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We are going to begin with my favorite number 5 and how things get easy with it. Fun with Number 5 Rules: This is normally used for multiplication and in this case both the units in digit i.e. the last numbers are 5. Also, the beginning numbers are same. You will understand this as you see some examples. 25 x 25 = 6 | 25 (This one's easy, right?) Explanation: As you see both the ending numbers are 5 and also they are beginning with 2. So we have satisfied both the rules. Now how do we get 625. The first step is that you write down 25 on the right hand side. Why we do so? Do not think about that. Just write 25 on the right hand side. Then we take the number on the left hand side (of 25), in this case 2. And we multiply it with the next higher digit number i.e. 3. Therefore we get the answer 2 x 3 = 6, which we write down on the right hand side. So we finally get the answer as 625. 45 x 45 = 20 | 25 Explanation: As this one satisfies the rule too. Both the left numbers are same - 4, and the ending numbers are 5. So without thinking the first thing we do is write 25 on the right hand side. Then we take the left hand number 4 and multiply it with 5, hence we get 4 x 5 = 20 (the left side of our answer). 135 x 135 = 182 | 25 (slightly tough) Explanation: Both the rule has been satisfied. We write down 25 on the right hand side. We multiple 13 with the next higher number and get 13 x 14 = 182. And there we have our answer 18225. Easy right? Slight Modification to the Number 5 method Rule: The ending number should add up to 10. Infact in the above case too the ending number 5 + 5 added upto 10. And the left hand side number should be same. This is very similar, so lets look at some example. 23 x 27 = 6 | 21 (I am sure that,not everyone finds this simple). Explanation: Ending numbers 3 + 7 add upto 10. So the first rule is satisfied. And both the number on left hand side are same i.e. 2. First thing you do is, you multiply 3 x 7 = 21. And then like earlier you multiply 2 with the next higher digit number i.e. 3. And we get 2 x 3 = 6 on the left hand side. Final answer is 621. 62 x 68 = 42 | 16 Explanation: Both the rule have been satisfied. We multiply 8 x 2 as the add upto 10 and we get the left hand side of the answer 16. Then we multiply 6 with the next higher digit number 7 and get 6 x 7 = 42 Fun with number 11 Now we try the tricks which can make calculation by 11 easy. We will also learn how to multiply by 22, 33, 44, 55, 66, etc. As it is nothing but 11 x 2, 11 x 3, 11 x 4, 11 x 5, 11 x 6 respectively. Steps involved: 1. To multiply a number by 11 put a 0 in the beginning and at the end of the number. 2. Starting from the digit’s place add two consecutive digit. Remember to go from right to left. Examples 342 x 11 = 03420 = 0+3 /4 + 3/2 + 4/0 + 2 = 3762 Explanation: We added 0 at the end and the beginning of 342 and got 03420. Now, as you see we broke the number by taking each consecutive number and added them. The first part is 0 + 2 = 2, the first number of our answer, then we took 2+4 and got 6, second number of our answer and so on. 4932 x 11 = 049320 = 0 + 4/4 + 9/3 +9/3 + 2/0+2 = 54252 Explanation: In this case when we add 3 + 9 we write down 2 and carry forward 1, we again get 4 + 9 + 1 we write down 4 and carry forward 1. We add 4 + 1 and get 5 In the above case if we need to multiply by 22. Then we first multiply by 2 and then by 11. 312 x 22 = 624 (312 x2) x 11 = 06240 = 0 +6/6 + 2/4 + 2/0 + 4 = 6864 Explanation: Here we are trying to multiply with 22. Hence we first multiply the main number with 312 with 2 and get 312 x 2 = 624. And then multiply the same number with 22. Rest of the steps remain the same. 312 x 22 = 03120 x 11 x 2 = (0+3 / 3 + 1 /2 + 1 /0 + 2) x 2 = 3432 x 2 = 6864 Explanation: This is the exact same sum as above. In this case you multiply the end answer by 2 and get the exact same answer. Similarly we can calculate any number into any multiple of 11. Even 11 x 11. We just repeat the process twice. Fun with number 9 Part A (Original number and the number of 9 is same, for example 234 x 999 but we cannot use this method for 234 x 9999 or 234 x 99) Rules: As usual there is a left hand part of the answer and the right hand part of the answer. When multiplying by 9 we subtract one from the original number and write it down on the left hand side. Then we subtract each number on the left hand side from 9 and write it down on the right hand site (this will become clearer with the examples that follow). 