All right, guys, good afternoon. Welcome to our webinar. I am really excited to have you. I really want to say, sorry, I'm really excited to see you, but I cannot. Anyway, so my name is Mahmoud. I am going to be your webinar instructor here. So in my free time, I like to read, play chess sometimes, not too much, like hiking, and that's it. So we have great teacher, Asim Soham. Would you like to introduce yourself, Soham, a little bit, talk about yourself a little bit? Hello, everyone. I'm Soham. I'll be your TA for these next 10 weeks. I've done Math Kangaroo TA in the past. I did it last year. So this is my second year TAing for Math Kangaroo. I've also participated in many math competitions, including the AMC series. And he's great, guys. You will see that. OK. So what's Math Kangaroo? If you do not know it yet, guys, you are going to give him like 30 questions for 75 minutes, OK? And there are going to be different levels for the questions, OK? Oh, we have another participant. OK. Another participant. OK. So we are going to talk about a little bit how to solve problems, because it's all about problems you will see in Math Kangaroo, and some of them are a little tricky. What I can say, guys, please make sure you understand the problem. So to be able to understand the problem like I did, you can either circle them, or you can underline the keywords. Little reminder, guys, you can turn off your camera since it's webinar. That's totally fine for me. If you have any questions, you will just put on the chat, and Soham is going to deal from there, OK? All right. So to be able to understand problem, make sure what they are specifically asking to you. Like, for example, they say, which one is the bigger number? And you find some of the numbers. Of course, you are going to look for the bigger one, all right? Or like some of the cases, they are going to say, which number represents the distance between, you know, like, for example, two cities? You make sure you find the distance between two cities, not the speed or not the time, OK? Make sure you know what they are asking you, basically. Second piece is plan and solve problem. So you can use algebraic ways. This is the most common. You can use geometry. You can draw the shape. You know, you can create some sort of chart. You can create table. You can create graph, whatever best works for you. But in many cases, guys, especially geometry, I recommend you to use some sort of visual. It's going to make your life pretty much easier. You will see that, OK? Also, you can just think about how can you tackle this problem or what do you need to solve it? These are the questions you need to ask yourself. Carry out your plan. That means you are doing actual calculations. But another piece, please, please, please, make sure, guys, you check your calculations a couple of times. That means, like, I have seen a lot of smart kudos. They are doing questions really fast. But because of a little tiny mistake, they just get the answer wrong. Like, it's really saddening for them. Think about it. You know everything, all of those topics. You learn everything. But because of a little tiny mistake, you get it wrong. And you get no point for that question if you're talking about actual contest. It's really saddening, man. Please, don't do that. Make sure you check your calculations. And also organize, not just on your thoughts, also organize your calculations as well, all right? And one of the most important part, look back and check and reflect. The reflection is really important because you have to ask yourself, does that answer make any sense? If they're asking you, well, there is an event. People are going to watch some sort of movie together in the city, whatever. So at the end, you find number of people is 45.5. I mean, we cannot have decimal represented by people, you know? I mean, this is really common sense. But still, for some of the answers, make sure distance is not negative. You know, sometimes they are going to ask you find absolute value. You cannot find those negatives. And anyway, make sure the number of answer or whatever answer you find it, it makes sense, all right? Okay, and we want the next one. If there is any question about those steps, please ask us via chat, guys, okay? Then Soham is going to check that, and he's going to go from there. All right, today we are going to talk a little bit about calculation iteration numbers. Then in the upcoming sessions, we're going to have multiples, algebraic thinking, percent, geometry, and so forth, okay? So what are rational numbers? If you have anything, you can just type it in chat, we can check that. So the rational numbers, guys, the numbers are we can kind of define as a fraction or decimal. But we cannot say every single decimal are rational numbers, because those decimals are supposed to be either terminating or repeating decimal. You can ask yourself or to me, but sir, what does terminating decimal means? Let's just check one example really quick. So terminating is, guys, we have 1.52. As you see, it's at the end after two, we don't have any more digits. I mean, you don't count those invisible zeros, just five and two. But if I just put here little tiny line, that means that two is repeating only. We call them repeating, but if there is nothing, and if the number ends, we call them terminating decimals. So you may ask me, so sir, what about neither terminating nor repeating, like pi? We know that pi is 3.14, but as you all know, we accept it as a 3.14. I don't