guys it's time let's start. I did ask you to check the warm-up question before I'm just asking again if you just join us. All right you can share your answer to with Soham please so we can just you know check your answers and tell us. So far when I just get one of the direct message one friend says it can be C. Let's just wait couple of minutes about other friends to solve that. Another friend says, answer should be C. And we still have people are joining us. Thanks guys. We have a lot of participants. And other friend says answer should be C. Look at that. Okay. All right, guys, let's just start. We have just a couple of answers, but I believe other friends are about to finish anyway. So in the question, it says a dragon has five heads. Okay. Every time a head is chopped off. Okay, five new heads grow. Look at that. If six heads are chopped off one by one. Basically, you are doing that process six times. How many heads will the dragon have at the end? So. Since it's rational numbers, we're going to talk today. It's also kind of creating equations if it is necessary. Think about guys. Every time one head is gone. That means we have negative one in one cycle. Then five new heads grow. That means. In that same cycle, five new heads come. That has been said. In one cycle, that dragon gets four heads. Interesting. Five minus one is four, you know. Or negative one plus five. And you do that process six times. In that case, we should have four times six. We just, you know, repeat the same process. Give us 24. Well. With that process. The dragon gets 24 heads. But at the beginning, you know that the dragon had five heads. 24 plus five. Gives you 29. Okay. If you have any questions about that, please ask us really quick. Many of friends says see anyway. Okay. All right. Then we can move on. Okay. Since we have a lot of new participants, I'm going to just review. Problem solving strategies, guys. Make sure you understand the problem. Make sure you know what they're asking. Sometimes they're going to say which one is not the right case. Which one is not the answer. You will find which one is the greatest value. Which one is the least value. You've got to be careful with those terminology. Make sure you know. You get what they are asking. Plan to solve the problem. Which method you're going to use. You're going to use algebraic weights. You're going to use some charts. You're going to create, you know, some pictures. If you have a question from warm-up, you can, you know, draw a dragon and just keep working with it if you like. Anything works, guys. Carry out your plan. Carefully complete your calculations and organize your thoughts and steps. This is really important, guys. When you work with like some problems, especially like requires a lot of steps. Make sure you didn't do any mistakes. And, you know, you are going to give them the scratch paper on the test. Make sure, guys, this is your paper. I would just kind of divide the paper. A couple of segments. So I will just make, you know, one of the questions. Let's say number one is here. Number two is here and so forth. If you start from here and at the end, your answer is somewhere here. You just do all of the calculations around and all of those empty, you know, space, it's going to be a mess. Anyway, we are going to require a lot of papers and you're going to confuse yourself. You don't want that. And the last part, look back and check and reflect. Does this answer make sense? If they are asking to find distance, you shouldn't find negative number, you know. If they are asking you to find the percent increment, you shouldn't find the number less than the original one. If you are adding text, you know, like with the same logic. If you add text, the price should be greater, not less than, you know. Or if you are looking for some certain items, like how many cars they can buy, you cannot find 1.5 cars, basically. It has to be whole number. You cannot have negative cars as well. All right. We got a 12-bit calculations with real numbers today. We put less than zero. It was last week. Since it was the introduction, we didn't put any, you know, number for that. What are rational numbers? Rational numbers are the positive and negative numbers that can be written as a fraction or either terminating or repeating decimals. Some of the examples, like all of the positive and negative whole numbers, besides those, you know, fractions, I mean, those fractions and these repeating and terminating numbers. Irrational numbers are the real numbers that do not meet that criteria. Examples include pi, e, and root of 5. You may say, sir, but we use pi as a, you know, 22 over 7. This is not pi, guys. This is just approximation of pi. We assume that pi is approximately that number. Because if we are talking about pi, that means we never know exact value of pi. We never know exact value of square root of 5 or logarithm e. These numbers, we just don't know. Because, like, after 