So yes, last time we did uta sutra multiplication. Any doubts? So Talani you were not there that time pushpa I hope so any questions you have? No sir it's clear it I think this was the most easiest thing right? Yes banan do you found any difficulty in solving any of the question? No sir, it's it's one pattern which we have to follow and thus it becomes very simple. We don't have to now sir remember anything any base subbase but but the point remains the same. If questions come like triple 9 into triple 9 then udwa sutra is difficult whereas in base method or niculum method you get the answer within 1 second right so that this both the methods have their own importance and this we will realize when we'll solve more special cases so today we'll start with special cases or specific techniques. Now just observe these two numbers. 37 33 44 46 52 58. So do you find anything common, uncommon, anything usual, unusual, anything which you find striking in these three numbers? These all questions are of multiplication. Two digit two digit by two digit. The third first digit same. Okay. Number of units place. Okay. What else? Yes. Ta. Infinently similar number. Which one? Similar number. Anything else? 334. So since I use the word similar unsimilar so you all copied the same word. Yes, this is similar. You don't base or same. Which base? So tense. Okay. Base is same. That is the reason our tense position is same. What else? The sum of the matter. H tell me. Okay. Now understand. See, start thinking beyond that also. Now if you observe 7 + 3 is 10. 4 + 6 is 10. 5 + 2 is 10. This is the first similarity between all these three. Second, this is common. This is common. This is common. See I told you in the very first day visualization which will be leading to logics reasoning. So if you are going Now we can solve these three questions by 2 into two multiplication the way we did last time but we'll take at least 30 seconds for every question because you'll make it 9 Then 9 + 21 30 then 21. Then balance here you'll have 16 you'll have 24 + 16 as 40 and 24. Here you'll be having 25 + 50 + 16 and then you'll be saying 22. This is 20 24. This is 3016. So that means you need at least 30 to 40 seconds to solve each question. But if you were able to visualize this, if you were able to understand this two points, then things would have been different. So what difference would have been? You were able to the do the same question in just 3 seconds. How? See whenever these two conditions are met one unit place digit just hold up. Uh that is a big issue. Now when unit plays digit to become 10 or power of 10. Hello. Let's write 10 only here. And second thing, the previous digits, I'm not saying tense position. Please understand the meaning. The previous digit or digits are same means identical. Then this is a case of anta dash per 10 multiplication. Pa 10 multiplication. And now we are able to just if we are able to visualize this finely. So what how do we solve it? The left hand side we multiply number with ikadica of number. Now what do you mean by kadika of number? Means number and number plus one. And here you are doing simple multiplication. I'll take the example again. 37 33. So 3 what is ikadena of three? That is 4. Ikadana means one more. So 3 into 4 is 12 and 7 into 3 is 21. Finish. 44 46. So 4 into 5 is 20 and 24. We did this 52 into 58. So 5 into 6 is 30. 2 into 8 is 16. Finish. So see, but it is only possible if you are able to visualize and you are able to find these two conditions are being met or not. Correct? Let's take two three examples more. Oh 62 68. So 6 7 are 42 16. How much time verbal otherwise we if we go for udwa sutra. So yes 36 60 and then 16. So 61 42 understand why it is happening. It is happening because here it is zero. This this concept we'll be learning in scaring also 65 65. So again 42 25 62 68 42 16 only one case if it is 29 21 so 3 2's are six but here it is not you're not going to write nine remember on the right hand side I I told you last time also whenever two single digits are multiplied on right hand side you have to write the answer in two digits Because 2 and 8 combination is giving you 16. 3 and 7 is giving you 21. 4 and 6 is giving you 24. 5 and 5 is giving 25 and others also. So only in case of 1 and 9 or 9 and 1, we may make a silly mistake. So please remember if it is 1 and 9, it is 0 9 or 9 and 1 0 9. Is this clear now? Yes sir. Yes. Ta. So no need of paper and pen for this. Try to answer immediately. Try to do it mentally. Then only you'll be 38 32. Yes. Give me answer. 