welcome get ready to learn all about math as you've probably never seen it before you'll learn how to do addition subtraction multiplication division fractions squaring and even algebra what's so different about that well you'll learn how to do it in your head with mathematics now get ready to learn all about mathematics with Professor Arthur Benjamin [Music] welcome to mathemagics if you think about it numbers are everywhere and the more sense you can make of numbers the more successful you will be I promise that if you watch the video listen to the audio tapes and practice with the workbooks you will improve your ability with numbers now the most commonly used mathematical operation is addition whether you're counting calories or figuring out the bill at a restaurant I have some valuable tips for you [Music] welcome to mathematics we'll start by talking about addition which is the simplest operation remember from the foothill Country Day School when Cory added that long series of numbers now this is Cory and I hear you're pretty good at adding numbers in your head is that right yeah could she really add a lot of numbers in her head okay I'm gonna give you some complex problems you add up all these numbers in your head I'm gonna use my trusty calculator here that I just bought you ready for this okay here we go what is 532 + 463 plus 837 how much is that 832 she's right look at that that's pretty good that was easy well by the end of this video you will also understand how to add numbers in your head just like that and with practice you'll be able to do it just as quickly I like to explain everything by example let's start with two-digit numbers plus two-digit numbers okay so the first problem we'll do will be 54 plus 41 now the most important thing to learn about mental calculations is that almost all of your calculating will proceed from left to right now that's the opposite of the way that we're taught in school where we learn how to do things from right to left right to left is fine for doing the math on paper but if you do the math in your head you'll see that it's much faster and much more natural to do the problem from left to right after all we read from left to right we pronounce numbers from left to right we say 54 not 4 and 50 and I claim that with practice it's a lot easier to calculate from left to right okay so let's do 54 plus 41 here we start with 54 plus 40 and then we'll add one now 54 plus 40 is 94 94 plus 1 is 95 and that's the answer now if that 54 plus 40 was giving you any kind of trouble at all then I recommend that you start with the audiotapes and listen as we warm up doing what I call one digit problems okay let's go on to the next problem 46 plus 33 okay well 46 plus 30 is 76 76 plus 3 is 79 and that's the answer here let's try a problem that's a little more difficult that involves carrying but it's not too difficult let's try 87 plus 52 now what's 87 plus 58 plus 5 is 13 so 87 plus 50 is 137 right now you add 2 to get 139 and that's the answer now if I'm going too fast you've got the remote control here use that pause button and use that rewind I've just got so much material that I want to cover I may be going a little quickly for you the first time also before we get on to any harder problems you should be exercising using our workbook okay do as many problems as it takes for you to get up to speed let's do let's do another problem with caries though let's say 58 plus 25 all right 58 plus 20 is 78 78 plus 5 is 83 and you're done here you try this one try doing 49 plus 32 okay so what do we do first 49 plus 30 that's 79 79 plus 2 is 81 and we're done let me give you another way to do this problem one of the great joys of doing math in your head is that there's often more than one way of solving the same problem look at the number 49 you know that's one away from 50 that's 50 minus one so let's do 49 plus 32 by adding 50 plus 32 to get 82 and then subtract one to give us 81 wasn't that easy here you try that same strategy with this problem 86 plus 79 okay 86 plus 80 is 166 and now you subtract 1 to get 165 okay now at this point I insist that you stop right here go to the workbook practice the two digit plus two-digit problems because only once you're good at doing that will you'll be able to go on to the next level when we start addressing three-digit numbers okay [Music] alright are you back are you ready for more I hope you've done the problems in the workbook but if you haven't well proceed at your own risk at least you'll be able to understand what we're doing it may take a bit of practice before you're up to speed and actually able to calculate the way we're doing it okay let's do 345 plus 23 that's a three-digit number plus a two-digit number okay we start by doing 345 plus 20 that's just a one digit problem which you should