 So, formally welcome everyone to your exam preparations. My name is Elizabeth Boyd, those who haven't met me before. Before we start with the session, please make sure that you complete the register, which I will post in the chat. Go find that register, which we will post in the chat. I have shared it in the chat. Please make sure that you complete it. Every week you come in, make sure that you complete that register. If you can see the register, please let me know so that I can I can repost it again. And that it's very important that you complete the register. And yeah, today's session, we're going to do an exam preparation by looking at different questions on how to solve fractions, how to solve things using lowest common multiples. And how to solve problems with powers and so on. How do you guys do the basic numeracy introduction questions that you get in your exam paper, which from part mostly of question one up until question five or 10, something like that. If you need to send me an email where you want to ask some other questions or you are unsure of in terms of the your numeracy, you can send an email to eboy. And if you have any technical issues, where to find the notes or there are no notes, so you won't have notes, but where to find the recordings. The same link way they shared with you probably they should be the schedule where you are able to go and find the recording. So the recording will be uploaded onto the website. So if you can find them, send an email to CT and CT and touch at unisa.ac.za and they should be able to send you the link on where you can access the recording for the session. Yeah. So that is it. So I'm going to share again with you. I'm going to stop sharing. I'm going to share with you all the, for some reason now, since I am the administrator, I can share the file through anyone of you who have already downloaded those who joined earlier today, if you are able to share, please share the files with everyone again because I'm unable to share or attach documents. It doesn't allow me to share a document. Is anyone able to share? Allow me. Unfortunately, because I am unable to share the same documents again, those who joined late. I don't know how you're going to receive the files, but anyway. Let's go and use the 2019, I think, 2019. 2019 6 E1. We will use that one. If we're done with it, especially the topics that we need to be doing today, once we're done with that, then we can go to the other three, the other two, which is 2018 10 and 2018 6. But we will start with one of those exam papers. So let me share my screen. And because this is exam preparations. So I'm not going to be doing some lot of explanations. You are going to be doing the lot of calculations and lots of activities. I will just be here to support you as well. So let's kick off. I'm not going to explain the type of questions that you're going to have and all that because I don't have access to your current exam guidelines. And you should receive that from your lecturer. My task today is just to look at equations and topics. So since we're looking at topics with fractions, topics with lowest common multiples and topics with powers, we can also we're going to skip the expression questions for now. We will do the expressions and equation later on next week in the next week session. So we're going to skip all this. We'll go to the fractions. So if questions like this comes up, you need to be able to know how to work with fractions. Right. And especially you need to remember and always know your botmas or your basic operations, how they work and the botmas rule as well. Because botmas rule says brackets first, operations or powers, division and multiplication, addition and subtraction where division and multiplication have the same priorities. You can work from left to right together with addition and subtraction have the same priority work from left to right. You also need to consider that when you work with fractions. Number two, when you work with fractions as well, you need to consider the following that in terms of addition and subtraction of fractions, if they have the same common denominator, you just add or subtract the numerator. What do I mean by numerator and denominator? So with fraction, the top parts of the fraction, we call that numerator. The bottom part, we call that denominator. You also get different types of fractions. I will get to that just now. So if you have addition and subtraction of fractions, then you need to check if they have the same common denominator and if they have the same common denominator, you add or subtract the powers. If they do not have the same common denominator, you're going to find the common denominator and there are different ways of finding the common denominator. And then once you have that, you solve your fractions. If you have multiplication and division. Now, multiplication, let's start with that one. For multiplication, there are different ways of multiplying. With multiplication, if you have fraction with A and B, multiply by, let's call it AB and AB. So, maybe let's call it C. AB, let's call it C. A divided by B and C divided by A. Let's say this is our fraction. Now, with multiplication, there are three ways that you can do your, you can solve your fraction. You can multiply the numerator with numerator, the denominator with denominator. So you will say A times C, that will give you AC. B times A will give you AB. And that's how you can solve your equation by doing numerator, multiply by numerator, or denominator, multiply by denominator. That's one way. The other way, you can simplify the equation by please crossing. If the numerator and the denominator can simplify, meaning they can go into each other, you can simplify that. If A can go into A, or A at the bottom can go into A at the top of the opposite. So, like this one, A and A will cancel