Mental Addition and Subtraction Tips Math Tricks with Arthur Benjamin

once we learn how to count the first rules of arithmetic that we learn are addition and subtraction and in order to do multiplication and division it's crucial that we be able to add and subtract numbers efficiently in lecture one we did some very easy addition problems and started the good habit of working from left to right in this lecture we'll continue to work from left to right starting with simple problems and working up to more complicated ones when doing mental addition we work one digit at time so let's begin with the addition of one digit number what is 52 plus 4 now in your head you should think 2 plus 4 is 6 so 52 plus 4 is 56 how about 56 plus 7. now in my head recognize that 6 plus 7 is greater than 10. so since i'm adding something to 50 know the answer will be 60 something and since 6 plus 7 is 13 and that ends in 3 the answer is 63. here let's try 76 plus 6. since 6 plus 6 is bigger than 10 the answer will be 80 something 80 what well 6 plus 6 is 12 which ends in 2 so the answer is 82 okay now let's start adding two digit numbers the easiest situation is when one of the numbers ends in zero let's first do 60 plus 27. now since 60 plus 20 is 80 60 plus 27 is 87 the 7 just comes along for the ride here now you try 81 plus 90. since 80 plus 90 is 170 81 plus 90 is 171. if we want to add any two-digit numbers we first add the tens digit then add the ones digit so for example the problem 62 plus 24. we first add the 20 then add the four okay so 62 plus 24. after adding 20 the problem simplifies to let's see 62 plus 20 is 82 so we have new problem 82 plus 4 and 82 plus 4 is 86 and that's the answer here let's try 73 plus 16. first we're going to add the 10 then add 6. 73 plus 16. 73 plus 10 is 83 so we have new problem 83 plus 6 which is simply 89. the next two the next few problems are just little bit trickier because some carrying occurs let's do 23 plus 58 first we add 50 then we add 8 23 plus 50 is 73 then we add 8 73 plus 8 though it has carry doesn't bother us that's 81. all right let's try 48 plus 37 what's our strategy first add thirty then add seven forty eight plus thirty is seventy eight then add seven giving us eighty five all right here's bigger one let's try plus 87 okay so we're gonna add 80 then add seven now 84 plus 80 see 80 plus 80 is 160 so 84 plus 80 is 164 plus 7. go ahead and say that 100 if you want 100 64 plus 7 is 71. there's the answer 107 with just little bit of practice you can add any two-digit numbers in just few seconds but it certainly requires practice here are some problems that you may want to try on your own before moving on recommend pausing this course for just minute and trying these problems and here are their solutions okay now let's move on to adding three digit numbers which is lot harder on your memory of course the easiest three digit number addition problems are when both numbers are multiples of like 500 plus 200 is 700 or 700 plus 600 is 1300 the next easiest situation is when one of them is multiple of hundred you try 400 plus 567. 400 plus 500 is 900 and the 67 comes along for the ride so the answer is 967. let's do 235 plus 800. now 200 plus 800 is 100 thousand so 235 plus 800 is 1035. the next easiest problems are when both numbers are multiples of 10. for example 450 plus 320 here our strategy is to first add 300 then add 20. 450 plus 320 this becomes after adding 300 750 plus 20. 750 plus 20 is 770 770 that's your answer we often see round numbers like this when we're counting calories for example if fast food restaurant sells hamburger with 410 calories and bag of french fries with 360 calories then the total number of calories would be 410 plus 360. we're going to add 300 then add 60. 410 plus 300 becomes 710 plus 60. you could pause you have simpler problem now 710 plus 60 is 770. when you throw in 560 calorie milkshake then how many calories would you have well 770 plus 560 we're gonna first add 500 then add 60. 770 plus 500 is 1270 plus 60. 1270 plus 60 that's going to be 1300 and something 70 plus 60 ends in 30 so it's 1330. now in order to add any two three-digit numbers together it's simple three-step process first add the hundreds digits then add the rest the tens and the ones digit let's try the addition problem three one four plus one five nine say it out loud three hundred fourteen plus one hundred fifty nine our strategy is to add hundred then 59. so 314 plus 100 is 414 plus 59. again you could pause take break you have simpler problem 414 plus 59. so next if you want to you could even say the 400 at this point you know enough about your answer to say 400 and focus on 14 plus 59. when we do that 14 plus 50 is 64 plus 9 is 73 and now you've said your answer okay the next problem is about as hard as it gets let's try 766 plus 489 even remembering the problem takes some work right 766 plus 489 first we add 400 then we'll add 89. 76 766 plus 400 is 11.66 and now you add 89. now know enough about my answer to say 1200 and i'll focus on 66 plus 89. so first we'll add 80. 66 plus 80 is 146. then i'll add the 9 to get 155 so know the answer ends in 55 and there's the answer 1255. now that problem was probably hard for you because you you need almost all of your memory just to hold on to the original problem and each addition step that we did involved carry but as you take each step the problem becomes simpler and you can start to say your answer along the way now as it turns out there's another way to solve the last problem that you might find easier to do 766 plus 489 notice that 489 is 500 minus 11. so if you want you can do 766 plus 489 by adding 500 then subtracting 11. and that's easier 766 plus 500 is 1266 and now what do you do subtract 11. 