be good at if you listen to the audio tapes 345 plus 20 is 365 now you add 3 to get 368 ok let's let's up the ante event let's do 345 + 523 adding to three-digit numbers now okay we're doing everything from left to right so first we're gonna add 500 then 20 then 3 ok ready 345 plus 500 is 845 right now we add 20 845 wait wait let me stop here what has the problem become the problems become 845 plus 23 right okay pause there and now go on 8 45 plus 20 is 865 plus three is 868 and that's the answer here let's try one that's a little more complicated and a little more practical too because we're going to be using money all right let's say I want to add 8 dollars and 46 cents to $7.25 ok just as usual we're gonna work from left to right start by adding the 7 dollars to the 8 dollars and 46 cents and we have 15 dollars and 46 cents right and now we have 15 dollars and 46 cents + 25 cents I add the 20 gives me 15 66 plus 5 is 15 71 ok let's do one more problem this is a special kind of problem but it's one that shows up a lot when you're using money I'm going to add six dollars and 37 cents plus 2 down 99 cents okay now $2.99 you see that a lot oftentimes when you buy items you're one cent away from an even dollar amount or maybe five cents away from an even dollar amount and I'm gonna take advantage of that let's do six dollars and 37 cents plus three dollars first because that's so easy it's nine dollars and 37 cents but we added one penny too much so we subtract the penny and now we're left with nine dollars and 36 cents okay let's now go back to the Foothill Country Day School and see Cory adding up her numbers now she added up more than two numbers but that's okay we're adding the digits one number at a time so it always seems like we're only adding two numbers at the same time let me explain go ahead Cory let's see what she's doing here what is 532 + 463 first she had 532 + 463 now freeze right there at this point in the calculation Cory should have been adding the 400 as Mike was saying the number so went 532 plus 400 and while he was saying and she should be adding that 400 right now so that she's in the instantly thinking of 932 plus the rest of the number which was 63 okay so she's heard 532 plus 463 she's now thinking 932 plus 63 which is 995 right now stop look at look at the number 995 that's a special number that's just five cents or five numbers away from a thousand nine hundred ninety five so she should be thinking nine ninety-five that's a thousand minus five I'm gonna add a thousand minus five to the next number the next number was what eight hundred and thirty seven eight hundred and thirty-seven so one thousand minus five added to eight hundred and thirty-seven that's 1837 - the five is 1082 which was the answer good for you Cory okay I'd like to end this module by giving one other practical application of addition and that's for calculating tips let's say you were having dinner and the bill came to forty two dollars and you want to decide to leave a tip of some sort now if you were a cheapskate you might leave a ten percent tip which at least is easy to calculate ten percent of forty two dollars that's easy you just move the decimal point over one and you have four dollars and 20 cents but that's not much of a tip you might decide to be generous and leave a 20% tip and that's good that's easy to calculate all you have to do is double the $4.20 that we calculated and that gives you eight dollars and forty cents suppose you want to live leave an amount that's in between let's say 15% 15% is also very easy to calculate take the cheapskate amount of $4.20 add half of that what's half of four dollars and 20 cents well it's two dollars and ten cents we'll talk more about doing division in a later module four dollars and 20 cents plus two dollars and ten cents add them up from left to right and you get six dollars and thirty cents which would be exactly a 15% tip and I hope these tips will be very useful for you [Music] have you ever tried to stretch your budget a little bit too far well hopefully if you learn mental subtraction you'll never write a rubber check [Music] now it's time to learn subtraction do you remember from the Foothill Country Day School when Chris - that subtraction problem in his in his head let's take a look at that all right Chris ready for a little mental subtraction here okay what is 1231 minus of 589 how much is that 642 yep in his head look at the heck can you pull it back by the end of the module you will also know how to do problems just like that with just a little bit of practice okay let's do start off very small with subtracting two-digit numbers and we'll work our way up let's say we wanted to do the problem 79 minus 43 well just like with addition we do all of our calculating from left to right so first we're gonna