234 x 999 = 233 | 766 Explanation: We subtract one from 234 and write it down on the left hand side i.e 234 - 1 = 233. Then we take each individual number and subtract that number from 9 in the order - left to right. So we subtract 9 - 2 = 7, then we subtract 9 - 3 = 6 and again 9 - 3 = 6. Hence, we get the right part of our answer i.e 766 Common Mistake: As you see from the above example we subtract 9 from each individual number, but we did so from the left hand side of our answer and not the original number. In simple words, we should subtract individually from 233 and not from 234. This is a common mistake I have found and would advice you to remember this crucial point. 948224243 x 999999999 = 948224242 | 051775757 Explanation: Wow! I can beat the calculator. Most of the calculator cannot perform calculation more than 12 digits and I doubt even your mobile phones can do that. Here is how its done, we subtract 1 from 948224243 and get 948224242 . Then we subtract 9 - 9 get 0, then 9 - 4 = 5, then 9 - 8 and so on. I am sure by now you get the idea. But now here is the question what if the number of 9's are more than the number which we going to multiply, for example, 273 x 99999 ( we cannot do this as of now) Part B (Original number is less then the number of 9's, for example 234 x 99999) Rules: Similar to above method, but just imagine 0's in front of the original number. But make sure the number of 9's are more and not less or equal.) 383 x 9999 will become 0383 x 9999 = 0382 | 9617 Explanation:It is simple, we imagine the number to be 0383. Subtract one from it and get 0382. Then as usual we calculate 9 - 0 = 9, then 9 - 3 = 6 and so one. 383 x 999999 will become 000383 x 999999 = 000382 | 999617 Explanation: In this case we had 3 extra 9's. So we imagine 3 extra 0's in front of 383. Therefore our first case will be 000383 - 1 = 000382, which becomes the left part of the answer. And like before we subtract 9 - 0 and get 9, then again 9 - 0 = 9 and again 9 - 0 = 9, then 9 - 3 = 6 and so on. Part C (Original number is more then the number of 9's, for example 234 x 9, this method is tough to undestand so pay close attention to this one) Rules:I will directly goto the example in this case, as the rules will confuse you at this level. 14 x 9 = 12 | 6 Explanation: From the number 14 we choose 1(the number in the ten's digit), increase it by 1 i.e. 2 or we add + 1. Subtract that from 14 and we get 12. Next we subtract 4 from 10 and and get 6 i.e. 10 – 4 = 6. The method is confusing but once a person gets a hang of it, it’s pretty easy. 24 x 9 = 21/6 Explanation: From the number 24 we choose 2(ten's digit); increase it by 1 i.e. 3 or we add + 1 (2 + 1). Subtract that from 24 and we get 21 (24 - 3). Next we subtract 4 from 10 and get 6 i.e. 10 – 4 = 6. 47 x 9 = 42/3 Steps involved: From the number 47 we choose 4(ten's digit); increase it by 1 i.e. 5. Subtract that from 47 and we get 42 (47 - 5). Next we subtract 7 from 10 and get 3 i.e. 10 – 7 = 3. 112 x 99 = 110/88 Steps involved: From the number 112 we choose 1; increase it by 1 i.e. 2 or we can say we add + 1 (1 + 1). Subtract that from 112 and we get 110 (112 - 2). Next we subtract 99 from 12 and then add 1 i.e. 99 – 12 + 1 = 88. 112 x 9 = 100/8 Steps involved: From the number 112 we choose 11; increase it by 1 i.e. 12 or we can say we add + 1 (11 + 1). Subtract that from 112 and we get 100 (112 - 12). Next we subtract 9 from 2 and then add 1 i.e. 9 – 2 + 1 = 8 So we have learned some simple methods of multiplying with 5 , 11 & 9. But that was all about scratching the surface. We have much more to cover. Today it might be slightly tuff. Add & Subtract Method Rules: This method is based on selecting a proper base and then using the Vedic maths method to calculate. Selecting a proper base is one of the most important things in this method, which makes major calculations very simple. See the example of different bases and make sure you use your wits to choose a proper base while calculation. It works in the following manner. 