1 00:00:00,500 --> 00:00:06,085 Once again we have three identical objects 2 00:00:06,085 --> 00:00:10,033 they all have the same mass, but we don't know what the mass is. 3 00:00:10,033 --> 00:00:13,005 But we do know that if you total up their mass, 4 00:00:13,005 --> 00:00:16,488 it is the same exact mass as these nine objects. 5 00:00:16,488 --> 00:00:19,633 and each of these 9 masses have a mass of exactly 1 kilogram. 6 00:00:19,633 --> 00:00:22,014 So we have a total of 9 kg on this side, 7 00:00:22,014 --> 00:00:24,300 and over here we have 3 objects that all have the same mass 8 00:00:24,300 --> 00:00:27,030 but we don't know what that mass is, we are just calling that mass 'X'. 9 00:00:27,030 --> 00:00:29,909 What I want to do is to tackle this a bit more symbolically. 10 00:00:29,909 --> 00:00:37,100 In the last video, we said "hey, why don't we just multiply 1/3 of this and 1/3 of that?" 11 00:00:37,100 --> 00:00:40,367 and then we would keep things balanced. 12 00:00:40,367 --> 00:00:44,500 keeping 1/3 of the same mass, so this total is the same as this one. 13 00:00:44,500 --> 00:00:46,813 and that is why the scale is balanced. 14 00:00:46,813 --> 00:00:50,343 Now, let's think about how we can represent this symbolically. 15 00:00:50,343 --> 00:00:52,567 The first thing I want you to think about is: 16 00:00:52,567 --> 00:00:56,833 Can we set up an equation 17 00:00:56,833 --> 00:01:01,813 that expresses these 3 things of mass 'X' 18 00:01:01,813 --> 00:01:05,900 and that in total their mass is equal to the total mass over here. 19 00:01:05,900 --> 00:01:07,698 Can we express that as an equation? 20 00:01:07,698 --> 00:01:08,952 I will give you a few seconds to do it... 21 00:01:08,952 --> 00:01:11,367 [silence] 22 00:01:11,367 --> 00:01:14,432 Let's think about it. Over here, we have 3 thing with mass 'X' 23 00:01:14,432 --> 00:01:22,087 so their total mass we could write as X + X + X 24 00:01:22,087 --> 00:01:26,414 And over here, we have nine things with a mass of one kilogram, 25 00:01:26,414 --> 00:01:39,620 which we could write as 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1. 26 00:01:39,620 --> 00:01:42,296 And actually, this is a mathematical representation 27 00:01:42,296 --> 00:01:45,036 We set it up as an equation, so it is an algebraic representation. 28 00:01:45,036 --> 00:01:48,220 It is not the simplest possible way to do it, but it is reasonable. 29 00:01:48,220 --> 00:01:54,835 I we want, we could say: "If have an X plus another X plus another X, I have 3 X's." 30 00:01:54,835 --> 00:01:59,479 So I could re-write this as 3X and 3X will be equal to... 31 00:01:59,479 --> 00:02:04,819 --If I sum up all these ones over here, we have 9 of them, 32 00:02:04,819 --> 00:02:07,220 So I have 3X = 9. 33 00:02:07,220 --> 00:02:11,832 Let me make sure that is right [counts to 9] 34 00:02:11,832 --> 00:02:13,554 So that is how we would set up. 35 00:02:13,554 --> 00:02:17,287 The next question is: What could we do mathematically 36 00:02:17,287 --> 00:02:21,213 to either one of these equations (but we will focus on this one) 37 00:02:21,213 --> 00:02:25,954 What could we do mathematically in order to solve for the X 38 00:02:25,954 --> 00:02:29,572 Or to figure out what that mystery mass actually is? 39 00:02:29,572 --> 00:02:31,420 I will give you another second or two to think about it. 40 00:02:31,420 --> 00:02:33,754 [silence] 41 00:02:33,754 --> 00:02:36,584 When we did it the last time, with the scales, 42 00:02:36,584 --> 00:02:40,354 we said "We have 3 of these X's here. We want to have just 1 X here," 43 00:02:40,354 --> 00:02:43,887 So we could say "whatever this X is, if the scales stay balanced, 44 00:02:43,887 --> 00:02:46,244 It will be the same as what we have there." 45 00:02:46,244 --> 00:02:50,220 There might be a temptation to subtract 2 of the X's from this side, 46 00:02:50,220 --> 00:02:53,287 but that will not help us, we can even see that mathematically over here, 47 00:02:53,287 --> 00:02:55,578 if we subtracted 2 X's from both sides, 48 00:02:55,578 --> 00:03:01,554 on the left-hand side we would have 3x - 2x, and on the right you would have 9 - 2x 49 00:03:01,554 --> 00:03:03,241 And you would be left with: 50 00:03:03,241 --> 00:03:05,888 3 of something minus 2 of something is 1 of something 51 00:03:05,888 --> 00:03:08,906 so you would have 1x there, if you get rid of 2 of them, 52 00:03:08,906 --> 00:03:11,832 but on the right-hand side, you will get 9 minus 2 X's. 53 00:03:11,832 --> 00:03:15,554 So the X's still don't help you out, you still have this mystery mass on the right-hand side. 54 00:03:15,554 --> 00:03:19,754 That doesn't help, so instead what we say is - and we did this the last time - 55 00:03:19,754 --> 00:03:22,560 We said "what if we took a third of these things?" 56 00:03:22,560 --> 00:03:24,835 If we take 1/3 of these things and we take 1/3 of those things, 57 00:03:24,835 --> 00:03:29,897 we should still get the same mass on both sides because the original things had the same mass. 58 00:03:29,897 --> 00:03:32,220 The equivalent of doing this mathematically is to say 59 00:03:32,220 --> 00:03:35,563 "Why don't we multiply both sides by one-third?" 60 00:03:35,563 --> 00:03:38,164 Or another way to say it is "we can divide both sides by three" 61 00:03:38,164 --> 00:03:41,739 Multiplying by 1/3 is the same thing as dividing by 3. 62 00:03:41,739 --> 00:03:44,572 So we multiply both sides by one-third. 63 00:03:44,572 --> 00:03:48,420 When you multiply by one-third, visually, over here, if you had 3 X's 64 00:03:48,420 --> 00:03:51,354 and you multiplied by 1/3, you would have only one X left. 65 00:03:51,354 --> 00:03:59,247 If you have 9 of these 1 kg boxes, and you multiply by 1/3, you will have 3 left. 66 00:03:59,247 --> 00:04:04,487 Over here you can even visually divide by 3 - which is the same as multiplying by 1/3 - 67 00:04:04,487 --> 00:04:15,408 -- so you divide by 3 - you have an X is equal to 1 + 1 + 1. X = 3. 68 00:04:15,408 --> 00:04:18,620 Or you see here that X = 3. 69 00:04:18,620 --> 00:04:22,887 Over here you do the math: 1/3 * 3 = 1 and you're left with one X 70 00:04:22,887 --> 00:04:29,898 so you are left with X = 9 * 1/3, or you can do 9 divided by 3, 71 00:04:29,898 --> 00:04:36,687 which is equal to 3.