out because A and A will cancel out and you will have one, one. And then you can solve the equation. We're saying one times C, which will give you C and B times A. Oh, sorry, B times one will give you B. Remember as well, the first one we said it is AC over AB. You need to make sure that when you solve fractions, I must just go back to this one. You need to make sure that when you solve fractions, you simplify them to their lowest form. Therefore, it means you need to make sure that you cannot do anything to that fraction, anything like that. So with AC divided by AB, A and A will cancel out and you will be left with C over B. And that is the lowest form. The same way as here, you will have your C and B, they cannot divide into each other. So if A and A cancel out, you are left with C divided by B and hence the answer will be C divided by B. That is one way of doing your fraction. The last way of simplifying fractions is that, so we have A divided by B times C divided by A. The other way of simplifying fraction is to see if you can simplify up and down. So it means if A and B can be simplified, you will simplify that. If C and A can be simplified, you will simplify that further. So those are the three ways of answering multiplication. So which is the thing that you need to always remember. The last thing is your divisions. So in terms of division, so let's say we have A over B divided by A over C. So now when you have an equation that has a division, we apply what we call it KCF. Keep the first equation, change the sign and flip the second equation. What do we change the sign to? We change the sign to a multiplication. We flip, the numerator becomes denominator, denominator becomes numerator. So we keep A over B. We change the sign to a multiplication. We flip C will be divided by A. That is when you have fractions, the way you need to make sure that you are finding the quotient or the division of the fractions. Simplifying them means you're going to apply the KCF method. And once you have applied the KCF method, then you can do your multiplication. But always remember, both must rule applies. Division and multiplication have the same priorities. If you have an equation where it has division and multiplication together, you just apply it from left to right. So let's say, for example, this is the equation that we are looking at. You can see that the equation has a division and a multiplication. Remember now, there are three fractions. The KCF says keep the first where there is a division. You keep the first, you change the sign and you flip the next fraction, not the other fractions, right, after that. So it means you're going to use your KCF on this and multiply by 3 over 7. You keep the rest. If the division was on this side, then you keep the other things the way they are. So if we say there was a fraction like that, it's multiplication and division. You just apply the keep, the change the sign and the flip on the one. So I hope that makes sense. So let's go through and work through this exercise. And then we will find more exercises that you need to do yourself. So simplify the following expression as far as possible. As far as possible means until the fractions can now or no longer or cannot be simplified. Therefore, it means if you can still divide by a second number, the fractions, you simplify that. So let's go on and do this one activity. So we have division and multiplication. What must says division and multiplication have the same priority. I need to work from left to right. So because it's division and multiplication and a division eventually changes into a multiplication. Then I can just do the KCF. So KCF will say 7 divide by 24. I must keep that change the sign to a multiplication and flip 14 divide by 12 multiply by 3 over 7. I've applied my KCF. Keeping the first change in the sign flipping the next fraction. Now I need to simplify because it's multiplication. And now you need to take into consideration that this is all denominators and numerators at the top. And things can, the first fraction can also simplify the last fraction or the second fraction or vice versa. So yeah, we go into apply the method of simplifying first because if I need, if I say 7 times 14 times 3, I'm going to get a huge number. And then I'm going to get 24 times 12 times 7. I'm also going to get a huge number. And then I need to think and figure out can they still be divided? Can they still be done this? Can this build the, you understand? So in order for us to simplify the equation for that, we can say 7 and 7 can cancel out because 7 goes one time into 7 and it goes one time into that 7. The other thing, we can divide into 12. 3 goes one time into 3 and it goes how many times into 12? It goes 3, 6, 9, 12. It goes 4 times. So it goes 4 times. So I keep the 4. And 14 and 24, something can divide into those. What is the number that can divide into 14 and also can divide into 24? I think the only number at this point is 2. How many times 2 go into 14? 