remember all the numbers, but we have a bunch of numbers, like let's see, five, zero, seven, six, whatever, I don't remember that. So anyway, but at the end, as you see, there is no some sort of pattern, so we can say all of those numbers are repeating, but it doesn't end. We cannot say it is terminating. It's neither. That means that number is irrational numbers. Look, this is a trick because many kids think, but it's a decimal, it's a rational number. No, it seems like a decimal, but it's not actually a decimal, okay? It doesn't fit the definition of decimal, basically. All right, guys. So you can perform standard operations, you know, addition, subtraction, multiplication, and division on rational numbers to get a rational answer. This is also important. You cannot get, you cannot do those calculations to get irrational numbers. When you do some of the operations with rational, the answer has to be rational as well, all right? We have also found this, I believe you remember that, p represents for parentheses, e is exponents. Multiplication and division comes together from left to right, if you have only those two, and a and s represents for addition and subtraction, from left to right, this is the order, okay? If you have a bunch of, all of those mixed operations, you have to follow this order, all right? When we go to the next... okay all the operations a little bit explanation you already see that current exponents multiplication division and addition or subtraction i really wanna solve some questions with you guys i'm going to pass it for that reason so some of those properties we are going to use guys associative property so if you have only multiplication or addition you can group them however you want like for example you have given let's say two times five times nine i wouldn't just group five and nine here guys because five times nine give me 45 then i will multiply by two but if i just first do two times five ten times nine as you see here my goal is to get ten or multiple of tens basically or ten or hundred whatever in that case i am having some sort of easy number to calculate you know look as i met people we all are smart but sometimes we are always looking for some sort of shortcut okay if you know the rules just find some shortcut and use them don't worry about that so commutative property says sometimes you can just switch them to make it works for you in that example as you see you wouldn't just guys start two plus five no if you just kind of switch eight and you know other five here you can make it two plus eight is easy to calculate which is ten also five plus five easy to calculate another ten at the end you get the idea okay some of the properties of exponents we have when you have the same basis and when you of course multiply the exponents you would add the powers because here x squared means basically x times x i assume you'll remember that x cube means one two three x just write them and multiply with each other at the end we have one two three four five x you see this is how you i mean this is why you add them okay so if you have power of power that means we have y to the fourth power rate by cube so in that case you have three of y to the fourth power basically y to the fourth times y to the fourth times y to the fourth remember it's look like the first property right now you will just add four plus four plus four or you will say four times three which is 12 as well so when we have exponent i mean when we have negative exponent guys the only thing you gotta flip the fraction because there is always invisible one here after you flip the fraction you will switch that power to the opposite which is negative one becomes positive all right is there any questions are you seeing anything on the chat oh i did receive a question on writing pi as a fraction so pi cannot be expressed as the ratio of two integers also okay so about where did the question or you'll explain i'm not sure what was that yeah so the question was but doesn't it count as a repeating decimal if you convert it if you convert pi from decimal to fraction it doesn't because pi the decimal expansion it does not have a repeating block okay thanks i will explain that part then i can continue or any any more question guys okay we can move on then all right again guys as i told you pi is kind of tricky but if you check the definition it's not gonna fit the definition so all right here i really need you to work on that problem and please share whatever answer you find on the chat then so when you share those answers with me then we can go from there remember you can underline the keywords or what specifically they are asking then you can plan then you can work on your solution and at the end and at the end the deflection check like what was that they're asking kind of understand you know plan and calculate then check anyone has anything yet yet or am I so not patient I don't think we have more chat here if I'm not mistaken Let's give a little more time guys no worries since i am sharing the screen you can check the chat yes buddy okay so do we have some answers about that yes i do the first person to answer was sylvia so can you please explain how you got your answer so they do not need to talk lower since it's a webinar webinar so here is a question can you just tell me how many kiddos says a or man one that said a on their first try okay one person says a what about b do we have anyone says b two people said b okay do we have anyone says c no one said anything on the