle -- I think it should be pretty 58 00:03:02,370 --> 00:03:04,330 straightforward when we have the same denominator. 59 00:03:04,330 --> 00:03:05,870 Remember, the denominator is just the bottom 60 00:03:05,870 --> 00:03:06,879 number in a fraction. 61 00:03:06,879 --> 00:03:08,409 Numerator is the top number. 62 00:03:08,409 --> 00:03:11,430 What happens when we have different denominators? 63 00:03:11,430 --> 00:03:15,099 Well, hopefully it won't be too difficult. 64 00:03:15,099 --> 00:03:24,329 Let's say I have 1/4 plus 1/2. 65 00:03:24,330 --> 00:03:27,190 Let's go back to that original pie example. 66 00:03:27,189 --> 00:03:28,259 Let me draw that pie. 67 00:03:28,259 --> 00:03:33,899 68 00:03:33,900 --> 00:03:37,250 So this first 1/4 right here, let's just color it in, 69 00:03:37,250 --> 00:03:40,469 that's this 1/4 of the pie. 70 00:03:40,469 --> 00:03:44,550 And now I'm going to eat another 1/2 of the pie. 71 00:03:44,550 --> 00:03:46,460 So I'm going to eat 1/2 of the pie. 72 00:03:46,460 --> 00:03:49,099 So this 1/2. 73 00:03:49,099 --> 00:03:51,400 I'll eat this whole 1/2 of the pie. 74 00:03:51,400 --> 00:03:54,289 75 00:03:54,289 --> 00:03:55,229 So what does that equal? 76 00:03:55,229 --> 00:03:57,189 Well, there's a couple of ways we could think about it. 77 00:03:57,189 --> 00:03:59,210 First we could just re-write 1/2. 78 00:03:59,210 --> 00:04:04,135 1/2 of the pie, that's actually the same thing as 2/4, right? 79 00:04:04,134 --> 00:04:06,949 80 00:04:06,949 --> 00:04:12,289 There's 1/4 here and then another 1/4 here. 81 00:04:12,289 --> 00:04:15,469 So 1/2 is the same thing as 2/4, and we know that from the 82 00:04:15,469 --> 00:04:17,519 equivalent fractions module. 83 00:04:17,519 --> 00:04:21,459 So we know that 1/4 plus 1/2, this is the same thing as 84 00:04:21,459 --> 00:04:27,109 saying 1/4 plus 2/4, right? 85 00:04:27,110 --> 00:04:36,160 And all I did here is I changed the 1/2 to a 2/4 by essentially 86 00:04:36,160 --> 00:04:38,820 multiplying the numerator and the denominator of 87 00:04:38,819 --> 00:04:40,409 this fraction by 2. 88 00:04:40,410 --> 00:04:41,740 And you can do that to any fraction. 89 00:04:41,740 --> 00:04:43,889 As long as you multiply the numerator and the denominator 90 00:04:43,889 --> 00:04:47,620 by the same number, you can multiply by anything. 91 00:04:47,620 --> 00:04:52,879 That makes sense because 1/2 times 1 is equal 92 00:04:52,879 --> 00:04:54,439 to 1/2, you know that. 93 00:04:54,439 --> 00:05:00,079 Well another way of writing 1 is 1/2 times 2/2. 94 00:05:00,079 --> 00:05:04,490 2 over 2 is the same thing as 1, and that equals 2 over 4. 95 00:05:04,490 --> 00:05:07,379 The reason why I picked 2 is because I wanted to get 96 00:05:07,379 --> 00:05:08,579 the same denominator here. 97 00:05:08,579 --> 00:05:11,229 98 00:05:11,230 --> 00:05:13,530 I hope I'm not completely confusing you. 99 00:05:13,529 --> 00:05:15,219 Well, let's just finish up this problem. 100 00:05:15,220 --> 00:05:19,220 So we have 1/4 plus 2/4, so we know that we just add the 101 00:05:19,220 --> 00:05:22,570 numerators, 3, and the denominators are the same, 3/4. 