16. Sorry. 1216 sir. 1 2 1 2 1 6 Very good. 3 4 are 12 and Yes. Yes. Very good. 47 43 20 21 Correct. 20 21 3931 pushpa 1 29 very good 129. So only this is the case where you have to take care otherwise everything is easy right? Only for nine only sir only for nine one and nine combination only now okay one or nine or nine or one it's for the same thing because the answer when whenever you multiply two-digit number answer has to take two spaces on left hand side also it is two spaces but since zero has no value on the left side so we ignore okay right 57 53 3 3021 82 888 7216 726 clear any doubt up till here no sir now I I say 108 102 now see here 10 is same if you remember I didn't use the word 10's position yeah I said the previous digit or digits. Got it? So that means even if I have three digits here, total three digits are there and this is 10. So 10 and 10 are same. So 10 into 11 is 110 16. Finish. Got it? And otherwise if I give you this question yesterday, if I gave you this question, you were doing this one then zero then 10 then 0 then 16. So 6 1 0 1 1. So see what is easy or if it is normal multiplication as in our school curriculum then we have to do three time multiplication and then addition. So just only only remember one thing observation is the key to success in Vic maths. You have general techniques you have special cases. Special cases are nothing but they have been taken they have been derived from those general cases only. But that makes your work much simpler. Okay. Then 112 118 1316 216 sir. Yes, we we did this was the first topic we did 11 11 into 12 is so one 1 and 11 9 209 sir 1 13209 Yes. Yes sir. 13209. Okay. Pushpa. Push by you're not answering. What happened? It's easy sir to do. Actually I tried this also many times but it's hard. But now in this very easy sir this one. to I I I could not understand why you tried and because because sir actually this for the students when I'm telling they will do in two steps same as two steps then they need to add it will goes to three to four steps but it's very easy see the thing is now see you have to it it all depends how you are teaching how you are explaining to them if you make them realize what is the thing if you have to make them visualize See what we do we start with the questions immediately why I gave you three questions first and asked you are there any similarities are they let the students think something if you'll not think if you'll not allow them to visualize right from the first class when we started with digit sum at that time only I said visualize nine ignore nine so what was that that was making you understand what do you mean by visualization second Class 11 multiplication start visualizing that this plus this if this is more than 10 it will be added initially. Correct? So visualization has to you have to make them visualize because it is not like a backus you have three years here here you have just three months no parent easily students can do sir now I no you understand no parents will give you more time here because the competitor is not giving a much time in the ads they are giving within two months 3 months so now you cannot take more time also So you have to start right from the day one. You have to make them visualize things. Correct. Okay. 125 125. Uh 15. Okay. Now similarly bigger one 132 138. So 13 into 14 182 1826. Yes. Yes sir. 18 16. So this this small numbers up till 14 15 12 13 this multiplication should be very much right. I don't say everything you should remember but these small 127 123 56 1564 1561 156 21 correct Okay. 1337. So 182 182 49 21 2 21 182 2 1 Okay then 167 163 So 16 into 17 first do it with any method to 17 to 2 27221. Got it. 145 145. So what is 14 into 15? 210. These small calculations you should remember. 211025. Correct. Now let's come some tough 1 84 186 3 4 2 4 3 4 224 So Uh do you have your own books or you are having some franchisee? Sir, do you have your own content material or you have franchise? Uh I am working with the sip sir. Okay. Okay. Okay. 193 197 38027 2 1 sir. Sir 21. Correct. 3821. Very good. 1 1 08 1 2 Now see again 11 1 into triple 1 it is concept of 11 also it is concept of triple 1 also we will do triple 1 later so if we understand this or even if we don't know anything now if I do it together I'll take seven steps. If I do this only I'll take only five steps. So at least two steps are saved. 