other three okay let's just focus those three then so think about it they say find the least possible product is equal to you have some positive numbers and some negative numbers in that case i assume that you are thinking when we multiply one positive and one negative it should give us negative number and answer should be negative also if you look at those answer choices we have only one positive in that case we should have answer should be negative number okay so if i am in a hurry and if i don't really check the question i would say answer is a but here is a thing so you just choose those two numbers you know nine times seven is actually positive i mean negative and negative seven i meant then you just think it's 63 but since it is you know it's supposed to be negative number or it's supposed to be smallest number you just do calculation mistake and get negative 63 but unfortunately it doesn't work guys okay that's the reason i am saying you just check your answer for that question you have to find one of the negative and one of the positive numbers from that set to be able to find least possible number here is i need to find one of the bigger positive number and one of smallest negative number all right otherwise you will get some numbers in the middle whichever other numbers you can try you will see that so then i should say that negative nine times six gives us negative 54 remember the questions are tricky they are gonna say that what yes negative 63 is less than negative four let's make negative 63 and if you do not pay attention to those signs you are gonna find negative 63 even when you multiply first two you get positive 63 actually okay so what i can say please make sure you are checking your answer okay and make sure you get it right one part i like about those questions are they are actually easy or they seems easy but they are tricky guys that's the case okay no worries i have another question for you guys number two when we get some answers so please if you can kind of tell me you know one person says a no one says b or whatever okay thanks so much and if you want to explain number two b y yes you don't have to it's fine okay if you want on me that's it All right guys I assume that you are working so hard to find this one but come on I still haven't get any answer yet. I mean in the chat okay I got I found someone answered it okay that's good. What about others? I have six participants here. Six kudos means I need more work, more answer. Okay two people make some answers. Okay, so can you share with me if anyone says the answer can be A? No one answered A? Okay, what about B? No one answered B? Oh, interesting. What about C? Okay, five people have C. Okay, everyone answered C now. Oh, interesting. Okay, you all think it's C. Okay, we gotta check that. So, let's just make it more fun for them since you are really fast chatters. So, it says when 29 is subtracted from the greatest two-digit number. What is the greatest two-digit number? Please share with us. Come on guys, you can do this. Quickly. You are typing ninjas, I guess. I still see five chats. You should show me more. Yeah, I have seen six. Okay, so someone says something. What was that number? What is the greatest two-digit number? 99. No one knows that? Okay, someone knows that. That's good. Someone said it in the chat. Someone said from chat this two-digit number is 99. Okay. What about smallest two-digit number? It's not a big deal. You got this. Give us some answer. We got 10. Yes, someone says 10 or a couple of people. Excellent, guys. I'm proud of you. So, these are the numbers. Remember, we are kind of planning right now. Then we can focus. It says when 29 is subtracted from the greatest number which is 99. This is also kind of a tricky part because you need to work with inverse. 29 is subtracted means, guys, you are not going to just do 29 minus 99. I am showing because I have seen that. So, you got to do kind of backward. It's going to be 99 minus 29. It doesn't make sense to you. Anyway. All right, 99 minus 29. I assume you get that answer. I will not ask you that one. Not yet. Okay, you should get 7. Then they say the difference is divided by the smallest two-digit number which is 10. Divide by 10. At the end, you should get 7, guys. I believe you all said 7 and I am proud of every single of you. Thanks so much, guys. Even I cannot see you, Ian. I cannot see your name. But it's going to be fine. I am still proud of you. You got this. All right. Clear everything. Since you can just watch that later. Okay. Next. Your next challenge is number 3. Let's see how many of you are going to find this one in a couple of seconds. Let me just start again. Your time starts. Let's see. In how many seconds someone is going to find some answer for this one? Come on. It's 15 seconds already. It's making me nervous. I am sorry, guys. 22 seconds. Two or five seconds No answer yet, but maybe someone is typing that Okay, I get some answers I mean After 52 seconds someone give us answer. That's nice Okay You Do we have only one answer guys You should get more. Come on Yeah Three answers. Oh, yeah, my chat is not working. Okay. Sorry. I just got a four Okay, what they say? Two people answered a on their first try And On one person answered B one person answered C on their first attempt Look at that. We have all different answers. Okay What about D and E? Maybe the bottle is gonna be empty. What do you think? Yeah, no one said D or E on the first try Or give us thumbs up if you need more time