102 00:05:22,569 --> 00:05:25,189 And if we look at the picture, true enough, we have 103 00:05:25,189 --> 00:05:29,379 eaten 3/4 of this pie. 104 00:05:29,379 --> 00:05:30,319 Let's do another one. 105 00:05:30,319 --> 00:05:34,029 106 00:05:34,029 --> 00:05:44,729 Let's do 1/2 plus 1/3. 107 00:05:44,730 --> 00:05:47,569 Well once again, we want to get both denominators to be the 108 00:05:47,569 --> 00:05:51,370 same, but you can't just multiply one of them to get -- 109 00:05:51,370 --> 00:05:53,850 there's nothing I can multiply 3 by to get 2, or there's no, 110 00:05:53,850 --> 00:05:56,510 at least, integer I can multiply 3 by to get 2. 111 00:05:56,509 --> 00:05:58,899 And there's nothing I can multiply 2 by to get 3. 112 00:05:58,899 --> 00:06:01,870 So I have to multiply both of them so they equal each other. 113 00:06:01,870 --> 00:06:06,230 It turns out that what we want for, what we'll call the common 114 00:06:06,230 --> 00:06:08,850 denominator, it turns out to be the least common 115 00:06:08,850 --> 00:06:11,129 multiple of 2 and 3. 116 00:06:11,129 --> 00:06:13,399 Well what's the least common multiple of 2 and 3? 117 00:06:13,399 --> 00:06:16,750 Well that's the smallest number that's a multiple 118 00:06:16,750 --> 00:06:17,860 of both 2 and 3. 119 00:06:17,860 --> 00:06:19,389 Well the smallest number that's a multiple of 120 00:06:19,389 --> 00:06:23,479 both 2 and 3 is 6. 121 00:06:23,480 --> 00:06:27,890 So let's convert both of these fractions to something over 6. 122 00:06:27,889 --> 00:06:30,339 So 1/2 is equal to what over 6. 123 00:06:30,339 --> 00:06:33,319 You should know this from the equivalent fractions module. 124 00:06:33,319 --> 00:06:37,500 Well if I eat 1/2 of a pizza with 6 pieces, I would have 125 00:06:37,500 --> 00:06:40,269 eaten 3 pieces, right? 126 00:06:40,269 --> 00:06:40,819 That make sense. 127 00:06:40,819 --> 00:06:43,949 1 is 1/2 of 2, 3 is 1/2 of 6. 128 00:06:43,949 --> 00:06:47,649 Similarly, if I eat 1/3 of a pizza with 6 pieces, it's 129 00:06:47,649 --> 00:06:50,729 the same thing as 2 over 6. 130 00:06:50,730 --> 00:06:57,700 So 1/2 plus 1/3 is the same thing as 3/6 plus 2/6. 131 00:06:57,699 --> 00:06:58,979 Notice I didn't do anything crazy. 132 00:06:58,980 --> 00:07:02,030 All I did is I re-wrote both of these fractions with 133 00:07:02,029 --> 00:07:03,219 different denominators. 134 00:07:03,220 --> 00:07:06,050 I essentially changed the number of pieces in the 135 00:07:06,050 --> 00:07:08,829 pie, if that helps at all. 136 00:07:08,829 --> 00:07:10,409 Now that we're at this point then the problem 137 00:07:10,410 --> 00:07:11,189 becomes very easy. 138 00:07:11,189 --> 00:07:14,829 We just add the numerators, 3 plus 2 is 5, and we keep 139 00:07:14,829 --> 00:07:16,430 the denominators the same. 140 00:07:16,430 --> 00:07:18,970 3 over 6 plus 2 over 6 equals 5/6. 141 00:07:18,970 --> 00:07:22,240 142 00:07:22,240 --> 00:07:24,740 And subtraction is the same thing. 