3.14, we have, like, those bunch of random numbers coming, and we have no idea what ends. I mean, it never ends, and we cannot say what the last number is, if there is any last number. You can perform the standard operations on rational number and get only rational answer. You cannot add two fractions to get pi. You cannot add two fractions to get square root of some numbers. Please be careful with that. You need to use order of operations or other properties to complete your calculations. You know, p is for parentheses, then exponent comes. From left to right, you can multiply or divide. And from left to right, you can add or subtract. Challenging problems with rational numbers are a good place to begin your math practice. Let's just move on. All right, order of operations. I already mentioned about that. First, work with parentheses, please. Then clear off exponents. Then keep working. Sound properties. We have associative property. Associative property. For this example, may or may not make so much of sense, but I would say, like, let's say you have given, instead of two times, three times, four, you have given, let's see. Three times. If we make... If we can make it two times five. Let's just make three times four times five. Is equal to three times four times five. You only change the place of the parentheses. So for those guys, if you just multiply first three and four, then multiply by five, three times four is 12. I wouldn't do that. Since any even number times five gives you multiple of 10, that would be easier. That's how you perform associative property. Commutative property. This is a good example here. If you just switch some of the numbers, of course, I would just add two and eight, not just two plus five. It's going to take some extra time. It might result to get some mistakes. Also here, instead of having four times seven times five, multiply four times five first. Then you can go from there. What are some properties of exponents? When you go to eighth grade, you'll just solve about that a lot. If you have the same base with the exponent, you add them because if you expand x squared, it means x times x. The same logic x cubed means x times x times x. You just add them. This is the literal proof why you add them. And power of power, you may ask, sir, why do you multiply them? Because here we have the same logic. We have y to the four power times y to the four power times y to the four power. This is how you get cube of any term. Well, from there, you just refer to the first property. That means y to the four plus four plus four or y to the 12th power, which is the same thing here. In many cases, they are going to ask you to find the answer as a positive exponent. In that case, you are going to just use the reciprocal and switch the negative exponent to the positive. Any question about that part so far? I believe we are good. No one put anything on the chat. Distributive property, we are going to use a lot as well. As you see, any number of parentheses needs to multiply each term inside of the parentheses. Also, we are going to talk about factoring as well. The opposite of distributing property is factoring. It's a great tool to break up calculations into mental math. For example, instead of 211 times 12, you should multiply 200 by 12 first, which is easier than 11 times 12. Because 212 can be written as 200 plus 11. Then you got to distribute that 12. Subaction is also adding a negative. We can use the addition property instead of subtraction. Here, we can, of course, switch the place of the numbers. Instead of putting negative 11, we can say plus negative 11. I'm going to move on. I know you are excited to solve some of the questions here. Number one, here is your first challenge. I don't think it's going to take too much of time, but let's see. At least maybe one friend give us some answer. Then we will go from there, guys. All right, someone says something. If I see. Okay, two new answers. Okay, two new answers. One friend says D. Two friends says C. We have D for one. And two friends says C. Guys, instead of sending me direct messages, please try to send to Soham as well. Okay, another friend found D as well. Okay, let's just. Two people say C. Okay, so two people says C and two of them D. Am I right? Send you. What was it? Four. Okay, one of the C people changed to D. So we have three on C, three on D. Okay. And one person say D. Okay, look at that. All right, let's just start, guys. So far, you are saying answer is going to be either C or D. Maybe none of them. Let's see. So here is the deal, guys. For those type of questions, before you solve them, I recommend you to just take a look at the question. Here, we have some of the zeros inside of the parentheses, as well. So here, just ignore zero. We have 20 times 6. Remember, I focus on the parentheses from PEMDAS, because PEMDAS says you've got to focus on parentheses first. Okay? So 20 times 6 is 120. And then since