what I'm gonna start solving that I just got a third person answered a on their first try Interestingly, so you are saying basically majority that says answer should be a All right so guys We have a bottle of a volume one third liter is Three four fifty juice. Okay, so we have some sort of bottle the volume of that bottle is one third liter Okay, then is filled three fourth of that Total filled juice my question here for you. How do you find three fourth of one third? What operation you should use are gonna add subtract multiply divide get squares or what are we doing here? What do you think? Hey, they want to multiply Multiply Something in the chat. Excellent. Okay, someone says multiply. I am proud of your many of you Okay, so guys when you find something of the number like let's see I'm gonna give you easy example for this one Even you find half of a number the number is is a number 10 you get 5 We both know that I mean we all know that half of 10 is 5 but algebraic How do you find it? You just multiply half by 10, you know Any Fraction of a number or any percent of a number means you just multiply them think about it the same Logic with the person as well. Then I ask you 25% of 80 I'm saying What would you do? You know that 25% of something is one quarter. So you just Do a divide by 4 which is 20 Also Guys, you can just take the multiply them 25 over 100 Times 80 also gives you 10. Okay These are just a reminder. All right, I am solving that So when we have one third of three fourth, you just multiply them I assume to those you do not just multiply numerators. You don't need to When it's possible just simplify make your life easier remember as I met people we always look for easy shortcut Okay, one times one basically one over one times four is four. All right, by the way Please keep chatting if you have any questions about how I explain or how the questions I mean how we find answer or whatever So we have one form it says how much juice will be left when they say will left What operation you are doing you think you're adding subtracting multiplying dividing That's what do you think So if you get any answer I will just I mean if you kind of just type the symbol or just type the You know by writing The people wish to use subtraction Exactly. I am proud of that people. All right. Thank you guys. So, all right, so then what we are gonna do How much juice left to be able to find that after pouring one about one four? I mean one fifth of a liter you are going to do these guys one over four Minus one over five Okay. Well as you see we have different fraction. I mean different denominators, we gotta get the common denominator. Well Since there is nothing between nothing coming between those numbers. We can multiply this one by five this one by four You know at the end you can get five times five four twenty Minus four times one is four over twenty at the end. You should subtract them to get one over twenty All right or for that part again guys if There is nothing common between the denominators of two fractions You can use that second way to if you like look whichever method is easier for you This is the best way for you. You are the boss of that. You know that I don't know I'm just showing you some ways so you can just group them diagonal And multiply them, you know five times one is five one times four is four, you know Five minus four over then multiply those denominators since nothing were coming between those so Twenty you should also get one over twenty Man at the end, you know, which is best for you guys Just solve whichever way you like or any question about number three Should I go to next one or you want any clarification about one or two or three? This is yes, no question you can put why as a yes and as I know that video come on give us something, please A one person says no Okay, so others what about others they will have a question and they are so shy to share it with me or what come on guys No questions Thank you so much. Whoever said that Thank you so much whoever said that Thanks for clarification guys. I Like okay number four Again, remember what I said about those medical questions. These are really great questions that they are easy But seems difficult it means whoever prepared those questions. Those people are smart to be honest. Okay All right. Number four is for you. Okay, I'm gonna start the time Let's see how many seconds first person can check and solve this one for us And give us some sort of answer Okay, first 30 seconds. Come on. We got an answer. It was B. Oh, someone says they find the answer. Okay, we get one answer. It's B. What about others? Okay, one person says B. Another person said B, actually. We got three total for B. One, no. They got C on their third try. Okay, someone says C. Okay, anyone else or anything else? Not yet. Okay, it's fine. Okay, someone corrected their answer. Oh. How do they know that's correct? Come on. Maybe you found the first answer was right and you are making mistakes. That's it. That would be tough, man. I would be sad for that. Okay. Okay, another answer. What about A? What about D and E? They are lonely. Come on. They are going to cry. No one picked up A? No. I guess the answer is no. Okay. You are so rude, guys. Okay, I'm going to stop now. We have only B and C as an option. Let's see which one or maybe neither of them answer. Okay, guys. So, my first question is for you. Remember, this is not going to be like regular lecturing. My first question is for you. When you look at the question, what