2 goes 7 times into 14 and it goes 12 times into 24. Now I have 1, so I can write out what is left. 1 over 12 times 7 over 4 times 1 over 1. That is what I have right now. In order for me to simplify this, then I can do multiply everything that is at the top with everything that is at the top, everything that is at the bottom with everything that is at the bottom. So 1 times 7 times 1 is 7. And 12 times 4 is 48. Can 7 divide into 48? No, it cannot. It leaves a remainder and therefore the answer is option 2. As I said, simplify your answer as far as possible. And that's how you will solve infraction. Are there any questions? Before I give you an exercise as well. If there are no questions, you see there is a... Thank you, Carol, for sharing the document. Okay, so let's move on to the next fraction. Simplify the following fraction as far as possible. Now, looking at this fraction, it has a subtraction and a plus. But must says addition and subtraction have the same priority. Because but must addition and subtraction have the same priority. Therefore it means we went from left to right. And addition and subtraction of fractions, we first find the common denominator if they do not have the same common denominator. The denominator is the value that is at the bottom. If we look at this, 6, 4 and 24 are the values at the bottom. They are not the same common denominator. And what is a common denominator? Common denominator is a number that it can divide into the other numbers and not leave a remainder. So if we're looking at 6, 4 and 12, what is that number that can divide into three of them and not leave a remainder? What is that number? Let me ask you that question. The question, the common denominator? No, all four of them needs to divide into that number. Sorry, not what is the number that can divide into all this. All, sorry, it's me. I explained it all wrong. What is that number that all three of this denominator can divide into and not leave a remainder? That is what I should have said it. A common denominator is that number that all the other numbers can divide into and not leave a remainder. So we have 6, 4 and 24. What will be the number that 6, 4 and 24 can divide into and not leave a remainder? 24. It's 24. So our common denominator will be 24. So you're going to write your common denominator 24 at the bottom. And once you have your common denominator, all what you do is simplify. Simplifying will say 6 goes how many times into 24? It goes 4 times and you multiply. So you will say it goes 4 times and you multiply with a 5 minus. So you're going to multiply with the numerator and you go and say 4 goes how many times into 24? It goes 6 times and you multiply with 3 plus. So you will keep the sign. 24 goes how many times into 24? It goes one time and you multiply with 7. And you can simplify. 4 times 5 is 20 minus 6 times 3 is 18. 1 times 7 is 7 divided by 24. Remember what my rule says? Addition and subtraction have the same priorities so I can work from left to right. So we keep from left to right 20 minus 18 plus 7 equals 9. The answer today will be 9 over 24. Is there a number that can divide into 9 and also divide into 24? 3 can go into 9 and it can also go into 24, right? 3 goes 3 times into 9 and it goes 8 times into 24. And the answer will be option 3. You see how easy it is to solve fractions? Very easy. Are there any questions? So let's move on to the next other fraction. I'm going to have to let you do this one on your own and then we can check if our understanding are correct. Remember always those who attended my sessions, they know that we always use Newman's problem solving framework or method, where it says ask yourself. What is the question asking you to do? It's asking you to simplify this as far as possible. What are the things that they are giving to you? What are the facts given in this? You have addition, multiplication and subtraction. What are the things that I can use or apply to help me solve this? But my rule, multiplication, addition and subtraction. So multiplication comes before addition and subtraction. So therefore it means I'm going to solve first the fractions where they are multiplying before I can do the addition. Okay. And then what now? Then I will be left with addition and subtraction. The Bodman's rule says they've got both have the same priorities. Therefore I'm going to work from left to right. And then because there are addition and subtraction, what is the data I need to do? I need to find the common denominator and then you go find the common denominator between the three fractions and then simplify. Now I'm leaving it to you. And you're going to keep three over four plus and minus one over three as they are and only solve what is in the middle. That is your first step. So your first step is going to solve this and get the answer and then solve the next. We remember solving multiplication. You multiply what is at the top with what is at the top at the bottom with what is at the bottom. And if you are able to simplify, you can simplify top top or diagonal diagonal. I'm giving you two minutes while I go get myself what I will be back. How are we doing? Are we winning? Is it better? Is it getting there? Ready? Yes, I've got it. What is your answer? One and one over 24. One over 24. So someone knows their way. Okay, so let's help those who don't know how to get there so that everybody is on the same page. So what is the first step we do? We keep multiplication. Yes. So I'm going to keep three over four. Plus, how did you solve that? It's five over eight. One times five is five and two times four is eight minus one over three. What is the next step? Lowest common multiple. What is, yes. 24. That will be 24. Four goes how many times into 24? Six times. Six times three? Eighteen. Eighteen. Eight goes how many times into 24? Three times. Three times five? Four of ten. Three goes how many times into 24? Eight. Eight times one? Eight. Eight. Eighteen plus fifteen minus eight. Twenty-five. Twenty-five over twenty-four. Now I must explain the types of fractions that we have. In fraction we have what we call a proper fraction and an improper fraction and a mixed fraction. So now a proper fraction it is when your numerator is small and your denominator is big. An improper fraction it is when your numerator is big and your denominator is small. And an improper fraction is not your lowest form of a fraction. There is another way of representing fractions so that you are able to use them in your calculation or when you're simplifying them. You can use the improper fractions. A lowest form of an improper fraction is what we call a mixed fraction and a mixed fraction has a whole number and a small and your big number