that parentheses was, I mean, since that negative sign was with that parentheses, it's gone anyway. We have plus 6 here, guys. Then you should have 126. Some friends found 114. Maybe they thought that negative sign applies here. It's not, guys. This is extra 6. It's definitely separated from that parentheses part. Okay? That's the reason answer was 126. I have been telling you, remember, read the question carefully. Don't make any mistakes. Please check your calculations again and again. All right. Number two, let's see how many people are going to get which one of these. Please try to send messages to Soham. I might not check the chat again and again. Even I am getting a lot of chat right now. Okay. Let's see. We also got one more with C. Two more with C, so four total on C. Okay, four Cs. Okay, what else? Another person says C. Another person say C. All right. Then you're saying answer has to be C, sir. There is no other option. Let's see, guys. It says, what number must be subtracted from negative 17? So the number is going to take it away from negative 17. Please make sure you get the question correctly. We have negative 17. And the number, let's say the number is x. We generally use x, remember? That number taken away from negative 17 to obtain, to get the result, is negative 33, all right? Here, well, we have some extra number next to the x, next to the variable. Remember, solving equations means you've got to make that variable alone by taking away everything next to it. So next to x, we have negative 17. To be able to cancel, you've got to do the opposite, remember? I'm going to add 17. So negative x gives us, when you do the calculation, should be negative 16. But we need positive x, remember? Cancel both of x or multiply each side by negative 1. x is 16, guys. Whoever gets 16, they got it right, OK? If there is any question, I will just put them in the chat. I get some chats, too, OK? Some messages. All people are giving me answers, OK. All right, guys. I am proud of you. Many of our friends getting the right answer really quick. OK, what about number three? Yeah, first two questions are easy. What about this one? It was number 12, but still, it was kind of mild. I'm in the middle. I wonder how many friends are able to solve this one really quick or slow, but give us some answer. I'm going to start timer, guys. No worries. OK, I hope we can find some sort of pattern. And your time starts. Okay, it has been 43 seconds, let me give you some hints. You are going to use a lot of properties, guys, remember previous slides. You got one vote for A. Okay, someone says A, look at that. It has been one minute, guys. I don't get too much of answer, let's just give you a little more time. I get, I believe the same person, Soham, same person give me, tell me, also A, I believe it's the same person who says A, I don't know. Guys, please try to direct message to Soham, not me, he is able to check the messages faster than I am. So we got two more for A. Two more for A, then maybe we can just start, so let's see if it is actually A or not, or maybe others. All right, guys, here is the deal, when you have this type of questions, are you going to add, start adding 1900 plus 1901 plus 1902, do you really do that, or we should have some different methods? Remember, mad people are practical people, or sometimes we call ourselves lazy, what does it mean? We got to find some sort of like easy shortcut ways, look, instead of focusing, you got to say, but sir, PEMNA says you got to focus on parentheses, I know, but for some cases, to make your life easier, you do not necessarily focus on the parentheses, because we have only addition here, not necessarily all of those different operations together, it's fine. So let's just ignore the parentheses here, we have 1900, this starts, and here, we have negative sign, and the parentheses, we have 100, 101, 102, and so forth. The trick you need to remember here is that negative sign means, guys, we have invisible negative one here, and that means that negative one multiplied by every single term here, look at that. All right. At the end, we got to get something like that, you know, 1900 plus 1901 plus 1902 until 1999, then we have negative one times 100, minus 100, then negative one times 101, minus 101, and oops, not the comma, my bad, then 102, and until negative 199. So right now, this still may or may not make any sense, but here is the catch, guys. Remember, we can just switch the, you know, placement of the numbers. Can I just look at those first numbers? I am getting one number from, I mean, the first numbers from those different parentheses. So I have 1900 and minus 100. With same logic, I have 1901 and minus 101. Interesting. Let's just group them, if we can change the color. So we have 1900 and minus 100 here, the first terms from the parentheses, you know, they are gone. Then second terms from the parentheses, we have 1901, and we have minus 101 here, they