8 2 16 anybody can do immediately. So 1 2 3 2 0 1 6. Okay. 1 2 2 1 0 6 1. So 1 six sir. Sorry 1 six. Okay. Now clear so this was the reason I said I didn't say 10's position it was digit or digits any any number of digits can be there one book question is there 5125 5125 so 512 is common five I know do it so at some places when you are not able to do mentally then at least your steps are decreasing so this is a benefit effect of antariki. done. Yes, sir. Okay. 1251 175 First digits different. Yes. Uh, who did it? So, 2187 21 87. 025. No. How you did push one or who was answering this one sir? First second one 125 175 see now only tuna was correct. See again we have to visualize 5 + 5 10 one condition is met but 12 is not equal to 17. So we cannot apply 10 multiplication. We cannot but observe again 25 + 75 is 100 and now 1 is equal to 1. 100 is again power of 10. So that time I was writing if power of 10 is. So what will happen now? Things are things are same here. Earlier we were taking two digits now four digits have to be done. Any question? And here the same thing 1 into 2 2 25 into 75 is 1875. So this is your answer. One more example I'll give you. 298 into 202. Now 98 + 2 is 100. So this is all observation. If you if you are observing you are able to solve questions easily. Otherwise normal steps are there. So 2 into 3 6 and 98 into 2 196. But I have to have zero because this answer has to be in four digits. Clear? Any doubt? Now just write down just see if you Have any question on this? Yes. Is it clear or not? Yes sir. So earlier we were looking for 10. Now we are looking for power of 10 that is 100. Both the things you have to now you'll say sir what did we save in this? See if you are doing it in normal way 125 and 175 you have to have five steps very clearly. You need to understand this. But if you are observing this so in 25 and 75 you are doing three steps. So at least two steps you are saving. But in this question instead of doing five steps you are able to do within verbal. So time at times you will be doing mentally also at times you'll be saving steps also. Both the ways you are in benefit only. Correct. 437 into 463. 20 23 23 31 2 33 1 1 Okay. 645 into 655 So 42 2475 422 42475 very good 29 229 into 271 Run 62059. Yes. Others everyone got that or not? Yes. Okay. Uh 358 into 342 So one and two 2 4 3 6 any doubt? Again 1 357 into 1 3 4 3. See 57 and 43 makes 100. 13 is common. Sorry 13 into 14 182. And now we just have to do 57 and 43. One night 224. Is it clear now or any doubt? No sir. Anything? Okay, one more 1271 1 22 29 Aruna I couldn't see. Yes, Aruna. Is it clear? 156 2 059. Okay. Yes, others. 1265 1 2 35 156 2275. Okay. 56 2275 Six 1 98 into 1 102. Now this is verbal. You should not take paper and pen. So this question is totally verbal. 11 into 12 132 and 98 into 2 is 196. 196. So zero zero. Correct. Because you have to do that, right? Okay. Let's 5 4 4 2 into 5 4 5 8 So 29702436. Okay. Yes. Others. Yes sir. 29702 436. So any doubt now? Okay, let's move further. Now now just now I did. Okay. Now it is 1 6 5 7 1 3 4 3. See again try to understand. 7 + 3 makes 10. But 10 is not possible because 1635 is not 57 + 43 makes 100. But it is not possible. Now 657 + 343 makes,000. So 1 into 2 and 657 into 343. So you are just saving two steps. But in some cases you might be able to do verbal do it clear. This is called as antia thousand multiplication. The previous one was 100 multiplication and the first one was 10 multiplications. So this is what is the difference. Is it clear now? 32 5181 triple 2 53 5181. Check. What's Yes. Did you check what is the answer? Triple 25351 sir. Correct. Aruna I'm doing please 5351 correct somewhere we you have made silly mistake Yes sir. Did you find your mistake or not? Yes sir. So where was it that mistake? Actually last second step from last I did uh 7 4 are 28. Okay. Instead 4 2 3 4 4 7 66 X the 2011 17924. Yes sir. 2011 1792 44. Okay. Now so up till here any doubt let's practice two three questions and then we'll move to the next topic very easy topic for today It's a 12 229 836. Okay. Yes. Go on. Yes sir. Same answers. Okay. Did you get 12 22 29836? Okay. So clear everything? Yes sir. One more. 1265 1735 2194775. Good. One last question. That's to 1771 1229. Nine. 21 765 91 7659. Okay. So everyone is