at the bottom. So to get to a mixed fraction you need to simplify this which will give you a whole number and a remainder. And then you keep your denominator as you have it and your denominator will become a small or sorry it will become a big value once you have simplified that. So how do we simplify? We say 25 goes how many times into 24. It goes one time and the remainder is one over and then we keep the numerator as it is and hence it is option four. Let's carry on with the next one. Now we've worked with numbers now if we're going to work with we're going to work fractions with variables. Remember variables are placeholders but we need to be able to also know how to simplify the placeholders if they are in a fraction. And that's where what we call lowest common multiples came in all lowest what are the common multiples and all that comes in. So in order for us to simplify variables, we need to find the common multiples or is the same as the multiples of those variables. So if we have variables like B and B squared is the same as B multiplied by B, right? If we have a variable like AB is the same as A multiplied by B which are called multiples. So in order for us to find the common denominator or the common multiples, we need to find the multiples of the values that we have. So at the moment we have, I just need to do this so that I'm able to write the values correctly. At the bottom we have A, we have B, we have B squared. Oh, sorry. That's a 4B. You also need to take that into consideration. The 4B. In order for us to find the multiples, the multiple of A is A. Multiple of 4B is 2 times 2 times B will give us the multiples of 4B. The multiple of B squared is B times B. Now to find the lowest common multiple, we're going to take the values that we see here and only keep the value once. If, sorry, if the value already exists in the multiple, we don't include it again. But if there are two of them, then we have to keep the other one as well. So let's look at this. So we have A multiplied by 2 multiplied by 2 multiplied by B. Coming to the last one already we have a B, so we cannot include the B again, but we also have 1B. So this B is not considered and we're going to multiply with the B. And hence we'll end up having 2 times 2. We always write the number first. It's 4AB squared. That will be our lowest common multiple of AB and B squared. That is if we solve the equation like that. I was just explaining and demonstrating to you. However, we need to take into consideration as well what we have here. We need to take that into consideration. It's division, multiplication. Because it's division, division we do KCF. So we need to first change this before we go find the multiples. So let's first solve this. We keep the first, which will be 2B divided by A. We change the sign, which will be multiplication and we flip. It will be 4B divided by A and A cubed divided by B squared. Now we have our multiples here. So we have A, A and B squared. Find the multiples A, AB times B. Therefore, if we need to find our LCM, we say A. We already counted A. So therefore A, the second A won't be considered times B times B. So because the two B's are on the same line, we need to consider both of them. If there was a B before, we would not consider the first B. So because there was no other B, so we keep everything as is. So our lowest common multiple here will be A times B squared. That will be our LCM, AB squared. Now, how do we solve these fractions like this? So going to say A goes that many times into AB squared. Therefore, it means A will cancel with A. You will be left with B squared times 2B. Come on. Times 2B. Oh, sorry. What am I doing wrong with all this? You can't find the LCM for a division and a multiplication, guys. That's one thing that I'm teaching you which is wrong now. You find the LCM only for addition and subtraction. We need to always remember that. Multiplication and division we simplify. Now, sorry, that's my bet for not reminding you of that. When we work with multiplication and addition, we need to remember the powers, the rules of the powers. It says if I have A multiplied by A, A is the same as A to the power of 1 and A to the power of 1. And if I have a multiplication of the same bases, they've got the same base, then I add the power. So it will be A1 plus 1 which will be equals to A squared. If I have A divided by A, A and A have to the power of 1 and it's a division, they've got the same base. Then with the rule says we're going to subtract what is at the power A1 minus 1 which is A to the power of 0. Any number to the power of 0 is the same as 1. So I've already touched the powers. I've already touched most of the things that you need to know about the powers. What else is there? The other thing that is there is if you have A to the power of 2, 2 to the power of 2 is the same as A to the power of 2 times 2, you've just multiplied the powers which will be A to the power of 4. If you have a fraction, A to the power of a half is the same as the square root of A. Those are the things that you need to always and constantly remember when you solve fractions. If they give you a root, you need to know if they give you a root of A. You must know that in front of that root there is a 2. Sometimes they don't write it for a square root but you must always know that there is that invisible 2. Same way as there is always an invisible 1 for any number where there is no power. So there is an invisible 2 there and the square root is the same as 1 over and 1 over the 2 in front of the power. So what if they wrote it like this? If they give you the square root of A, sorry, the cube root of A. So because it's a cube root, we know that the root is the same as 1 over, so the denominator will just be 3. The same way if they give you the quadruple root of A, it will be A to the power of 1 over 4 and so