9 x 9 9 – 1 [ We assume the base as 10 ] 9 – 1 8 /1 Steps involved:- We select the base as 10. Then we check the difference for each number as compared to 10. In this case the difference is -1. So we write -1 on the right on side of this number. Then we multiply the right hand side. In this case -1 x -1. So we get 1 as the answer. Then we do cross addition/subtraction. In this case we would subtract -1 from the number 9 and then we get the answer as 8. And the final answer as 81. 9 x 8 [ We assume the base as 10 ] 9 – 1 8 – 2 7 /2 Steps involved:- We select the base as 10. Then we check the difference for each number as compared to 10. In this case the difference is -1 & -2. So we write -1 & -2 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case -1 x -2. So we get 2 as the answer. Then we do cross addition/subtraction. In this case we would subtract -2 from the number 9 or we subtract -1 from the number 8 and then we get the answer as 7. And the final answer as 72. 11 x 12 [ We assume the base as 10 ] 11 + 1 12 + 2 13 /2 Steps involved:- We select the base as 10. Then we check the difference for each number as compared to 10. In this case the difference is +1 & +2. So we write +1 & +2 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case +1 x +2. So we get 2 as the answer. Then we do cross addition/subtraction. In this case we would add +2 from the number 11 or we add +1 from the number 12 and then we get the answer as 13. And the final answer as 132. It is simple, isn't it? 91 x 91 [ We assume the base as 100 ] 91 – 9 91 – 9 82/81 Steps involved:- We select the base as 100. Then we check the difference for each number as compared to 100. In this case the difference is -9 & -9. So we write -9 & -9 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case -9 x -9. So we get 81 as the answer. Then we do cross addition/subtraction. In this case we would subtract -9 from the number 91 and then we get the answer as 82. And the final answer as 8281. Now, we have moved on to higger numbers too. You may practice higher numbers like 120 x 102 or 105 x 108. But remember so far our base has only been 10 & 100. We will move on to slightly complex task tomorrow. You may check the next few examples. Solution is not given, but by now you should be used to doing these simple calculation mentally. 888 x 998 [ We assume the base as 1000 ] 888 – 112 998 – 002 886/224 598 x 998 [ We assume the base as 1000 ] 598 – 402 998 – 002 596/804 Add & Subtract Method Rules: This method is based on selecting a proper base and then using the Vedic maths method to calculate. Selecting a proper base is one of the most important things in this method, which makes major calculations very simple. See the example of different bases and make sure you use your wits to choose a proper base while calculation. It works in the following manner. Importance of choosing the RIGHT BASE. 25 x 98 [ We assume the base as 100 ] 25 – 75 98 – 2 24/50 Steps involved:- We select the base as 100. Then we check the difference for each number as compared to 100. In this case the difference is -2 & -75. So we write -2 & -75 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case -2 x -75. So we get 150 as the answer. But since the base is 100 we can only use 2 up to 2 digit and we will carry forward 1 to the left hand side Then we do cross addition/subtraction. In this case we would subtract -2 from the number 25 or we subtract -75 from the number 98 and then we get the answer as 23 but due to carry forward we add 1 and the answer turns out to be 24. And the final answer as 2450. So far we have looked at numbers where after addition/subtraction from the base, both the numbers were either positive or negative. Now lets consider the other way round. 