143 00:07:24,740 --> 00:07:31,460 1/2 minus 1/3, well that's the same thing as 3 144 00:07:31,459 --> 00:07:35,109 over 6 minus 2 over 6. 145 00:07:35,110 --> 00:07:39,530 Well that equals 1 over 6. 146 00:07:39,529 --> 00:07:43,059 Let's do a bunch more problems and hopefully you'll 147 00:07:43,060 --> 00:07:43,990 start to get it. 148 00:07:43,990 --> 00:07:47,120 And always remember you can re-watch the presentation, or 149 00:07:47,120 --> 00:07:49,629 you can pause it and try to do the problems yourself, because 150 00:07:49,629 --> 00:07:53,250 I think sometimes I talk fast. 151 00:07:53,250 --> 00:07:55,100 Let me throw you a curve ball. 152 00:07:55,100 --> 00:07:59,330 What's 1/10 minus 1? 153 00:07:59,329 --> 00:08:01,629 Well, one doesn't even look like a fraction. 154 00:08:01,629 --> 00:08:04,149 But you can write it as a fraction. 155 00:08:04,149 --> 00:08:07,969 Well that's the same thing as 1/10 minus -- how could we 156 00:08:07,970 --> 00:08:11,020 write 1 so it has the denominator of 10? 157 00:08:11,019 --> 00:08:11,589 Right. 158 00:08:11,589 --> 00:08:14,829 It's the same thing as 10 over 10, right? 159 00:08:14,829 --> 00:08:16,329 10 over 10 is 1. 160 00:08:16,329 --> 00:08:20,899 So 1/10 minus 10 over 10 is the same thing as 1 minus 10 -- 161 00:08:20,899 --> 00:08:24,949 remember, we only subtract the numerators and we keep the 162 00:08:24,949 --> 00:08:31,170 denominator 10, and that equals negative 9 over 10. 163 00:08:31,170 --> 00:08:34,379 1/10 minus 1 is equal to negative 9 over 10. 164 00:08:34,379 --> 00:08:35,919 Let's do another one. 165 00:08:35,919 --> 00:08:36,469 Let's do one more. 166 00:08:36,470 --> 00:08:38,680 I think that's all I have time for. 167 00:08:38,679 --> 00:08:47,319 Let's do minus 1/9 minus 1 over 4. 168 00:08:47,320 --> 00:08:53,770 Well the least common multiple of 0 and 4 is 36. 169 00:08:53,769 --> 00:08:55,590 So that's equal to 36. 170 00:08:55,590 --> 00:09:00,070 So what's negative 1/9 where we change the denominator 171 00:09:00,070 --> 00:09:02,010 from 9 to 36? 172 00:09:02,009 --> 00:09:05,029 Well, we multiply 9 times 4 to get 36. 173 00:09:05,029 --> 00:09:07,230 We have to multiply the numerator times 4 as well. 174 00:09:07,230 --> 00:09:11,860 So we have negative 1, so it becomes negative 4. 175 00:09:11,860 --> 00:09:16,860 Then minus over 36. 176 00:09:16,860 --> 00:09:20,110 Well to go from 4 to 36, we have to multiply this fraction 177 00:09:20,110 --> 00:09:23,070 by 9, or we have to multiply the denominator by 9, so you 178 00:09:23,070 --> 00:09:25,190 also have to multiply the numerator by 9. 179 00:09:25,190 --> 00:09:28,370 1 times 9 is 9. 180 00:09:28,370 --> 00:09:35,769 So this equals minus 4 minus 9 over 36, which equals 181 00:09:35,769 --> 00:09:39,579 minus 13 over 36. 182 00:09:39,580 --> 00:09:41,850 I think that's all I have time for right now, and I'll 183 00:09:41,850 --> 00:09:44,000 probably add a couple more modules, but I think you might 184 00:09:44,000 --> 00:09:47,519 be ready now to do the adding and subtracting module. 185 00:09:47,519 --> 00:09:49,019 Have fun.