subtract 40 then we're going to subtract three now 79 minus 40 is 39 39 minus 3 is 36 and that's the answer now if any part of that problem gave you trouble you should go back to the audiotapes where we warm up subtraction with one digit problems okay let's go up a level and try you try this one 87 minus 35 all right now 87 minus 30 is what 57 good minus 5 is 52 and that's it 52 is the answer let's do let's do another problem 83 minus 49 this will be a little bit more difficult first we do 83 minus 40 well 83 minus 40 is 40 3 minus 9 is what 34 now I'll bet you had a little bit of trouble with that last subtraction 43 minus 9 is 34 you might remember that 13 minus 9 is 4 but let me suggest an even easier way of doing this problem in fact we'll be doing a lot of our subtraction problems this way let's instead of subtracting 40 at our first step let's subtract 50 instead so 83 minus 50 is what 33 but you've subtracted one too much so you have to add the one back and you're left with 33 plus one is 34 wasn't that simpler let's try another problem you try this one 91 minus 37 so start by subtracting 40 91 minus 40 is 51 right and now we add three back to that 51 plus 3 is 54 wasn't that easy let's try another one how about 71 minus 25 we'll start by subtracting 30 then we'll add back five 71 minus 30 is 41 41 plus five is 46 and you're done now you should do some exercises in the workbook before we do the three digit subtraction problems make sure you're good at those before you go on to the next step [Applause] [Music] okay welcome back at least I hope I'm welcoming you back if you've just been watching this that means you haven't been doing your exercises and the only way to build those mental muscles is to use is to work those exercises okay let's try 357 - 23 okay well we'll start off with 357 - 20 okay three fifty seven minus twenty is three hundred and thirty seven that should be easy okay 337 minus three is three hundred and thirty-four okay alright that was our transition problem that took us from three-digit numbers - two-digit numbers now let's do a three-digit number - another three-digit number let's do a similar problem 457 - 123 okay well the first thing again we do all of our calculating from left to right so we'll subtract off a hundred 457 - 123 well after I subtract a hundred my problem becomes 357 - twenty-three hey and that was the problem we just did and we got the answer of 334 that way okay let's um let's try another problem that will seem hard at first but once you see the sneaky way of doing it it'll be very easy let's try 743 - 295 now if you subtract 200 and make the problem 543 minus 95 this point it might get a little bit scary so I have another suggestion let's start the problem all over again 743 - 295 let's subtract too much let's subtract 300 ok 743 minus 300 is what it's 443 and but we've subtracted too much how much too much five too much so we have to add the five back to get 448 you see the strategy this is a useful thing to know because lots of things prices are often end in 95-99 and so on let's try you try this one 826 - 399 all right so what are you gonna subtract not 300 but subtract 400 that's right 826 - 400 is 426 add back the 1 and you've got 427 let me remind you again I'm going too fast for you that's what the pause button is for that's what the rewind button is for you need to go over this several times be my guest spend as much time as you need [Music] now before we begin doing even more complicated subtraction problems we need to learn a very valuable tool for the practicing Mathemagician this is called the method of compliments first we have to answer the question how far are these numbers from 100 look at them okay and this these are the answers now what do those answers have in common well notice that the leftmost digits always add up to 9 and the rightmost digits always add up to 10 did I say always well almost always there's one exception okay the exception is if the number ended in zero as was the case with 60 then we already knew how far that was from a hundred that was 40 and that was very easy but if the number doesn't end in zero then the left digits add up to 9 and the right digits add up to 10 and if you think about it that makes sense here you try these how far are these numbers away from 100 okay got them thinking of them all right what were our answers we got 51 72 1409 21 and the last one was 20 got it all right let's try a more complicated subtraction problem now 824 minus 386 okay I'm not gonna subtract one digit at a time because that 80 is gonna really give me trouble what I'm gonna do is I'm going to subtract too much I'm gonna subtract 400 ka 24 minus 400 is 424 but I've subtracted too much how much too much well how far is 386 from 400 well it's the same as how far is 86 from 100 and we