on and so forth. So those are the things that you always need to take into consideration when you solve multiplication or division of powers. Say different, difference true when you have addition and subtraction. So those are the basic concepts that you need to learn. Now let's go to this. We have division, multiplication, we do KCF on the first one, KCF. We keep 2B over A, we change the sign to a multiplication, we flip 4B over A, we multiply by A cubed over B squared. What's next? Next, we have a multiplication. We've learned about multiplication of equations. Remember, multiplication, you can multiply what is at the top with what is at the bottom with what is at the bottom. That's what we did. We also said something to the effect that you can cross or simplify. So for example, if you look at A cubed and we spoke about this, it's A multiplied by A multiplied by A. So I can rewrite A cubed as, I can rewrite it as A multiplied by A multiplied by A so that I can simplify it with those other ones at the bottom. And B squared, I can also simplify it and write it as B multiplied by B in terms of multiples, right? Now, if I look at this, A and A will cancel out, A and A will cancel out and I will be left with A at the top. So yeah, we have 2B and 4B. So we can also say B and B will cancel out or let's leave it like that. I just want to demonstrate something as well that how you can also simplify this. Now, we can also multiply what is at the top with what is at the top and continue. So we already canceled out the A's. We're not going to get them back. The reason why I'm leaving the 2B and the 4B is so that I can explain something when we get to the end. So we're going to keep 4 times B like the way they are. So 2B times 4B is 8B. Remember, they've got the same base, B to the power of 1 and B to the power of 1, right? So 2 times 2, we multiply numbers as numbers. It's 8. B to the power of 1 plus 1 will be B squared, right? I'm just going to keep B squared the way we have it so that I don't have to do anything to this B at the bottom. And A is what is left at the top. Divide by... We didn't have to even simplify it to B times B so we can put it back to B squared because it doesn't really make any difference. And now, B squared and B squared will cancel out. You will be left with 8A. That is one way. The other way you could have done it is to say B cancels out with B. And B cancels out with B. Then you are left with 2 times 4 is 8A. And at the bottom we are left with 1, which is the same as 8A. So I just wanted to show you and demonstrate to you that you can do it in many different ways and still get the same answer that you have. So I know that I'm complicating things, but that's the purpose of skills literacy. It's to show you many different ways of untieing things so that you feel comfortable to choose whichever one you want to use. Are there any questions? Before we go, find more fractions. Are there any questions? No questions and nobody is saying anything to me. We're almost left with 35 minutes. Let's look at no questions. Okay, so now we have subtractions. And remember with subtraction we said we find the LCM first. Right? So you can apply the LCM. So let's go and apply the LCM. We have 5AB and 2B. What are the multiples? 5A times B, 2 times B, LCM will be 5 times A times B times 2 and we already have B so we don't have to read another B. The answer here 5 times 2 is 10A times B is AB. So our lowest common multiple will be 10AB. Now we're going to say 5 goes how many times into 10 because a number can divide into a number, not a variable. So 5 goes how many times into 10? It goes 2 times. 2 times 3 is 6. A goes how many times into A? It goes one time so there's nothing. B will also cancel out with B so the answer there will be 6 minus. Then we come to the top one, to the next one. So the next one is minus 2 goes how many times into 10? It goes 5 times. There is no A there therefore we're going to keep A times 1. We're going to keep that. 5 times 1 is 5 and because there is no A so we're going to keep that A there. B goes one time into B so then it's nothing. So the answer is 6 minus 5A divided by 10AB. That's the answer. Which will be option number 4 and that's how you will simplify and answer the questions. Are there any questions guys? You are so quiet. Okay. So that covers almost everything that I wanted to showcase and show you how to simplify fractions. With powers, lowest common multiples and so on. So we have covered almost 1, 2, 3, 4, 5, 6. 6 questions that will appear in your question paper probably. The others are more about expressions and equations. We will solve them next week when we deal with that. Okay, so in the next session or next question paper we can look at 2018-10. So let's look at similar questions and on this one I'm going to start first with the fractions with letters. Because I think we did a lot in terms of those with all these fractions. Okay, we can start with this one. Even though it's not visible but at least it's something. So this is 18x squared divided by 4y minus into bracket. 3x divided by 4y multiplied by 9. Now, but must always important, very, very important. But must brackets first. Everything with the bracket must be solved first. So in the bracket we have 3x times 3x over 4y times 8 over 9. Now remember multiplication you can simplify numbers can simplify numbers. So here we have 3, 3 goes one time into 3 and it goes 3 times into 9. If we simplify this it will go 3 times into 9. 4 can go one time into 4 and it goes 2 times into 8. And then we can simplify what is inside the bracket which is 18x squared divided by 4y minus. What is inside the bracket 1 times 2 is 2 over 1 times 3 is 3. So now you have two fractions. Remember always this must always be your go to LCM because we've got fractions. Sorry, letters, variables. So 4y and 3 is the same as 4 times y. So we can say our LCM will be 4 times 3 times y which is 12 times y. So your lowest common multiple will be 12y. Simplifying 4 goes as many times into 12. That will give you the answer. It goes 3 times. 