clear. Anyone has doubt? So this was and dash 10 multiplication, 100 multiplication or 1,000 multiplication. Further also can be done. Any doubts, any questions you have? Till now we have done two types of general multiplication and first case of specific technique or special methods. Any doubt in any question you have please ask before we move further. It's all clear. Yes sir. Okay. So now we'll come to the most popular method which you can say it is very interesting. Very interesting. When multiplier is 99 999 or means series of nine and you can learn within one minute very easy to do I'll just give you one first question I'll take three questions simultaneously to make you understand 9 into 7 99 into 77 or let's say 75 triple 9 into 6 2 3 first see there are two ways to do these questions though the method does not is there is not big difference But if you are teaching this to class third, fourth then first we'll do for them and then another. So very first thing whatever is this number other than triple 9 applyad a on that means less one. So a unit on this side and here 9 minus new number see now 7 - 1 is 6 and 9 - 6 here only you see 9 - 6 3 got the answer 75 25 - 1 is 74. 9 - 7 2 9 - 4 5 finish. 6 2 3 First is -1 6 2 9 - 6 3 9 - 2 7 9 - 2 7 and if you are doing it with see there are again understand what is this what we did the previous antiash that was a special method for ura sutra But this is a special method of basically your nikulum. Remember if one number is 999. So what is what happens? Minus one. So that is the reason you are subtracting one. On the other side what you are doing whatever now whatever is number one all from nine last from 10 here that you are multiplying. So you need to understand these small small things. So this is basically based on nikilam sutra. The earlier what we did was based on uh ura sutra. So every specific technique will have a background of either one of the method. Did you understand or not? First thing so please write down these three examples. First on the uh left hand side you find ekayun that means one less than the previous and on the other side you are I'll tell you what this is for class three students first write down these three then I'll tell you we will solve five questions then I'll tell you what other method is Yeah. Done. Okay. Now see what in actual if you do understand we are applying a Kai You know on this compliment on this side but small children class third they don't understand compliment see again take the example 9 into 7 don't touch 9 what is one less six what is complement of 7 99 into 75 let's Talk again. 74 1 - what is complement of this? 9 - 7 2 10 - five four five four five four. So but now complement word a class three student does not understand because you have not taught them nikulum remember to fourth you have not taught them nikulum even subtraction view we are not t teaching them nikulum we teach them addition first then we teach them cross multiplication then we teach them subtraction that two the other so they cannot understand complement initially. So that's why for them first subtract one and then 9 minus that new number. But you have to understand this is nothing but complement all from nine last from 10. Clear any doubt? Yes. Now we have three case studies in this also. Every time it is not that the same kind of numbers would be coming. Case one. Case one is a general one which we just did. When number of 9th are equal to number of digits of other number. For example, yes I did seven. So 7 was single. So 9 was also single. 77 was double. So 99 there were two 9ines. Means number of digits and number of nines are equal. Try to give these answers now verbally. What is 30 into 99? Yes. 30 into 99. So 29 and then 70 30 compliment yes bhan 57 into 99 Aruna 56 and what is comp and what is complement of 572 43 42 compliment All from 9, last from 10 questions. Why 71 into 99 fast? Everybody is busy writing. You'll be getting 7029 7029 53 into 99 50 to 48 47 Yeah. Now we are finding just complement of this. It cannot exceed. Now 86 into 99 85 8514. Correct. Now you are on the track. 37 36 63 63 63 sir correct. Now last 89 into 99 one 881 811 sir very good. Now with triple 9. So first 123 into triple 92. It's double sir. 1 2 2877. Clear. 3464. Correct. 5 6 7 4 33 876 124 124 What? 124 sir. 875124. I can't understand sir. What is the answer? 