know the complement of 86 is 14 so we have to add it back 14 so we're holding on to 424 we add back 14 and the answer is yes 438 okay let's try one problem for you to do let's try 717 - 268 so subtract 300 that's right 717 minus 300 is 417 add back the complement of 68 what's that right 32 417 plus 32 from left to right is 449 you got it okay let's go back and see how Chris did his subtraction problem what were the numbers again 1231 - 589 let's watch this for a moment what is 1231 - of 589 how much is that 640 - yep in his head look at the heck can you pull it back okay 1231 - 589 the first thing he did was he subtracted 600 he probably subtracted 600 as soon as he heard the 589 being pronounced okay 1231 - 600 is 631 add back the complement of 89 which is 11 631 plus 11 is 640 - whoo you got that one give yourselves a pat on the back if you did [Music] before we begin doing even larger subtraction problems we have to do three-digit complements how far are these numbers from 1000 here are the answers now except for the last two problems what do these answers have in common well the first digits always add up to 9 the second pair of digits add up to 9 and the last pair of digits will add up to 10 so the complement of 2 for 6 would be well what number adds to 9 7 what number from 4 do you have to add to get 9 5 and what number do you have to add to 6 to get 10 4 so the complement is 754 which we did from left to right if the number ends in 0 like with 480 the complement will end in 0 and now we just take the complement of 48 which we already know how to do that's 52 ok and the complement of 700 is of course 300 you think you have the hang of it if not do some problems in the workbook but let's give it a try anyway how far are these numbers from 1000 the answers should be 471 139 253 630 and of course 200 now do you remember from the show when little Taylor found the exact change from a hundred dollar bill for $14 and 39 cents let's take a look at that I already know mathematics really can we test you well I know how much change is supposed to get back wait a second you can you figure how much change for my hundred how much change eighty-five dollars and 61 cents is that right right on the box we explained how he did that problem in effect let's look at our explanation from there now if you tried to do that problem the old-fashioned way it would just be a nightmare you could never do that in your head I don't think so but let me show you how easy it is to do using mathemagical let's say the purchase price was 14 dollars and 39 cents right I make change from $100 all you need to know how to do is subtract numbers from nine and subtract numbers from 10 will subtract every number from nine except the last one which you subtract from 10 let me give you an example okay what's nine minus one eight and nine minus four five and nine minus 3 6 and 10 10 minus 9 1 and that's the answer eighty-five dollars and 61 says that's it that's all Taylor was doing was taking a complement of the number 14:39 to get 85 61 now Mike asked well what if the amount of change had been different suppose say a $50 bill had been left on the table instead of a hundred dollar bill no problem all you'd have to do is subtract $20 from 50 subtracting too much giving us 30 dollars and then we add back the complement of 439 the complement of 439 is 561 so the change would have been thirty-five dollars and 61 cents if he had left a $20 bill in the envelope then when we subtract $20 we'd be left with nothing we add back the complement of 439 and that's our exact change five dollars and 61 cents [Music] as long as we're on the subject of money I'd like to conclude this module by talking a little bit about checkbook math unfortunately most of the math that I do in my checkbook is subtraction let's start with the problem a hundred and twenty three dollars and forty-five cents and subtract sixty-seven dollars and 18 cents now what I'm about to give you is a pencil and paper method this is I would not do this entire problem in my head that would just be a little bit too much of a memory burden let's start I always break the problem into two parts I do the cents and then I do the dollars suppose I want to start by subtracting forty-five cents minus eighteen cents well 45 minus 18 is 27 so I write down the cents to get 27 cents next I'll do 123 dollars minus 67 dollars and I'll do this much in my head I'll subtract a hundred dollars then I'll add back the complement which is 33 so I have $23 plus 33 dollars is fifty-six dollars and I'll write that in so that's my answer 56 dollars and 27 cents but let's suppose now the problem was just a little different suppose there was one hundred and twenty three dollars and forty-five cents and now we were subtracting 67 dollars and 89 cents instead so once again I'll