are gone too. Then with the same logic, we have 1902, minus 102, until we have, the last number was 1999, minus 199. All right, guys, here is the deal, when you do 1900 minus 100, you get 1800, I believe or I don't know that. So 1901 minus 101, huh, it's 1802, interesting, we get 1902 minus 102, oh gosh, it's 1802. Look at that, every single term, at the end, you are going to give, you're going to get 1800, guys. That's awesome. So then the question here is, how many 1800s do we have? This is the catch, and please count the numbers carefully, we start from 1900, or if you only focus the, you know, last two digits, like ones and tens digits, you start with zero until you get 99, you know. For those cases, you gotta do last term minus first term, 99 minus zero is 99, but since you include 99 and zero itself as well, that means we have 100 different numbers, 100 terms here in that sequence. That has been said, you need to do 1800 times 100. And whatever you get is going to be your answer, guys. Let's see, how many zeros do I have? I should get two zeros here, then I am going to get 180,000. I don't remember how many friends says what number, so do we get A most likely, oh, you get A most likely, my bad, I missed that part. I thought- Everyone said A. Okay, good. I thought these numbers are kind of part of social, my bad, guys, okay. So the answer would be 180,000, I hope we are good, I hope you get what's going on here, then we can move on to the next one. All right, guys, check number four, please, and then you can share your answer with us. Let's just start the live, we're one minute started. I will have someone send me the message via chat if you please don't do that guys am I not able to check the messages all the time one more reminder please send messages directly to Soham not me okay we got four for C and two for D okay one of the friends says D as well here on my end I don't know if the same person or not okay it has been one minute 11 or 12 seconds guys okay so I'm sorry how many of them says what but I can just hear one more thing a 6 4 C 2 4 D okay 6 4 C and 2 4 D look like it's going to be C let's see what's gonna get what we're gonna get okay guys X is a negative number which of this expression is it great first thing first if we would like to find some sort of greatest number we are talking about positive number guys you know when you compare negative and positive numbers and that positive number should be the greatest possible among those so for those if it requires you know if it doesn't even require you to just use some algebraic base I would just plug in some numbers guys to be able to you know check them what what's going on X is a strictly negative number we can say like for example negative 1 for X I'm just making up any number you can choose but it has to be negative so then just let's just plug them negative 1 plus 1 give us 0 negative 1 times 2 is negative 2 negative 2 times negative 1 is positive 2 oh that's interesting 6 times negative 1 negative 6 plus 2 should be 4 and negative 1 minus 2 should be negative 3 again I choose the greatest negative number but you can still choose other numbers I just plug in negative 1 because it's the one of the you know easier number to deal with you can choose other numbers as well and at the end guys no matter which number you choose you are going to see that answer should be C because remember you have negative number you have to multiply negative by negative to get positive this is the main idea for this question actually to remind you negative times negative you positive I hope there is no question or if you had you would ask us yeah if you have something you don't get it still via chat please ask and you can just go from okay the next one here from 2017 and was number 11 let's see how many of our friends says which one should be the answer and let's start the timer guys Okay it has been one minute. Let's just wait a bit more. Maybe we can get at least one answer, guys, you know. Just one answer. Well, four people say in. Okay, we get some answers already. Four people says in. Alright, thanks. Let's start, guys. We said so. And in the end, end is here, as you see, at the left of a pole and start here and come back out here, two thirds of the whole length. And Bob is the middle, start at the right of the same pole and crowd to the three fourths here. Alright. What fraction of the length of a pole are Annie and Bob now apart? We need to find that little pink distance, guys. We need to find this. Before we do that, guys, well, we have those fractions. We have two thirds and I mean two thirds and three fourths. Think about that. Which one is easier to work? Since we are going to look for only the difference between those two, the bugs. I really need to subtract one from one of those fractions to get that little piece. That extra parts I am going to show us the green. Either on the right or left, that green part we need to find. Then