3 times 18 is 54, right? That will be 54 and y goes y times. It goes one time into y. So then we are left with x squared minus 3 goes as many times into 12. It goes 4 times 4 times 2 is 8. What is the remainder here is y. So you also need to keep that into consideration when you answer this question. There might be some error with this equation. It might be that this is a division and if this is a division, then it changes everything because I can see that I'm not seeing the answer here. I'm going to convert that, especially with the past exam papers at UNISA. You must be very careful with them. The dot sometimes disappears when they scan the exam papers. So we're going to treat this as a division and multiplication. And therefore if it's a division and it's not a subtraction that, then it means it changes everything. So we're going to change this to a division because we realize that that is not the answer. Then if it's that, we're still going to keep the same because it's got brackets. We still do the bot mass. So this will be 18 x squared divided by 4 y divided by, we still need to simplify this. It goes 3 times and it goes 2 times. And therefore the answer here will be 2 over 3. Now, because it's division, remember we keep x squared over 4 y. We change the sign and we flip the equation. Now when we flip the equation, 2 can go into 18. 2 goes one time into 2 and it goes into 18. 9 times. Therefore 9 times 3 is 27. Should have been easier. X squared divided by 4 times 4 y times 1 is 4 y. Gosh, you know what I forgot is the x and the y. Sorry, my bad. There are x and y at the bottom. That's the other thing that we didn't take into consideration when we were simplifying this equation. Sorry. When we were simplifying this equation, we're going to have here at the bottom, it's going to be 3 y and 2 x. So therefore this will be 2 x and they will be y squared. And therefore it means that is not a division. Oh, yeah. I bet because in the original one that we had, I forgot to put the x and y. Hence I converted it to a division. Now I just realized that I'm now even thinking. But it's correct because it should be the same. So we can still simplify this. x squared and y and y over that. So y and y will cancel x and x squared. Remember x squared is the same as x times x. So one x will cancel out. Therefore the answer here will be 27 x over 4 because one of the x will cancel out. If you have x squared divided by x, we can also convert it and say because it's dividing. We can say x squared divided by x, they've got the same base, which is the same as x 2 minus 1, which is the same as x. That is the other way of solving it. So then it means you will be left with only x. And the same way y divided by y, which is y to the power of y, which is y divided by y to the power of y, which is y 1 minus 1, which is y to the power of 0, which is equals to 1, hence at the bottom. And y's disappeared. So the answer here will be option 2. So if it's true, this will be a division, not a subdirection. Okay, I'm going to give you this as your exercise. Simplify as far as possible. So yeah, we have 3 over x plus 4 over 2x plus 5 over 3x. Remember to go find your LCM first. I'm giving you a chance to do the activity on your own and then you will tell me what is the answer you get. And we will do it together. Are we winning? No question. Are you guys winning? No answer, no response. So I don't know whether you are getting it or you need help. No, you are done. Let's do it together. So because it's addition, we need to find the LCM. We have x, 2x, and 3x. So we'll have x, 2 times x, 3 times x, which in terms of LCM, it will be x times 2 times 3 times x. Oh, sorry. We cannot count x again because there is only one x on there. Sorry, my bad. So those will be the x's because we already counted x. So that x1 count, that x1 count because we already have an x in the first instance. So our LCM will be 2 times 3 with 6x. So that will be 6x. x goes how many times into 6x? It goes 6 times 6 times 3, 18. 2 goes how many times into 6? It goes 3 times. 3 times 4 is 12 because already x will cancel out. 3 goes how many times into 6? It goes 2 times, 2 times 5 is 10, and x will cancel out. 10 plus 12 is 18 plus 12 plus 10 is equals to 40. Divide 6x. What is the number that can go into 40 and also can go into 6 because we need to simplify this further. Who can go into both of them? Because 40 divided by 6, 6 cannot go into 40. So 2 goes how many times into 40? It goes 20 times and it goes how many times into 6? It goes 3 times and the answer will be option 4. Happy? Are you guys happy? The happiness in the house, are we all good? Yes, ma'am. We left with 15 minutes. Let's see if there is one more question. So this is just a fraction. Let's see if you still remember how to work it out. You have 3 over 8 plus 4 over 7 minus 9 over 14. If you have or if you own a cashier calculator, when it comes to fractions, you can use your fraction button to calculate in the exam. There is nothing wrong. You don't have to do it manually like we're doing it. The only manual thing you will do is when you are using variables. Otherwise, on your cashier calculator, you just put the equation as you see them using your fraction and it will calculate them the way and you will get the answer as quickly as possible. And it might be that this question also is one of those that has an error. The subtraction might be a division. Yeah, this is a division and you won't be able to find the answer if you don't use it as a division. And that is the problem with the past exam papers. In the exam, it will be clear because you get the original paper, not a scanned copy. Option 3. Okay, so how we got to option 3. Remember, board mass is the key to everything. 