12 x 8 [ We assume the base as 10 ] 12 + 2 8 – 2 10/-4 = 96 Steps involved:- We select the base as 10. Then we check the difference for each number as compared to 10. In this case the difference is +2 & -2. So we write +2 & -2 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case +2 x -2. So we get -4 as the answer. Then we do cross addition/subtraction. In this case we would subtract -2 from the number 12 or we add +2 from the number 8 and then we get the answer as 10. However since the we have – 4 on the right hand side we subtract it from the base and -1 from the left hand side. And we get the answer 96 It is slightly confusing but you will get the hang of it pretty soon. 108 x 97 [ We assume the base as 100 ] 108 + 8 97 – 3 105/-24 = 104/76 Steps involved:- We select the base as 100. Then we check the difference for each number as compared to 100. In this case the difference is +8 & -3. So we write +8 & -3 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case +8 x -3. So we get -24 as the answer. Then we do cross addition/subtraction. In this case we would subtract -3 from the number 108 or we add +8 from the number 97 and then we get the answer as 105. However since we have – 24 on the right hand side we subtract it from the base and -1 from the left hand side. And we get the answer as 10476 We have also been using the base such as 10, 100 & 1000. Now, lets try some different base. Play close attention to the difference as it will help you master this method. 49 x 49 [ We assume the base as 50 ] 49 – 1 49 – 1 48/01 24/01 Steps involved:- We select the base as 50. Then we check the difference for each number as compared to 50. In this case the difference is -1 & -1. So we write -1 & -1 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case -1 x -1. So we get 1 as the answer. Then we do cross addition/subtraction. In this case we would subtract -1 from the number 49 or we subtract -1 from the number 49 (same thing in this case) and then we get the answer as 48. But since the base is 50 which is nothing but 100/2. So we divide 48 by 2 And we get the answer 2401. Similar Example: 46 x 46 [ We assume the base as 50 i.e 100/2 ] 46 – 4 46 – 4 42/16 21/16 23 x 23 (BASE AS 20 i.e. 10 x 2) 23 + 3 23 + 3 26/9 52/9 Steps involved:- We select the base as 20. Then we check the difference for each number as compared to 20. In this case the difference is +3 & +3. So we write +3 & +3 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case +3 x +3. So we get 9 as the answer. NOTE: We write 9 as the answer and not 09 as the answer because here the base is 10 x 2. That means the underlying base still remains 10. And that’s why we just write one number on the right hand side i.e. 9 Then we do cross addition/subtraction. In this case we would subtract +3 from the number 23 or we add +3 from the number 23 (same thing in this case) and then we get the answer as 26. But since the base is 20 which is nothing but 10 x 2. So we multiply the base 26 x 2. And we get the answer 529. 23 x 23 (BASE AS 20 i.e. 100/5) [ Notice the above sum has the same numbers] 23 + 3 23 + 3 26/09 5.20/09 5/29 Steps involved:- We select the base as 20. Then we check the difference for each number as compared to 20. In this case the difference is +3 & +3. So we write +3 & +3 on the right on side of these numbers respectively. Then we multiply the right hand side. In this case +3 x +3. So we get 9 as the answer. NOTE: We write the base as 09 as our basic base or main base is 100/5. The whole idea is if the basic base or main base is a multiple or division of 100 then we will have 2 digits on the left hand side. And when the basic base of main base is a multiple of 10 then we take only one digit on the left hand side, rest we carry forward to the left hand side after doing the division. Then we do cross addition/subtraction. In this case we would subtract +3 from the number 23 or we add +3 from the number 23 (same thing in this case) and then we get the answer as 26. But since the base is 20 which is nothing but 100/5. So we divide 26 by 5 and get the answer 5.20 Now whenever we get answer in decimal form after division we transfer it to the right hand side by adding it. Therefore we add 20 to right hand side and get the answer as 29 And we get the final answer as 529. GOLDEN RULES FOR ADD SUBTRACT METHOD. Choose the base according to your common sense so that the calculation becomes easy. If the base is 100 there will be 2 digits on the right hand side, if the base is 10 there will be 1 digit on the right hand side, if the base is 1000 there will be 3 digits on the right hand side and so on. And the extra digit you carry over to the left hand side. Remember to always divide the left hand side or multiply it first, if required. And then minus right hand side (if negative) or carry forward from left hand side (if positive). If your answer is something like 24/-26 (with your base being 10). You carry forward -2 to the left hand side. Then your answer will be 22/-6. And then you subtract. This will give you the answer 21/4. However, you can do it directly too, since you know by subtraction of 24 by 3 will give you 21 and you will be able to carry 30 to the left hand side and then you will be able to get your answer easily. i.e. 30 – 26 = 4. 