we can look at, you know, either of those fractions, then take out that green piece to be able to find answers. But we have two thirds and three fourths. You know, two thirds, when we compare, I mean, when we convert, it's going to be 0.67 and this one is 0.75. In that case, it seems that two, three fourths is easy to deal with. And I'm going to do one minus three over four. One minus three over four gives us. Now I got to do, you know, same denominators, four over four, minus three over four, give us one over four. That means, guys, since that middle started and came here, that little piece on the right, I mean on the left, should be one fourth. Okay, this piece is one fourth. And when you look at that end, end came until here. And the total was, you know, two thirds. Well, we have the total two thirds and we need to take out that one fourth to get the answer, you know. Two thirds minus one fourth should give you the answer, guys. We have different denominators. Yes, we're going to get the same denominator here. So instead of doing like regular methods, guys, since three and four are relative to prime, there is nothing between them. I mean, nothing common between them. I will just show you another method. Just multiply the numbers closely to subtract. Subtract, what does it mean? You do four times two is eight. And since we have subtraction, I just minus three times one is three, divided by four times three is 12. Then you should get five over 12. I hope it does make sense. If it's not, you can still use other methods like, you know, regular, getting the common denominator. If that method doesn't make sense, do you just text us via chat, either me or Suhan, just say how. No worries, I'm not going to say who is that. It's fine. Then I can explain one more time in another example. But that method, guys, works only if those denominators are relatively prime. That means nothing is common between three and four. We don't have any common, you know, divider. That is the case. If you have other fractions, it may not really work. I got a new message, but I cannot read that. Okay. Some friend says how. Okay. Let's just explain to you about it. Not just you, all of you, actually. So regularly, let's see, guys. It can work with either subtraction or addition. Let's see, four over five, and I'm going to subtract one-third, you know. As you look at the denominators, like regularly you should do, you know, getting common denominators. To be able to do that, I'm going to look at five and three, and they are relatively prime. That means I have to multiply this first fraction by three, you know, the other number. And I have to multiply the second fraction by the five, the other denominator of the, you know, other fraction, basically. Then I got to do what? Four times three is 12. Then I got to do what? 15. And then I got to do five times one. Not equal, my bad. Five times one is five over five times three, 15. That gives us 12 minus five over 15, guys. So this is how you do regularly. Then what I am saying here is instead of doing that, can we just use the same method with like little shortcuts? Can we just multiply four by three to get 12 here? And also, can we just multiply five times one, which would be five, but also, you know, since we have negative sign here, I'm going to make it negative five in the second step. Then you should have 12 and five here at the end, guys. That 12 comes here and minus five is here. You know, we literally do the same thing, but I'm just showing you shortcut methods, if these are the right ones. Multiply them closely, do the operation. At the end, multiply denominator anyway, because when you multiply the denominator, you get the common denominator anyway. I hope it does make sense. This is not a different method. This is just a shortcut method of the same thing. Do I still have a question? I hope our friend says I get it or I don't get it or something else. Then we can move on. Okay, but I hope you get it. I don't know what you are thinking right now. I'm going to move on. Also, if you say nothing, that means I am going to assume that you got it. All right. Then number six. All right, you got to work with integers, and that's what you will get. Forty seconds so far. so one person who says b is that b yes this one okay we have only one answer right now there's been one minute and 13 seconds maybe a little more time for other friends There's one more for B and two for E. Okay, and two person, two people says E, okay. All right, guys, since we get a couple of answers, let's start what we should get. Look, constant pairs of positive integers, okay, here, with sums no larger than one or three. Okay, I will not keep reading the question. I am gonna say, maybe I can eliminate some answer choices here. Look, seven plus seven is 14, two, seven, nine, it's already one of four, ignore this one. Six plus seven is 13, this might be because sums no larger than means less than or equal