875 1. Uh 1 2 4 sir. Only 1 to476. No sir. Oh sorry. We need to add one more sir. What are you doing? Eight. What there? Six three digit into three digit. You are saying it's 124. So it it is 875 and then 124. No or only 124 is the answer. What happened aan? What is 7 into 9? Three. Yes sir. Three and nine is ones. Wow. 9 sevens are three. That means you're not listening. Sorry. No sir. No. No sir. I'm doing how I can guess. What? What you are doing? All questions. Sir, what? What is the first digit? Seven. How? Seven. See? Six. 63 is the answer or three is the answer? Tell me. 63. So you are saying three. Here also you are saying only 124. Who will say this 875 before? Three digit into three digit will be less than that. Nice. You didn't understand that means bhan got confused and you also got confused. Yeah. Yes. This one. Everybody take two minutes. Don't worry. Relax and take two minutes. Water break. Yes. Now fine. It's fine. Yeah. Yes. Did you understand what what was the Okay. 167 into triple 9. Yes, Aruna. 6 m 8 2 3. Yeah, sir. 1 66 833. Daruna, how? 823. 833 83. We will uh if I'll not take care the you'll speak wrong every time. Huh? 8 654 into triple 9 653 653 346 sir 3 6 Yeah 3 3 4 6 I understood somehow that was a concept of compliment we did there then we use that in subtraction. Then we used actually we had to use in multiplication but we were not using because we were directly subtracting 100 minus and all those it was the same there also right? Okay. Now if it is 4 9ths into 1 2 3 4 123 876 Yes sir 1 2 33 8766 clear okay into 5 04 500 499 96 53 949 Sorry 5003 4996 clear Oh, 31 4 0 3 1 3 9 6 6860 sir. Clear. Yes sir. See both the ways are open for you. One is find the complement. 6 8 6 0 one less. This is the answer. Correct. Otherwise first put this and then subtract from this. Both options are within. Now it is clear. Now case two. Case two is number of n are less than number of nines. For example, 6 into 99, 72 into triple 9, 712 into 4 9. So what will you do? Yes. Kalyani. Tuna. Aruna. Pushpa. Goan. Sir, we need to less sir in case. So I I'll give the answer in one minute. But first I'm just thinking gas 05 94 4 071 928. Why don't you think? Yes sir. Zero or balancing? We did balancing in Yes. Yes sir. So why did you do here? Huh? Yes. 711 9288 I mean 0 711 no zero will not be there 0 is just written to find so that you don't forget this nine if you'll not write zero you will never be able to find this you understand now yes this is wrong Huh? You people don't think that is a problem. Ch. Let's do it. So, we'll be doing with triple 9. What is 69 into 9? 68 931 931. Yes. I don't know. I don't know. Which 71 70 929. Yes. Ta. Aruna pushba no answers from your side. So 7029 70929 70930 86. 85 914. What? 85 914 85 914 914 91 sir. 914. Sir 969 3 now with four nines. So first is 1 2 3 1 2 1 2 2 9 8 8 7 1 2 9 8 7 3 46 5 3 4 5 9 9 6 5 4 5 4 762 7 6 1 9 2 3 9 1 9 8 3 8 7 6 1 9 2 38 876 875 9124 8 7 5 9 24. Okay. 234 23 59 7 2 33 9 1. How? 235 23 97 6 anyone has any doubt please ask clear now comes the last is number of number of digits ns you can say are more than number of nines. 33 into 9 313 into 99. Now what will you do? And think before you speak. Yes. Tell me. Daruna, Aruna, Kalyani, Bhan, Pushba. Yes. Yes. Aruna. Okay. Thank you. Now this is very diffic typical first I'll tell you for small kids if it is 9 make it 10 minus one If it is 99 make it 100 minus one. If it is triple 9 make it 1,000 minus one. This is one way and this is also very useful. Why? So for this you can do 3 0 - 33. Correct. This is one way 313 - 313. And here uh whatever the question may be you can do. This is one approach. Why this is important? Because okay let me erase everything. I'll again write wait see when we are doing let's say 31 into 99 so the you can do 3 1 - 31 means this will be converted to 10 - 1 313 into 99 we can write it as 31300 minus 313. Why? Because we are making 100 - 1. Now this mental method is important also because whenever they'll be getting not only 9 or 99 let's say they got into 19. Now multiplying with 19 is tough but multiplying with 20 - 1 means 2 is easy. 660 - 33. If it is 33 into 39. So 33 40 - 1 means wherever you have 9 in the uh unit place. So you can do like this. This is a part of mental maths when when they learn. So one way one approach is this and with kids you can follow this with small kids but if we have already taught compliment now what to do now it is yes it this cannot be done