do you notice? Is there any pattern? If there is a pattern in that question, please reply by yes, you know, Y as a yes, and kind of give us a little explanation. Really quick, please. Do we have any answers for them? I mean, how do they find if they don't know if there's a pattern or not? Are you guessing the answers? Guys, no! We're not doing that. I hope. Okay, no answers so far, I believe. I don't see anything in the chat. Alright. So my question was, again guys, is there any pattern here? Like you said answers like 18, A plus B is 18, how? Or A plus B is 35, how? Do they get the calculator and just start multiplying them three times four times five times six. What was it? Someone said the numerators cancel Yes, something like that. Thank you so much. Whoever said that I am proud of you guys. Okay, so look guys here When you before you start the question, you just look at that look at to find some sort of pattern We have here three at the top Huh three at the bottom as well. Yeah For at the top and for at the bottom it doesn't have to be at exactly below three, you know, we can still kinda Cancel what I meant guys. This is related some of the other property of multiplication of fractions what I mean, can I convert those modifications like only as a one fraction like three times four times five times six and so far until a Over two times three times four and so far until That's kind of work as that and that's the same thing and the answer is nine Think about it. We have to at the top to at the bottom. There is only one operation which is multiplication then just cancel for and for five and five six and something and We have to have some sort of B here and B is gone because of the pattern Well, we have left over on the a at the top and two at the bottom. We have a over two is equal to nine I'm not gonna ask you. What's a then? I assume you all know that to be able to cancel that division You've got a multiply both sides by two to get a equals 18 guys. All right So if a is 18, how can you find B what do you think about B guys? Please give us something on the checkbox. I am waiting for you Like let's see you get B's for the whites for them, how do you find that little explanation for this? Since you're not talking here at this check guys, come on. I Know you text a lot in your life. So give us something fast Yeah somebody said B equals 17 how do you know that I am talking to you somebody how do you know that buddy please give us another answer come on I need some explanation if we know the total and one number we can subtract I believe what they meant here is guys when you look at here every single fraction here yeah of course we use that properties yes but here if you look at original question guys every denominator is one less than numerator and the last one is a a will be as you see so we know that B is equal to a minus 1 because we get those numbers you know always 2 is less 1 less than 3 3 is 1 less than 4 and so forth well if A is 18 that means B is 18 minus 1 let me plug it in which is 17 well we found 18 we found 17 at the end you just add them okay kiddos remember I told you before the questions are tricky they put answer A represents B as you see answer B represents A but answer is A plus B if you think for some reason they are asking us B yes answer is A no even you figure out even you know everything about the question you still make a mistake please don't okay or if you think yeah answer is A because I mean answer is B answer is 18 because they ask us A no they ask us to find the total okay please be careful when you learn how to solve that I believe at the end you will also check your answer and make sure it makes sense okay we are giving us some thumbs up I mean in the chat not here all right all right guys here is your next challenge let's see what we think about that Oh, I forgot to start timer, man. OK. Let's see how many seconds it's going to take at least one of you to figure out. Swami, I believe someone is going to... So do we have anyone says A as an answer? No. What about B? No. C? We got one person. We got two people have C. Okay, what about D? D, one. Okay, what about E? And one for E. Oh, look at that. Okay, so I think it's A, maybe you think it's B. We choose every single of them, you know. Okay. So guys, let's start. All right. First thing first, what are they asking us? All four digit positive integers with the same four digits as in the number 2013. One, guys, I do not continue to reading the question. I know there is something going on here. Remember, they say written on the blackboard in English. Okay, how many digits numbers we need? We need four digit numbers, of course positive integers. But we have 2013, there is a zero. That means, can I use numbers in that order like zero, one, two, three? No. No. Remember, you really need to be careful when you just, you know, work with the problem. All right. This is the one of the reminder. Again, please make sure you underline or circle. Four digit is important because we have to be careful with the zero. Zero cannot be in the first, on the left parts, on the thousandth place, basically. Okay. Then they put them in increasing order. Okay. What is the largest possible difference? Remember, after we put them in order, increasing order, we are going to look for some difference, but not just random difference. We are looking for the largest possible difference. Okay. Between two neighboring numbers on the blackboard. Okay. Let's just start, guys. So, we will use those numbers, those