3 over 8 plus change, flip, and 7 goes one time and it goes two time. There is no other number that can divide into 9 and 4. So we will have the answer as 3 over 8 plus 4 times 2 is 8 over 1 times 9 is 9. Find the common denominator, it's 72. And that is some number that is very difficult to get to. Alternatively, you can say if you want to find the common denominator between two values, you can multiply those two values together, which is 8 times 9 is 72. So our common denominator will be 82. That is the easiest way of finding a common denominator. Like for example, sometimes it's easy, sometimes it's not easy. Like if you have 6, I'm just going to repeat the other one that we had, was it 6, 4, and 24. Now finding a common denominator between those three, it's easy because you can say it is 24 because 6 can go into 24 and 4 can go into 24 and 24 can go into itself. If you are going to multiply them, you will get a very big number. So you just need to be very careful when you use this method of multiplying the denominator, right? So I was just explaining that. But in this instance, we can use 8 times 9 will give us 72 because it's the LCM, which is the lowest common multiple between the two numbers. Okay, so 8 goes how many times into 72? It goes 9 times 9 times 3 is 27. Plus 9 goes how many times into 72? It goes 8 times and 8 times 8. It's 64 and 27 plus 64 is 91 divided by 72. 72 goes how many times into 91? It goes one time. The remainder will be 19 over 72. How do we get the remainder? You say 71 goes how many times into 92? Sorry, into 91 it goes one time. The remainder will be 91 minus 72. That's how you will find what the remainder is. The remainder will just be the subtraction of your numerator and the denominator. And that is 19 over 72, which is option number 3. So I've given you an additional question paper. Oh, sorry. In this question paper that we are looking at October, November, there are also some questions that we didn't cover. Like not this one, where it has the equation, we will cover next week. So not that. So actually we've covered everything. So all the other questions on this one, we will cover next week and the following weeks. So we've given you the other question paper, which is this one. You can go and practice. If you need any help, there is a WhatsApp group that I've created that you can also use to share ideas and share your calculation and find assistance from other people. But always remember when you do that, when you share your questions or your options, do not just share your options. Show us how you solved it. Take a picture of how you worked it out. Like I've been working out. So you will work it out the same way as you work it out and you take this picture, you email it to us so that we can tell you, oh, you went wrong here. Just fix that. Or you missed this step. Then you are able to know where your challenges are. So don't just send us the... I got the answer and answer is option three. We are not going to answer that for you. So we're just going to look at your question and leave it as it is. So you need to send us your work things as well. Okay, so the other thing, what we didn't cover because on the other question papers, they didn't have this type of things. It's if you get questions like this or questions that say, simplify this fraction but do not use a calculator or something like that. Find a decimal, change this to the C. It's easy to do. You should be able to answer that. So this is one of the LCN questions that you should have perceived in your assignment or when you were busy with your assignment or when you were practicing. But normally they don't ask this type of questions in your exam, but they do. They will ask you. So you'll never know because the lecture can change the way they ask questions. So if you get a question like this, you know how to solve it. Always write your X squared, Y squared and your 14 X, Y squared and your 12 X, Y cubed. And just write the multiples. So what is the multiple of eight? It's two times two times two. Make sure that you get the lowest multiples. The multiple of X is X and X and the multiples of Y squared is Y and Y. You do the same here. 14 is two times seven times X times Y times Y. I don't have to put multiply by X, multiply by X because X and Y, X and X is the same as X multiplied by X. You don't have to put anything in between. Multiples of 12, so 12 can be two times six and six can, so it can be two times six and six, it's got its own multiples, which are two times three. So this will be two times two times three, which will give us the multiple of 12. So always start with the highest one and then see if you can find the multiples of the lowest one and then X times Y times Y times Y. So let's go find our LCM. Finding the LCM, we're going to keep all the values that we have right here at the top. The first line always will keep everything. So that will be two times two times two times X times X times Y times Y. Now when we go to the next one, anything that we already have, we don't have to include. But if it appears two times, we have to include the other one. So two, we already covered two into this. So we don't have to write two again. Seven, multiply by seven. X, we already covered the first X. Y, we already have Y. We already have the second Y. So we don't have to include anything else. Then we come to the next line. Two, we already have the first two. Two again, we already have the second two. Multiply by three, which we don't have. X, we already have the first X. And Y, we already have Y. And Y, we already have Y. And the remainder of the others would be multiplied by Y. So now we solve the numbers first. Two times two times two is eight times seven. So we'll say eight times seven times three equals to 