mentally you need paper and pen so I'll explain let's say first question was into 9 and another question is 91 1 into 99. So what were we finding on this side? We were finding complement. Correct? Now I have only one nine. So I'll underline one digit. I have two nines. I will underline two digits. And what I'll do in this side is complement not of number but of underlined number underlined digits you can say. So what is complement of three that is seven. Here what is complement of 13 that is 87. Any doubt up till here? No sir. On the on the front side on the left hand side you will do number minus non underlined + one not underlined + one. So here not underlined is three plus 33 - 3 + 1 4 to 29 297. You can check 330 - 33 is 297. Correct. Here it is 313 minus what is not underlined 3. So 3 + 1 4. So this becomes 309 87. So this is bit typical uh other than two cases. So we should not thank thank god you nobody said add zero because then the answer would have been different. But yes, I would. Now I'll tell you I even I have not taught you one method is not ever defined ever thought ever claimed ever you can say told but in my last batch 52nd one teacher did it from Delhi itself but I don't want to reveal right right Now that is very easy for this third case. Very easy. Nobody thought like that. Even I have I never thought no teacher ever thought no book has ever claimed not even Swami G's book. So we we thought we should release in a big way. So I'm not disclosing it to you. The meaning why I'm telling you is these two number these two methods your ura and your nikulum try to devise your own concepts from it. So many people are doing if in between number is this do this this are making shortcuts. So how they are able to do because they practice on that now. So you have to practice on that. Right? Is this clear? First write down because then you'll be having a problem. You can write down first. What happened? Nothing. Did you understand or not? Listen, I ask you come. I'll show you more examples first then you'll practice. Let's say I take a number 1 2 3 4 into 9 1 2 3 4 into 99 1 2 3 4 into triple 9. Now see the difference one digit so one digit so what will I have 123 minus oh sorry 1 2 uh 3 4 minus 123 + 1 124 and what is 6 so 1 1 1 0 6 Is it correct? Check. Check. 1 2 234 - 124 is triple 1. Correct? Yes sir. Ch. Good. Now second case 299 2. So 1 2 3 4 - 12 + 1 is 13. And here 34 means 6 and 6. So this will become 1 2 2 1 6. Is it clear? Is it uh correct? Check please. Correct sir. Third case. Now here three nines. So 1 2 3 4 minus only 2 because 1 + 1 is 2 and on the other side 2 3 4 means 7 6. So answer is 1 2 3 2 7 6. Yes or no? Yes sir. So your wish this method once you understand then it is very easy and the last case I could have given you 1 2 3 4 into four 9ines. So you already did it right? Anyone has doubt? Yes sir. The last one. Can you tell again? Which one? The last one. 1 2 3 4 into 999. Three lines. This one. Yeah. So what is the complement of 234? It's 7 6. Correct. And what is not underlined? only one. What is the concept? Total number minus we need to add one every time add one. Yeah. I'll give you one question. Solve that then everything would be clear because practice will only make you clear in Yes. Do it. Let's hear the answers. Just Sir for first one 58 five sir second yes others first answer first answer the second one 9868 let everybody answer first second 7 It's 77 77 735. Sorry, now it's 977 7735. 98 6 235 sir. 98 6 2 35 Then so 9875 1 2 3 5 987 7 5 1 2 3 5 3 Yes, Aruna. Check please. Pusha now. Do you have any questions? Yes. Anyone? Yes. Any doubts? Any pointi? No sir, it's clear. I completed clear. Yes. So it we we did two special cases today, right? One two more are left. Then chesh rest you can devise your own also. It's not that other than this there are not but these are major ones. Tuna any question? topic. Yes. Push. No sir. Push. Second time move. Huh? You don't ask any questions. Sir, it is clear. It is clear. So what the teacher said, sir? 62nd batch the easiest to we we we are thinking on making a like declaration on that so just announcing or just think I didn't think over it and rest I know what people are doing they are just I'm also teaching from four years never gave a thought like But very good. See when everybody is working then you also feel good to teach otherwise it's routine. Yeah, thank you. Okay. Good night, sir.