66 minus 11 is 55 so there's your answer 1255. the lesson here is that hard addition problems those involving lots of carries can usually be turned into easy subtraction problems that don't involve any borrowing so let's turn our attention to subtraction how are we going to do mental subtraction well just like with mental addition we do it one digit at time from left to right we'll start with two digit numbers let's look at the problem 93 minus 41. all right we're going to first subtract 40 then subtract 1. 93 minus 40 is 53. so you have new simpler problem 53 minus 1 which is 52. now that problem was easy because of the ones digit the 3 is bigger than the 1 and that made things easier let's try 74 minus 29 that'll be little trickier now there are two ways you can do this in the first way you subtract 20 then subtract 9. so the problem becomes 74 minus 29. subtract 20 and have 54 minus 9. 54 minus 9. that's going to be 40 something 40 what well we know that 14 minus 9 is 5 so it's got to be 45 but prefer to do this problem different way let's treat 29 as 30 minus 1. so to subtract 29 we're going to first subtract 30 then add back one okay 74 minus 29 we do 74 minus 30 to get 44 then add back one to get 45. did you find that second method easier you try it this time with 82 minus 48 okay so what are we going to do to subtract 48 we're going to subtract 50 then add back 2. 82 minus 50 is 32 when we add back two we get 34. in the addition section we saw that complicated addition problem one that had lots of carries can be converted into an easy subtraction problem with no borrowing here we see that subtraction problem that would normally involve borrowing can be turned into an easy addition problem with no carrying the advantages are even clearer as the numbers get larger here you try the problem 121 minus 57. what's our strategy we're going to first to subtract 57 let's subtract 60 then add back 3 okay so we start by doing 121 minus 60. 121 minus 60. well 120 minus 60 is 60. when we add back one and when we include the one we have 61. okay so 121 minus 60 is 61. add back 3 to get 64 and that's the answer now let's subtract three digit numbers if you're lucky every digit in the first number is at least as big as the corresponding digit of the second number that happens only if you're lucky but when that happens you'll subtract the hundreds then the tens then the ones for example let's do the problem 846 minus 225 we're going to first subtract 200 then subtract 25. 846 minus 225 the hardest part about doing this problem is keeping it in your memory now do this by saying the problem at least in my head 846 minus 225 and as say this i'll focus on the hundreds digit 846 minus 225 8 minus 2 is 6. so 846 minus 225 becomes 646 minus 25. take pause take break this is simpler problem 646 minus 25 and that's easier to remember you can even say 600 if you want get that out of your memory and now we have 46 minus 25 to do so first we subtract 20 then we'll subtract 5. 46 minus 20 is 26 minus 5 is 21. go ahead and say it 21. you've now said your answer 621. the bad news is that most three-digit subtraction problems require some sort of borrowing but the good news is that they can be turned into easy addition problems let's start with very easy one it would be hard for you to do it on paper but in your head it's really easy let's do 835 minus 497 again not fun to do on paper but it is fun in your head because 497 is simply 500 minus 3. so to subtract 497 we'll subtract 500 then add back three okay so 835 minus 500 is 335 and now we add back three to get three hundred thirty-eight here let's change the problem slightly to eight thirty-five minus four hundred seventeen new problem here eight thirty-five minus 417 first we're going to subtract 400 okay so 835 minus 417 becomes 435 minus 17. you could even say the 400 at this point say it 400 okay now we'll do 35 minus 17. how do we do that well we'll we'll subtract 17 by subtracting 20 and adding back three 35 minus 20 is 15 when you add back 3 you have 18 and there's your answer 418. okay raising the stakes little what if the problem were 835 minus 467 before answering this let's do very important warm-up exercise how far are these numbers from 100 okay 75 49 67 33 and 80. how far are they from 100 here are the answers we say that the complement of 75 is 25 because 75 plus 25 is hundred likewise the complement of 49 is 51 because 49 plus 51 is hundred and so on now how do you find the complement the pattern is very simple and we do it from left to right the first digits add up to nine and the last digits add up to 10. so for the complement of 75 notice 7 plus 2 is 9 and 5 plus 5 is 10. look at the number 49 how do you find its complement well 4 plus 5 is 9 and 9 plus 1 is 10. so the complement of is 51. right 67 how would you find its complement well what do you have to add to 6 to get 3 what do you have to add to 7 to get 10 3 so the complement of 67 is 33 and similarly the complement of 33 would be 67 now wait wait see an exception here look at last the last problem 80 plus 20. if the number ends in zero like 80 then the complement will also end in zero and the first digits instead of adding up to nine they're going to add up to 10. so the complement of 80 is 20 but we knew that already right how far is 80 from 100 it's 20. okay now let's go back to our original problem 835 minus 467. since 67 is bigger than 35 i'm going to do this problem by subtracting 500 then adding back something okay so here we do 835 minus 500 is 335 plus something now what is that something well how far is 467 from 500 here's the same question how far is 67 from 100 we know the complement of 67 is 33 so 335 plus 33 right is from left to right 368. okay here's another problem let's do 621 minus 274. okay how are we going to subtract 274 we'll subtract 300 and add back what the complement of 74 which is 26 so when subtract 300 621 minus 300 is 321 when you add back 26 you get 321 plus 26 from left to right is 347. personally think that it's easier to do this problem mentally using complements than by the pencil and paper method we are turning hard subtraction problem into an easy addition problem here let's do one more consider the problem 729 minus 256 what's our strategy we'll subtract 300 and add back the complement of 56 44 okay so 729 i'm going to subtract 300 then add back 44. 