digits, two, zero, one, and three. So, since we have zero, one, two, and three, the smallest four digit numbers should be, I'm going to start with one. Remember, one. Then, since we have smallest, I can put zero, and I can make it two, three. Am I right? I believe you are saying yes. I don't want to believe that. Okay. So, this is the smallest number. Then the next number, I can still put the first two digits, you know. I can keep ten. Then just switch those, which is 32. Okay. After that, I can still have thousands. Then I can keep thousands. Maybe I can only switch two here. Two. Then, zero, three. Then I can have 12. I mean, 30. Okay. Then, let's see. Do you think I can have anything else with the thousands? Yes. We forget to put, you know, three in the next one. So, we can have one, three, zero, two, and one, three, and ten, 13. Okay. So, I believe there is no number which is in thousands. Am I right? Maybe we can contribute to two thousands, guys. After that, we can contribute two thousands, 13. With the same logic, two thousands, 31. You know, two thousands, one hundred, or three. Two thousands, one hundred, 30. And what else? Two thousands, three hundred, one. Two thousands, three hundred, ten. All right. I don't think we have another two thousand number. Am I right, Suham? If I'm missing something, please remind me. Okay. Then, contribute three thousands, guys, you know. So, we have three thousands, 12. Three thousands, 21. Then, three thousands, one hundred, or two. 31, 20. And, 32, one. 32, ten. All right, guys. At the end, we are going to look at that. We are going to look at the numbers. Do you really think that we need to look for the difference between every single of them? No, guys. So, we have one thousand, three hundred something. And the second number in the sequence is two thousands. As you see, we have some sort of jump here between one thousand and two thousand. Okay. This is the first jump. Also, when you look at here, we have two thousand something and three thousand something. Well, as you see, guys, there is another jump. At the end, you are going to find the difference. I would say I would find both of the difference, you know. Two thousand, 13. Minus 13, 20. Please find that answer and share it with us via chat. You are not just here to watch me, you know. We are not watching TV, guys. Come on, please be active. Thank you so much. And we have 30, 12. Minus 23, 10. Okay. Please find both of the answers and share with us in the chat. Then I am going to put those answers. Then we are going to figure out which one is bigger, biggest, or largest possible difference. No answer yet? Come on. Grab a calculator and do that. Or, I mean, in your mind. So, do you get any answer, Bobby, or not yet? About those calculations? Okay. So, the first one, somebody said 793. And the second one, they said 702. So, this one is 793. Are you sure about that? I would say, whoever the person is. The first one. Is it 793 or something else? Okay. 693. Oh, they changed it. Okay. You get 693 then. Another one is 702. Another one is 702. Yes. Okay. So, make sure you do not do miscalculations. As you see, if you just thought this is 793, then you would just get this answer. Unfortunately, if you know everything, you get it wrong. Please don't. Okay? Make sure you do the right calculations. The right answer was A, as you see. Any questions for this one? You can type Y as a yes, N as a no. You can say, we are good. Sometimes, you know. No questions. That works good. Soham, do you have any questions about it? That means no. Okay. Thank you so much. I'm going to move to the next one then. And guys, here is your next challenge. Number six. Let's see how many seconds it's going to take for them or someone to give us an answer first. But again, guys, please check your answers too. Don't make calculation mistakes. Alright, one minute. Do we have an answer yet, Suha? Or I can give you some more time. We got one person who answered C. Okay. It's fine. Use your time, guys. Okay, let me give you a hint, guys. Whoever says C, please check your answer one more time, guys. That means others are discussing answer may not be C. Careful, okay. You got one answer for D. Okay one answer for D. Okay we got a second one for D. Okay D is popular. Anyone else? all right guys i believe you have been working and i gave you enough time so let's see how can we approach this question every second equation okay this one is equal to zero is replaced either with plus or minus okay so the equation is correct first thing first guys think about you have those bunch of numbers you can only play is plus or minus and the answer is zero well if the answer is zero you have some group of numbers and the total of those numbers is equal to another group of numbers and at the end you just subtract them does make sense like let's say you have one group which is total is 15 another group which is negative 15 you just combine them get zero basically okay because you have you can only add or subtract and we all know that even if i put plus or minus zero it doesn't change anything i will just ignore them okay these are just here to confuse you anyway so look what about the left of our numbers we have two one five two one five two one five i have three of those two one five if i say two plus one plus five times three basically because we have three of those numbers i am finding total gives us 24 guys okay because this total is 8 8 times 3 is 24 but remember 24 is total at the end we are going to find