to one or three, you know, this is what no greater, no larger means. That works, five plus six, 11, that works, five plus seven, 12 works, five plus six is 11, okay. These are works, at least we eliminate one answer choice. They say, and the question is, or I mean, the result is less than one third. Look, guys, for those cases, some of you may think, let's just divide 77 by 26, or 77 by three to get 26. Or maybe, hopefully, some of you think that, what if we multiply the numerator by three to get the numerator? So when you multiply 26 by three, guys, here should be 78. As you see, this fraction is already greater than one third. So this shouldn't be answer, whoever says B, I'm sorry about it. What about E? Let's just take 26, guys. Look, 26 times three should give you, what, 70, oh, 78, I already did it, my bad. 78, and look at that, it is also, look at that, this is 75, it's still greater than, it shouldn't work either. Guys, we are looking for the smaller than one third. That means that denominator should be, like, maybe 79, who knows? Or this one, like, maybe 80, you know. Any number greater than 78. So we have only two options left over. When we check numerators, both of them has the same thing, 25, of course, you're gonna have 50% chance of eliminating some of them. And when we think of one third in terms of 25, 25 times three, 75, the perfect number would be 75 over 75. But we have only two options here. They say, what is the largest possible quotient of any pair is? Think about it, guys. You are comparing two fractions. Whichever has smaller denominator means that fraction has bigger value, which is larger. In that case, we are looking for the largest possible, of course, after 75, we can get 76, anyway, there is no 75.5 to be able to put as a denominator, so answer should be C, guys. No one were able to give us right answer. I hope you get what's going on. If any of friends couldn't get it, please ask us. I can just try to explain again, or maybe in different way, hopefully. Okay, no one text me anything. So I believe no one text you as well, so I'm gonna move on, guys. A lot of people change their answer from B to C. Oh, okay. One of the friends says, maybe this doesn't make sense. Explain one more time, hopefully differently. So for those type of questions, what you should do here, read the question and make sure you get what's going on. Look, we have given two piece of information first. We are gonna look for the pairs of positive integers. Yes, and we are gonna use those two positive integers, pairs means two of them, to get some sort of fraction. So when you add those fractions, because sum means you add them, but their total shouldn't be greater than 103. That's the first reason was, guys, I just add the numbers. You know, when I add 27 plus 77, I should get 104, you know. Since we both know that 104 is greater than 103, we eliminate that. If you add other numbers, guys, you cannot get, you know, greater than 103, they should be fine. I hope this piece is fine. I hope you get this part. So they say, and the quotient is smaller than, first thing first, quotient means you gotta divide, you gotta use division, guys. That result should be smaller than 103. And we are looking for the greatest possible number with that value. Okay, so if you want, you can compare like some of the numbers with the same numerator or same denominator. There are like all different ways. So you can say, so since B and D has the same denominator, I can compare them by looking for numerators. We have 26 here and 25 here. We both know that B is greater than, you know, D. But the question here is, if B is any fraction smaller than one third. So we have 26 over 77. There is no easy way to, you know, find out if that number is smaller than one third or not. But at least we can think of 77 is not exactly divisible by three. We can think of 78 or 75, you know, since 75 is one product of three anyway. When you divide 25 by three, you should get 25, guys. But we have 77. That two extra should gives us 25.6. You know, 0.3 for every one of them. So you should have 25.6 divided by 77 is exactly one third. But we have 26 over 77. This number is even greater than one third. They say smaller than. This doesn't work. So this one is little bit, you know, close, but we still don't know if it's gonna work or not. Okay, so then you gotta check others. We have 25 over 76 and 26 over 75. Here you have 26 over 75. We both know that the number should be 25 over 75 to be able to get, you know, one third. But it says 26 over 75. Well, it's also greater than one third. With that has been said, you shouldn't choose E or B anyway. At the end, you gotta compare only two of those fractions, either C or D. Well, from here, I can say, guys, we know that if we had, since we have 25 on the board, you know, numerators, we can work with 25 instead of denominator. When you multiply 25 by three, you should get 75. So right now, you will compare