26 x 24 = 317 x 22 = 125 x 125 = 999 x 294 = 99999 x 93 = 112 x 99 = 98 x 97 = 110 x 95 = 49 x 49 = 23 x 23 = Basic Multiplication But Mentally and Faster. The reason we are learning this method is because it is not possible to do all calculation using the add and subtract method. There are many other methods involved but this is simple and easy. Rules: This method simplifies our normal method of multiplication. Just remember the following diagram when you multiplying 2 digit number with a 2 digit number. Multiply Conventional Calculation 54 x 84 216 4320 4536 By simply Calculation 54 x 84 40/52/16 40/53/6 45/3/6 Steps involved: We follow conventional method but we try to skip the extra work to reduce time and effort. First we multiply the right hand side number 4 x 4 (straight line). Write down 16 as our answer Then we multiply 5 x 4 and add it with 8 x 4 i.e. 5 x 4 + 8 x 4 (the cross). We get 52 as our answer. Then we multiply the left hand side number 5 x 8 (straight line). Write down 40 as our answer. Then we carry forward 1 from 16 to the left hand side as we will have only one digit. So we get 53 in the centre. Then we carry forward 5 from 53 to the left hand side as we again will have only one digit in the centre. So we get 3 in centre and on the left hand side we get the answer 45 Our final answer is 4536 27 x 35 6/31/35 6/34/5 9/4/5 Steps involved: First we multiply the right hand side number 5 x 7 (straight line). Write down 35 as our answer Then we multiply 5 x 2 and add it with 3 x 7 i.e. 5 x 2 + 3 x 7 (the cross). We get 31 as our answer. Then we multiply the left hand side number 2 x 3 (straight line). Write down 6 as our answer. Then we carry forward 3 from 35 to the left hand side as we will have only one digit. So we get 34 in the centre. Then we carry forward 3 from 34 to the left hand side as we again will have only one digit in the centre. So we get 4 in centre and on the left hand side we get the answer 9. Our final answer is 945. 67 x 4 This is 67 x 04 0/24/28 0/26/8 2/6/8 Steps involved: First we multiply the right hand side number 7 x 4 (straight line). Write down 28 as our answer Then we multiply 4 x 6 and add it with 0 x 7 i.e. 5 x 2 + 0 x 7 (the cross). We get 24 as our answer. Then we multiply the left hand side number 6 x 0 (straight line). Write down 0 as our answer. Then we carry forward 2 from 28 to the left hand side as we will have only one digit. So we get 26 in the centre. Then we carry forward 2 from 26 to the left hand side as we again will have only one digit in the centre. So we get 6 in centre and on the left hand side we get the answer 2. Our final answer is 268. Now, if we want to multiply a 3 digit number with a 3 digit number we keep in mind the following diagram. 342 x 425 12/22/31/24/10 12/22/31/25/0 12/22/33/5/0 12/25/3/5/0 14/5/3/5/0 Steps involved: 1. First we multiply the right hand side number 2 x 5 (straight line). Write down 10 as our answer 2. Then we multiply 4 x 5 and add it with 2 x 2 i.e. 4 x 5 + 2 x 2 (the first cross). We get 24 as our answer. 3. Next we multiple 3 x 5 add it to 4 x 2 and again add that to 4 x 2 i.e. 3 x 5 + 4 x 2 + 4 x 2 (the star). And we get the answer 31. 4. Then we multiply 3 x 2 and add it with 4 x 4 i.e. 3 x 2 + 4 x 4 (the second cross). We get 22 as our answer. 5. Then we multiply the left hand side number 3 x 4 (straight line). Write down 12 as our answer. 6. Then we carry forward 1 from 10 to the left hand side as we will have only one digit. - We keep carrying on each number to the left hand side and we leave one number in each case and hence we get the number 14/5/3/5/0 1. Our final answer is 145350. 862 x 35 we can add a 0 to 35 and then solve as following 862 x 035 0/24/58/36/10 0/24/58/37/0 0/24/61/7/0 0/30/1/7/0 3/0/1/7/0 Steps involved: 1. First we multiply the right hand side number 2 x 5 (straight line). Write down 10 as our answer 2. Then we multiply 6 x 5 and add it with 3 x 2 i.e. 6 x 5 + 3 x 2 (the first cross). We get 36 as our answer. 