zero as an answer that means we have two different groups guys and they have to be equal so 24 divided by 2 gives us 12 all right that means i am going to find some of the numbers and add them which the total give us 12 then i'm going to put minus sign in front of some of the other numbers and their total should be also 12 then they are going to cancel each other since remember they said what is the smallest number well well that is the tricky part of osiris that master base with the plus we are going to look for the smallest amount of number we can add them together to get to that so think about that if you are looking for the smallest amount of numbers where you are going to use those numbers we just start with one and two think about it would you say one plus one plus one plus two plus two or would you say let's just start with the five because five is the biggest number here and we can get as much as amount of five then we can find a way to get total is 12 think about it if we choose every single five year the total should be 15 you know five times three but we need 12 well maybe we can choose only two of those fives you know i'm trying to be reasonable five plus five then i am adding another two to get 12 it gives us the total 12 that means that works again think about right now you can say okay so we found plus five plus five plus two that means if we have if we just put plus sign for three numbers we would get that equation is correct because others can be all negative we don't care about that but guys here is the trick part it is the catch remember the first number is already positive that means you don't have to put plus sign here can you see that again one more time i told you some questions are especially tricky that means you don't have to count that too so how many plus sign we use we use only two of them guys and answer was two okay i know many of you you say answer cannot be one or two it's so small but actually answer was two guys any question for me right now please check something say something so someone is asking why we divide 24 by 12 excellent anyone would like to answer that question via chat or so on you can do that too i am tired man come on someone explained to him or her then i can check some other friends were listening or not so we divide 24 by 2 because we want to find basically if we're going to have a bunch of pluses and minuses that make the numbers sum to zero then all the positives have to sum to 12 and the negatives have to sum to 12. thanks so much that means we have two group of the numbers which is same results does it make sense guys whoever asked that question please say yes or no then we can move on do we have another question come on No, are you good? Okay, we can move on next one. We still have some times, come on, we can solve one more. Why not? All right, another interesting question for you, just for you guys. You check this one, and I'm gonna give you one minute to think. All right, almost one minute guys, come on, someone gives us some answer. One person answered B, another person answered D. Okay, look at that. Okay, since we don't have too much of time left, let me start. So guys, as you see, all of those integers from 1 to n, look, this part is important, 1 to n, we start with 1, not 0, are inclusive, are equally spaced in order around the circle. Okay, so the diameter through the position of integer 7 also goes through the positive entry. As you see, this is the diameter, that means this is half of the circle, you know. Can I say that since that line, you know, 7 and 23 are in those edges, that line is diameter. Can I say that we have semicircle here? That means here on the right side and here on the left side, we have same amount of numbers. Can I say that? It's logical, you know, okay. What is the value of n? They are saying basically how many numbers we place here. Think about it, guys. After 23, we are going to keep going with the numbers, 24, 25, whatever. And at the end, we are going to go n and we will start with 1, 2, 3, until 7. Since n is here, we do not know how many numbers here, we cannot just find it easily. But when you look at on the right side, I just put here, you know, 8, 9, so far, and until 22. Do you think we can find how many numbers do we have here? We can find it, am I right? So how am I going to find it? I'm going to do 22 minus 8, then I'm going to add 1 because 22 and 8 is inclusive. So 22 minus 8 is equal to 14, 14 plus 1 is equal to 15, guys. If you like, you can count, you know, 8, 9, 10, and you are going to see that there are 15 numbers. If you do 8 and 22 inclusive, okay, we get 15 numbers. Well, if I have 15 numbers here, 15 numbers, since that's a semicircle, I have to have, guys, 15 numbers here as well on the left side, okay? 15 numbers. All right. Well, we need to ask ourselves, how many total numbers do we have here? Well, 15 here, 15 here, plus we need to include 7 and 23, those two numbers as well, gives us 32, guys. That means, remember, that's the tricky part. Since we start integers from 1, well, answer should be also n, should be 32, guys, okay? Is there any question for me about this one? Please ask me really quick, it's time already, but I can just answer really quick. No questions. All right, guys, I want to say that thank you so much for joining our webinar. I hope you learned something today, I hope you had fun, and I'm going to see you next time, okay? Take care, guys. Bye-bye.

Math Kangaroo

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