168. And I don't even have to go and do the others in the exam. Or you can just go and circle the one that you have. And then let's do the X times X. We've got only those two X. So that will be X squared. How many Y's do we have? One, two, three. That will be Y cubed. And you can see that is the answer. So that is how you will answer the questions in the exam. And that concludes today's exam prep. I know that the session is two hours. And I really want to apologize for us starting late. But at least we have covered almost most of the concepts that we planned out to cover for today. And we did a whole lot of activities and exercises. Which should also give you some ideas in terms of preparations for your exam. So you can go and find more other activities to do. Go back to your past, your tutorial letters again. And look at those questions that they gave you initially. If you struggle to find the answer, try them again. If you still can get the answer right, we are here to support you. You can just send an email or use the WhatsApp group. And we should be able to help you with anything you need in order for you to prepare for the exam. Other than that, are there any questions? I didn't cover everything, right? So there are some questions here. So you will need to solve those fractions to get the answer. And I think that is the only fraction question that you have in this exam paper. And you can also go and find more question papers. You know, UNICEF students are very resourceful. There are question papers everywhere. So those are the ones that I have access to. If you haven't completed the register, please make sure that you do complete the register. Before we go, remember next week, we go into cover expression. Let me go to the content. You did receive the email, so you know which one we are covering. I think it's expression equations and any other equations that we have. Those who are doing QMI as well, you must also check what other sessions are we doing on QMI because we're not going to cover everything, everything, everything. We have limited time and limited hours as well. But I will try by all means to make sure that the difficult concepts that you are struggling with, you are able to go through and solve. So the next session that I have after this one is the QMI session. And I think it starts at 12 o'clock. Let me just give you a brief idea in terms of the type of... But then it will not benefit you because the concepts that we are discussing there are totally different to what you guys deal with. Other than that, those who are doing QMI, I will see you at 12 o'clock. Are there questions? Are there questions? I cannot add anyone on the group. You need to add yourself. I will send you now on here. Just stay on there. I will send you the link to the WhatsApp group. I cannot add you because if I add you, then it means I must add you on my phone and I'm not going to contact you anytime soon, which is pointless for me. So you can join the group by yourself. WhatsApp, right? That's where we need to be at. If there are no questions or comments other than the WhatsApp group, then we are done with today's session. But guys, someone needs to say something. Let's reflect. Was the session useful? Helpful? Can I hear some feedback? I found it really helpful. Thank you. Me too, ma'am. Thank you. Thank you. Next time when I ask, is there a question? I expect someone to at least say some comments. And usually I... I don't know whether you guys understand the concepts. You understand what we went through because when I ask a question and you are all quiet, you don't respond. You don't tell me we're still working. We are still busy or we are stuck. So it's a little bit frustrating, but I really appreciate some feedback and some engagement and some... I get the energy from you guys. But if you guys are quiet, I'm also going to be like, the session is going to drag along. And I will feel exhausted at the end of the session. But if you guys engage with me, talk to me, discuss with me. Ma'am. Sorry, ma'am. The reason I think why we're quiet is because when we filled out the form or something, it says that you agree to mute yourself on the calls. So I think that is why some of us are quiet because we don't know if we can unmute ourselves and talk. But when I ask if there are questions, then it means I expect you to unmute. At the beginning of the session, we ask you to mute yourself during the session and when we're doing some explanation and all that. So that then there is no noise in the background. But when it's time for engagement, when I ask is there a question or any comments, I expect you to unmute and talk to me. Because that is the opportunity that I'm giving you to talk to me now. Yes, ma'am. So next week is going to be a hyped session, right? Okay, ma'am. So you prefer us to unmute ourselves rather than put it in the chat? Yes, I prefer you to talk to me. Okay, ma'am. So you will talk and then you go back and mute yourself. And then, yes, only when we engage. Because now what happens when you place it in the chat, I don't see it until I go back to the chat because I don't see it. But if you engage with me and talk to me, it also makes the session interesting when we publish the session for the recording as well. Because now it's like you guys are not doing the work. I'm doing the work, right? Whereas I'm reading what you guys are giving me. And it's always nice to hear another voice than this husky voice that is on there. So I really appreciate that. You will unmute yourself next time I ask, what is your answer, any comments and all that. Okay, just a little bit. All right, I've got the link. I've just posted the link there on... This is BNU, right? Yes, BNU. You can join the WhatsApp group. And I will see you next week. Enjoy the rest of the weekend. Thank you, Mam, you too. All right, bye. Bye-bye.