3. Next we multiple 8 x 5 add it to 0 x 2 and again add that to 6 x 3 i.e. 8 x 5 + 0 x 2 + 6 x 3 (the star). And we get the answer 58. 4. Then we multiply 3 x 8 and add it with 0 x 6 i.e. 3 x 8 + 0 x 6 (the second cross). We get 24 as our answer. 5. Then we multiply the left hand side number 8 x 0 (straight line). Write down 0 as our answer. 6. Then we carry forward 1 from 10 to the left hand side as we will have only one digit. - We keep carrying on each number to the left hand side and we leave one number in each case and hence we get the number 3/0/1/7/0 Our final answer is 30170. We have already covered a lot and today seems to be one of my favourite topics and I am sure you are going to enjoy all the neat tricks I am going to teach. Have fun! Let’s learn square using the add and subtract method. In this method we again choose a base. This method is nothing but the above method in a more simplified manner for calculating squares. 7 x 7 = (7-3)/3 x 3 = 4/9 By subtract and add method. Taking the base 10 7 – 3 7 – 3 (7-3)/ (-3 x -3) = 49 Look at the following examples 12 x 12 = (12 + 2)/2 x 2 = 14/ 4 14 x 14 = (14 + 4)/4 x 4 = 18/16 = 19/6 (remember the base is 10) 91 x 91 = (91 – 9)/9 x 9 = 82/81 95 x 95 = (95 – 5)/5 x 5 = 90/25 108 x 108 = (108 + 8)/8 x 8 =116/64 988 x 988 = (988 – 12)/12 x 12 = 976/144 (The base is 1000. Therefore we do not need to carry 1 of 144 on the left hand side. 45 x 45 = (45 – 5)/5 x 5= 40/25 = 20/25 (The base is taken as 50 (100/2). Therefore divide by 40 by 2) 45 x 45 = (45 – 5)/5 x 5 = 40/25 = 200/25 = 202/5 (The base is taken as 50 (10 x 5). Therefore we multiply 40 x 5 = 200 and the carry forward 2 from left hand side to right hand side as the base is 10) Squares using the formula: (a + b) (a + b) = a2 + 2ab + b2 (a - b) (a - b) = a2 - 2ab + b2 This is simple enough to understand. 97 x 97 = (100 -3)(100-3) = 10000 – 600 + 9 108 x 108 = (100 + 8) (100 + 8) = 10000 + 1600 + 64 29 x 29 = (30 – 1) (30 – 1) = 900 -60 + 1 = 841 49 x 49 = (50 -1) (50 – 1) = 2500 – 100 + 1 = 2401 78 x 78 = (80 – 2) (80 – 2) = 6400 – 320 + 4 = 6084 Let’s try a different method to find squares now: Method: 1. Square the units digit. 2. Tens digit x units digit x 2 3. Square of tens digit Examples 42 x 42 = 16/16/4 = 1764 (we carry forward 1 from the centre calculation to the left hand side) Method: 1. Square the units digit. We get 4 2. Tens digit x units digit x 2. We get 16 3. Square of tens digit is 16. 23 x 23 = 4/12/9 = 5/2/9 (we carry forward 1 from the centre calculation to the left hand side) Method: 1. Square the units digit - 9 2. Tens digit x units digit x 2 - 12 3. Square of tens digit - 4 62 x 62 = 36/24/4=38/4/4 Method: 1. Square the units digit - 4 2. Tens digit x units digit x 2 - 24 3. Square of tens digit – 36 73 x 73 = 49/42/9 =53/2/9 26 x 26 = 4/24/36= 6/7/6 (We carry forward 3 to 24. That makes it 27. We put down 7 and carry forward to the left 2. Thus we get 6 on the extreme left. And our answer is 676) 34 x 34 = 9 / 24 / 16 = 1156 42 x 42 = 16/16/4 = 1764 63 x 63 = 36/36/9 = 3969 Consider base 10 Cube for 13 13 = (13 + 3 x 2)/(9 x 3)/3 x 3 x 3 = 19/27/27 (But since the base is 10 only one number will remain in each left hand side. Therefore answer is = 19/29/7 = 21/9/7. Steps involved 1. First we double the excess number from the base 10 and add to the original number i.e. 13- 10 = 3. Then 3 x 2(double the excess) + 13 (original number) = 19 2. Then we subtract 13 from base, get 3, which we multiply with 9 ( the excess number of the left hand side number minus the base i.e. 19 -10 ). So basically - excess from original number from 10 x excess from the left number which we got in step 1. We get 3 x 9 = 27. 3. Then we take the cube of the unit digit of the original number. 4. And doing the required calculation for the base 10, we get the answer 2197. Cube for 15 15 = (15 + 5 x 2)/(15x5)/5 x 5 x 5 = 25/75/125 =25/87/5 =33/7/5 Steps involved 1. First we double the excess number from the base 10 and add to the original number i.e. 15- 10 = 5. Then 5 x 2 + 15 (original number) = 25 2. Then we subtract 15 from base, get 5, which we multiply with 15 ( the excess number of the left hand side number minus the base i.e. 25 - 10.) So basically, 15 x 5 = 75. 3. Then we take the cube of the unit digit of the original number. 