those fractions with 25 over 75. We know that both of the fractions, both of the numerators are less than, I mean, both of the numerators are greater than 75. That means both of the fractions are less than one third. The condition works, but which one is the largest possible? Well, the largest possible should be 25 over 76 because this one has, you know, smaller compared to 25 over 76. I hope that makes sense right now because if one of friends ask, I don't know about it. You can say yes, or if you say nothing, I assume you get it. I'm gonna move on. There are a lot of words for these questions as well. This is kind of tricky as well, guys. I'm gonna move on to next one. Number seven, please check this one. This is kind of a logic problem, but still, you can figure it out, I'm pretty sure. Let's start time-lapse. It has been one minute and I'm not sure if we have any answer yet. One person says E. Alright. E gets one point. What about others? another person says E. All right I'm gonna put E is the winner then and it has been one minute and 40 seconds so we can start guys. All right guys it's Andy that person that enters all the digits from 1 to 9 into cells here. We have similarly 1 2 3 and 4 is you know placed. Two numbers are considered neighbors if their cells share an edge. Edge means guys those little sides you know I can say one is going to be neighbor with like let's say there's X and that's Y. Right now I can say one is neighbor with X and Y that's it no other numbers because that piece in the middle number only share vertex guys that's middle point. But if they share edge that means these little sides I'm showing with the red lines. Anyway we have three more people who say E. All right look at that they all get E I guess. So E is now another person says E. All right after entering all the numbers he not said the sum of the neighboring neighbors of 9 is 15. What's sum of neighbors of 8? Look guys since we have given 9 and neighbors sum of the neighbors of 9 is 15 we can play with 9 like let's see 9 is here if it is possible. In that case neighbors of 9 would be this number this number and number in the middle as you see. But 1 plus 2 is 3. To be able to get 15 well I need 12 more to be able to have 15 as a neighbor of 9. Well it's not possible guys they say you're gonna put all of this from 1 to 9. 12 is not really possible choice for us to put. Anyway we can just put 9 in another space we can just put 9 here. With same logic 2 plus 4 is 6. I had to have another 9 is here to get 15 you know. Well it doesn't work because we already choose 9. Maybe I can choose 9 here in the you know here. By the way you can ask me sir why don't you put 9 in the middle? Guys if I put 9 in the middle that means all of those four numbers are going to be neighbor of 9. That's not really a good option because we cannot get 15. 15 is not really a big number. That's the reason I first think maybe we should just put 9 in the you know those sides with less amount of you know neighbor. So 3 plus 4 is 7. Well then I can just put here 8 because 7 plus 8 is 15. Yep in that case 9 is here and 8 has to be in the middle guys. Well after that I can just place all other numbers. It doesn't matter which one I put where. So we have missing 5 maybe goes here, 6 here and 7 here. No one cares you know. But what's the sum of the neighbors of 8? When we are talking about neighbors of 8 I'm gonna use those little pink color. All of the numbers shared with that pink side those numbers are neighbors of 8. So we have 5 plus 6 plus 9 plus 7 guys. So instead of adding 5 and 6 you can say 6 plus 9 is 15. 15 plus 5 is 20 and 20 plus 7 is 27 guys. So reset E. I am proud of you guys. You got this. I hope other friends get it. I'm gonna move on. Okay I get more check that means I will be arguing. All right then we get number 8. Let's see what we get from here. It has been two minutes. Soham, do you have an answer or not yet? Anyone said anything, Soham, or not yet? Okay, I got some E's. Oh, how many? A lot of them. Okay, it looks like many friends said E. It can be. Alright, let's see if it's E or not. Alright guys, so here is the deal. Elisabeth, or that Elisabeth, whoever person is that, had a large bag of 60 chocolate. She started by eating 1 tenth of them on Monday, then 1 ninth of the remainder on Tuesday, and 1 eighth of the rest of them Wednesday and so on. Look, she started with 60, yes, and she started eating 1 tenth of them on Monday, we know that. So then 1 ninth of the remainder. So if we are talking about, you know, just remainder of the 1 ninth, so we are going to multiply 60 by what, guys, here? She already ate 1 tenth, so we get 9 over 10, am I right? On Monday. Then with the same logic, she get what? Then she ate 1 ninth of the remainder. With the same logic, I'm going to get 9 over 8. I mean, 8 over 9, my bad. Then another, we get 1 over 8 and 1 over 7. I mean, 1 seventh of, you know, 1 over 7 to the rest of today. Then