729 minus 300 is 429. now add 44. take breath take break 429 plus you have new problem it's simpler problem 429 plus 44. let's see is 469 plus 4 is 473 okay let's do one more just for fun let's do one two three four minus five six seven okay how do we do that we're going to subtract 600 and add back the complement of 67 33 so 1 1234 minus 600 is 634. add 33 from left to right and we get 667. look what we did we took what would have been an ugly subtraction problem and turned it into an easy addition problem sometimes it's convenient to know your three digit complements how far are these three digit numbers from thousand 675 849 777 223 and 670. how far are they from thousand check it out 325 151 223 777 and 330. do you see the pattern the complement of 675 is 325 and if you look at the digits six the digits on the left six plus three is nine then the middle digits seven plus two is nine and the digits on the right five plus five is ten they add up to nine then nine then ten this makes sense by the way because if you were to actually do that addition problem on paper from right to left like you're like you've usually done 675 plus 325 how would that go you'd start on the right with 5 plus 5 is 10 which would put down the 0 carry the 1. then that one you'd add that to 7 plus 2 which is 9 1 plus 9 is 10 you'd put down 0 carry the 1. right then you'd have another 1 to add to 6 plus 3. 6 plus 3 is 9 add that to 1 is another and that's how you get thousand so for that reason if we sum the columns of of the other three digit numbers we'll have the same pattern right 9 9 10 9 9 10 9 9 10. but wait there's another exception if the original number ends in zero then so will its complement so with problem like 670 the complement will also end in zero and that's going to be preceded by the complement of 67 and the complement of 67 is 33 so that gives us complement of 330. you will use three-digit complements all the time when making change for example if an item costs 6.75 and you pay with ten dollar bill and the amount of change you get back will be the complement of 6.75 namely 325 3.25 cents if the item costs let's say 8.49 then change from 10 would be what what's the complement of 849 let's see they have to add up to 9 9 10 so we get 1 5 1 dollar 51 would be your change the same strategy works for change from hundred dollars what's the change for 23.58 well here we have the numbers have to add up to 9 9 9 and 10. for the complement of 2358 that's going to be 7 6 4 2. we did that left to right 9 9 9 10. adds up to 76.42 that would be your change now in practice when hear 23.58 cents what actually goes through my mind is think the dollars add to 99 and the cents will add to hundred so if you have the problem 2358 you think what adds to 99 76 what adds what what do have to add to 58 to get 142. to get 76.42 when making change from 20 the idea is essentially the same but with tiny change if you will here when making change from 20 the dollars add to 19 and the cents add to hundred right so for example if an item cost 13.57 and you pay with 20 bill what would your change be okay the dollars add to 19 and the cents add to hundred so thirteen dollars and fifty seven cents if the dollars add to nineteen that gives you six dollars and the cents add to hundred what's the complement of 57 it's 43 so there's your answer 6.43 cents okay your turn if the bill came to 7.56 what's the change from 20 what's the what what do we get for for 7.56 the dollars add to 19 so it's 12 and complement of 56 44 cents the mathematics of making change is very handy but hope more importantly that it has started to change the way you think about mathematics in this lecture i've shown you nearly everything know about mental addition and subtraction let me sum up the main ideas for you which are pretty simple we work from left to right go one digit at time and look for opportunities to use complements that turn hard addition problems into easy subtraction problems and vice versa now we did problems as large as adding and subtracting three digit numbers together you might wonder why didn't we go higher when the problems get very large we usually resort to mental estimation and besides to add pair of four-digit numbers it can be challenge just to remember the problem the human memory can only hold on to about seven to ten digits without resorting to special memory tricks that i'll cover later of course some four digit numbers are very easy to add especially when they have lots of zeros you can do 1234 plus 5000 right to get 6234 since there are not really eight digits to remember it's really more like five digits to remember it's also not hard to add four digit number to three digit number when they over when they only overlap in one place for example twenty seven hundred plus one hundred eighty is listen to the numbers twenty eight hundred eighty we see this all the time when we do multiplication problems which is the topic of our next lecture recommend taking little time now to practice your addition and subtraction and then you should be prepared for the next lecture to go forth and multiply you