4. And doing the required calculation for the base 10, we get the answer 3375. 97 = (97 – 2 x 3)/(-3 x -9)/-3 x -3 x -3 = 91/27/-27 =91/26/73 Steps involved 1. First we double the excess number from the base 100 and subtract to the original number i.e. 100 - 97 = -3. Then -3 x 2 -3 (original number) = -9 2. Then we subtract 97 from base, get -3, which we multiply with -9 ( the excess number of the left hand side number minus the base i.e. 100 - 91.). So basically, -3 x -9 = 27. 3. Then we take the cube of 100 – 97. 4. And doing the required calculation for the base 100, we get the answer 912673. In similar manner, 52 = (52 + 2 x 2)/(2 x 6)/08 = 56/12/08 = 14/06/08 104 = (100 + 3 x 4)/3 x 4 x 4/4 x 4 x 4 = 112/48/64 This is just a introduction to this method and advance method for this has not been taught in this lesson. Straight Division 4096/ 64 4 / 940 6 6 / 4 1 . 6 4 Answer is 64 Steps involved 1. First separate the divisor into two parts, write the first number down and then above it write the second number. Therefore, we write 6 down and then 4 above it. 2. Then we manually check the closest number divisible by 6 from 4096, as we would do in our normal division, if we were dividing 4096. 3. Therefore, we divide 40 by 6; write down the quotient 6 down, and the remainder below the next number, in this case we write 4 below 9. 4. Now, we multiply 4 into our quotient - 6. The 4 we are using is of 64 or the one which we wrote above 6. 5. This we subtract from 49 i.e. 49 – 24 = 25 6. Next we divide 25 by 6 and get our quotient 4, which we write below, and remainder below the next number, in this case 1 below 6. 7. We again repeat the procedure. We multiply 4 with the next quotient i.e. 4, therefore 4 x 4 and we subtract it from 16. 8. We get the answer 0. Therefore our answer is complete which is 64 996/64 4 / 9 96 6 / 3 5 1 5 .6 Answer is 15.6 Or 4 / 9 9 6 6 / 3 5 1 5 36 Where 15 is quotient and 36 the remainder. Steps involved 1. First separate the divisor into two parts, write the first number down and then above it write the second number. Therefore, we write 6 down and then 4 above it. 2. Then we manually check the closest number divisible by 6 from 996, as we would do in our normal division, if we were dividing 996. 3. Therefore, we divide 9 by 6; write down the quotient 1 down, and the remainder below the next number, in this case we write 3 below 9. 4. Now, we multiply 4 into our first quotient - 1. The 4 we are using is of 64 or the one which we wrote above 6. 5. This we subtract from 39 i.e. 39 – 4 = 35 6. Next we divide 35 by 6 and get our quotient 5, which we write below, and remainder below the next number, in this case 5 below 6. 7. We again repeat the procedure. We multiply 4 with the next quotient i.e. 5, therefore 4 x 5 and we subtract it from 56. 8. We get the answer 36. Now either we can leave it as it, because this will become the remainder. Or we can divide this by 6 to get the answer in decimal form. Therefore our answer is complete which is 15.6 5100 / 25 5 / 3 1 0 0 2/ 1 2 2 1 2 4 . 0 Therefore we get the answer as 124 Steps involved 1. We first divide 3 by 2, get quotient 1, which we write down and carry over the remainder 1 below the next number of 3100, which would be 1 2. Now we subtract 11 from 5 x 1. Get 6 which we should divide by 2 and write down the quotient as 3. 3. But we don’t do this. Reason being, if we take quotient 3 our remainder will be 0 and we won’t be able to continue with the calculation. 4. Therefore, when ever writing down the quotient, we should check the next calculation should not give us a answer 0 or negative. 5. Hence, we write down 2 us our quotient and carry forward 2. 6. Now we again repeat the procedure, 20 - 5 x 2 (quotient) = 10 7. 10 we divide by 2, since again our next number should not be 00, we take the quotient as 4 and write the remainder as 2 8. And 20 – 5 x 4 = 0 9. Therefore our answer is 124. Remember the last number is always the remainder. If the last number is 0 that means it is perfectly divisible. 