we're going to get another, not 8 over 9, 7 over 8, my bad. 7 over 8. And if you just keep continuing with the pattern, guys, it says until she eats half of the chocolates. Well, that should give us 1 over 2, you know. At the end it says, how many chocolates does she have left? Do you think something is going on here if you just look at the fractions? Well, if you just look at the pattern, guys, you can say easily that some of the numbers are going to cancel each other, you know, at the end. So then we have left over with 6 over 10, which is 6, guys. Also, if you just, you know, look for every single day, you're going to figure out that each day that Elizabeth person, that kiddo, she eats 6 of the chocolates. Because she started by eating 1 tenth of them. 1 tenth of 6 is, you know, 6. Then she had 54. Then she ate 1 ninth of the remainder, which is also 6. 1 ninth of 54 is also 6. We took out another 6, you know. They get 48, I get 18. You can just go, you know, step by step. At the end, you're going to get, until you see half of the chocolates remaining from the previous day, you get 6 at the end, guys. I hope it doesn't make sense. But if you just keep working with the numbers one by one, it might take some time. You really need to figure out what fractions you're multiplying. After you start with 60, what's going on? After 1 tenth of them are 18, we have 9 over 10 left from the total. Then 7 over 9, I mean, 7 over 9, yeah, 8 over 9 left from, you know, previous day. Then 7 over 8 left from previous day and so forth. Okay? I hope you understand the answer. Any questions for us here? Cool. Okay, we can just maybe solve one more question really quick. It's almost time, I guess. All right, check this one really quick, please. This one from 2005 and 22. It says it's time, guys. If you can kind of share, I hope at least one of the friends says something. Or I can just start, actually, since it's time, guys. Look. In each of the four small squares shown in the picture, yes, different nature odd. Odd number, less than 20 was written. In that case, first thing I would just say, I would just write that down, all of the odd numbers, less than 20, guys. These numbers are 1, not 2, what am I doing? 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19, guys. You know, because this is less than 20. Only one of the statements below is true, which one is it? Let's check one by one. The sum of the numbers that are inscribed in the square equals 66. Since 66 is a really big number, let's just assume that we are using all of the biggest numbers. So we have 13 plus 15 plus 17 plus 19. When you add them, 13 and 17 give us 30 plus 15. 45 plus 19 should be 64, guys. As you see, the biggest number can be 64, but this is 66. No, it's not possible. It's not possible. Second statement, the sum of the numbers that are in the squares are 12. Well, this is a really small number. Maybe we can just look at the smallest numbers. Look, 1 plus 3 plus 5 plus 7. We have 7 plus 3 is 10 plus 5 plus 1 is 16. As you see, it cannot be 12. Nope. CCS products of all of the numbers in the squares are equal to 2005. For those, since we are looking for the products to be able to get 2005, guys, I would work backwards. What I mean here, what are the factors of 2005 we can use? Look at 2005. It is divisible by 5 at least. It's obvious because the bounce digit is 5. When I divide 2005 by 5, what number I should get? Let's see. I don't really know what's that. I will use paper. I'm going to use calculator now. 5 should give us 401. One friend says, give me the chair. You are smart. I know. 2005 is, I can just put it somewhere here. 401 times 5. If you think about 401, I don't think 401 is divisible by another number. We can just test some of the numbers like 13, 17, whatever. We definitely know that it's not divisible by 3, not 4, not 5, not 6, maybe 7. I don't think so, to be honest. It shouldn't be the case. Some of the friends are sending us. 401 is prime. Thank you. I don't check that, but it must seem no other divisible number. Since 401 is prime, we cannot divide anymore. Right now, they are saying the numbers, all numbers less than 20 was written. 401 is definitely greater than 20. It's not possible to get products of that equal to 2005. This says products of the numbers on each diagonal is 21. Look, we got to look at each diagonal. Still, we got to look for the factors of 21. We can have 7 times 3 or 21 times 1. That's it. Well, we cannot put 21 since 21 is already greater than 20. Well, it's not the case. And it says that the answer has to be on the last one, which is E. Any questions for today? I hope you get it, guys. Even I have my idea. Okay, that means take care, guys. This is the end of our session and I'm going to see you next time. Okay, bye.

mathematical problems

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order of operations

properties of numbers

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