80742 / 53 3 / 8 0 7 4 2 5 / 3 2 2 3 1 5 2 3 23 Therefore our answer is 1523 with the remainder 23 Steps involved We first divide 8 by 5 get the quotient 1, which we write down and carry the remainder below 0 * Next we subtract 3 (3 x 1) from 30 = 27 * Divide 27 by 5, write down the quotient 5 below and the remainder 2 below 7. * 27 – 15 (3 x 5) = 12 * We divide 12 by 5, write down the quotient 2 and carry forward the remainder 2 below 4. * 24 – 6(3 x 2) = 18 * 18 divide by 5, gives the quotient 3 and the remainder 3. * 32 – 9 (3 x 3) = 23, which becomes our remainder. We are going to begin with my favorite number 5 and how things get easy with it. Fun with Number 5 Rules: This is normally used for multiplication and in this case both the units in digit i.e. the last numbers are 5. Also, the beginning numbers are same. You will understand this as you see some examples. 25 x 25 = 6 | 25 (This one's easy, right?) Explanation: As you see both the ending numbers are 5 and also they are beginning with 2. So we have satisfied both the rules. Now how do we get 625. The first step is that you write down 25 on the right hand side. Why we do so? Do not think about that. Just write 25 on the right hand side. Then we take the number on the left hand side (of 25), in this case 2. And we multiply it with the next higher digit number i.e. 3. Therefore we get the answer 2 x 3 = 6, which we write down on the right hand side. So we finally get the answer as 625. 45 x 45 = 20 | 25 Explanation: As this one satisfies the rule too. Both the left numbers are same - 4, and the ending numbers are 5. So without thinking the first thing we do is write 25 on the right hand side. Then we take the left hand number 4 and multiply it with 5, hence we get 4 x 5 = 20 (the left side of our answer). 135 x 135 = 182 | 25 (slightly tough) Explanation: Both the rule has been satisfied. We write down 25 on the right hand side. We multiple 13 with the next higher number and get 13 x 14 = 182. And there we have our answer 18225. Easy right? Slight Modification to the Number 5 method Rule: The ending number should add up to 10. Infact in the above case too the ending number 5 + 5 added upto 10. And the left hand side number should be same. This is very similar, so lets look at some example. 23 x 27 = 6 | 21 (I am sure that,not everyone finds this simple). Explanation: Ending numbers 3 + 7 add upto 10. So the first rule is satisfied. And both the number on left hand side are same i.e. 2. First thing you do is, you multiply 3 x 7 = 21. And then like earlier you multiply 2 with the next higher digit number i.e. 3. And we get 2 x 3 = 6 on the left hand side. Final answer is 621. 62 x 68 = 42 | 16 Explanation: Both the rule have been satisfied. We multiply 8 x 2 as the add upto 10 and we get the left hand side of the answer 16. Then we multiply 6 with the next higher digit number 7 and get 6 x 7 = 42 Fun with number 11 Now we try the tricks which can make calculation by 11 easy. We will also learn how to multiply by 22, 33, 44, 55, 66, etc. As it is nothing but 11 x 2, 11 x 3, 11 x 4, 11 x 5, 11 x 6 respectively. Steps involved: 1. To multiply a number by 11 put a 0 in the beginning and at the end of the number. 2. Starting from the digit’s place add two consecutive digit. Remember to go from right to left. Examples 342 x 11 = 03420 = 0+3 /4 + 3/2 + 4/0 + 2 = 3762 Explanation: We added 0 at the end and the beginning of 342 and got 03420. Now, as you see we broke the number by taking each consecutive number and added them. The first part is 0 + 2 = 2, the first number of our answer, then we took 2+4 and got 6, second number of our answer and so on. 4932 x 11 = 049320 = 0 + 4/4 + 9/3 +9/3 + 2/0+2 = 54252 Explanation: In this case when we add 3 + 9 we write down 2 and carry forward 1, we again get 4 + 9 + 1 we write down 4 and carry forward 1. We add 4 + 1 and get 5 In the above case if we need to multiply by 22. Then we first multiply by 2 and then by 11. 312 x 22 = 624 (312 x2) x 11 = 06240 = 0 +6/6 + 2/4 + 2/0 + 4 = 6864 Explanation: Here we are trying to multiply with 22. Hence we first multiply the main number with 312 with 2 and get 312 x 2 = 624. And then multiply the same number with 22. Rest of the steps remain the same. 312 x 22 = 03120 x 11 x 2 = (0+3 / 3 + 1 /2 + 1 /0 + 2) x 2 = 3432 x 2 = 6864 Explanation: This is the exact same sum as above. In this case you multiply the end answer by 2 and get the exact same answer. Similarly we can calculate any number into any multiple of 11. Even 11 x 11. We just repeat the process twice.

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