start with the cents I'll do 45 cents minus 89 cents well gosh 89 cents that's too big let's just take the complement of 89 and add that to 45 45 plus 11 is 56 cents I write that down then I know I've borrowed a dollar in doing this calculation so I either have to say 122 minus 67 or sometimes I think of it as 123 minus 68 take your pick we'll do it the first way 122 minus 67 122 minus 100 is 22 add back the complement of 67 which is 33 giving us 55 dollars and 56 cents okay finally on an upbeat note let's end it with an addition problem for the checkbook let's say I had 123 dollars and 45 cents I wanted to add 67 dollars and 89 cents first we do the cents then we do the dollars 45 cents plus 89 cents would be a dollar 34 cents so we write down the 34 and the dollar to the 123 gives us 124 dollars which when we add it with to the $67 gives us a hundred and ninety one dollars so that's the answer one hundred and ninety one dollars and thirty four cents hopefully you'll be doing a lot more adding than subtracting in your checkbook as well how would you like to learn how to multiply you can even watch these guys or listen to me [Music] now we come to my favorite operation multiplication I assume you know your multiplication tables through ten and are fairly comfortable with multiplying one digit numbers together if not you need to consult the booklet and the audio tape let's do is our first example forty-two times nine now just as with addition and subtraction we do all of our multiplying from left to right also so we'll start with nine times 40 well 9 times 4 is 36 so 9 times 40 is 360 next we do 9 times 2 which is 18 add those numbers together from left to right of course and we get 378 here you try this one 64 times 3 what's 3 times 60 180 3 times 4 is 12 add those numbers together and you get 192 again if this is going too fast for you please make judicious use of that pause and rewind button that's what I'm here for ok here's another one for you to try these are very important to master before we can do very big problems let's try 37 times 7 ok what's 7 times 32 hundred and 10 right 7 times 7 is 49 add 210 plus 49 and you get 259 if you practice these enough you'll be able to do these problems very very quickly in a matter of seconds here are a couple that you could probably do very quickly already these are some easy ones how about 83 times 3 well 3 times 80 is 240 3 times 3 is 9 notice there's really no overlapping digits 240 plus 9 is 249 okay here's another easy one 67 times 5 5 times 60 is 300 Hey look at those two zeroes there they're gonna help us out enormous ly because there's nothing I'm going to be able to add to that that's going to change that 300 into a 400 or anything we can even say the 300 right now say it 300 now we do five times seven which is 35 say it 35 and that's the answer 335 I'll do a couple more problems that are a little more difficult because they involve some carrying let's try 18 times seven okay from left to right of course 7 times 10 is 70 7 times 8 is 56 70 Plus 56 is 126 all right how about trying this 176 times 7 this is tougher 7 times 70 well 7 times 7 is 49 so 7 times 70 is 490 7 times 6 is 42 and 490 plus 42 well 49 plus 4 is 53 so 490 plus 42 is 532 let's do one more example 26 times 9 all right 9 times 20 is 180 9 times 6 is 54 add 180 plus 54 gives us 234 I'm going to do another multiplication problem 59 times 7 but I'm going to do this one two different ways so you can see the variety 7 times 50 is 350 7 times 9 is 63 350 plus 63 from left to right is 413 but let me show you another way of doing that problem which you might even find easier this first time around let's treat 59 as 60 minus 1 and now do 7 times 60 and then subtract 7 times 1/7 times 60 is 420 - 7 gives us 4 her 13 was that easier than the first way it probably was let's do another example of that sort let's do 79 times 8 okay this time I'll do 8 times 80 which is 640 minus 8 times 1 640 minus 8 is 632 okay let's do one more two digits multiplication problem let's try 68 times 7 we'll do this two ways 7 the first word I'll call the addition method then the subtraction method 7 times 60 is 420 7 times 8 is 56 420 Plus 56 is 476 the subtraction method we would start by doing 7 times 70 and treat 68 as 70 minus 2 7 times 70 is 490 minus 7 times 2 which is 14 490 minus 14 hmm that's 476 frankly I prefer doing the addition method for that problem in fact I always use the addition method because by the time it takes me to decide whether I should be adding or subtracting I could have done the problem already so I recommend doing the addition method primarily for two-digit numbers times one digit numbers okay now go in your workbook and practice practice practice two-digit numbers times one digit numbers because you won't be able to do the next group of problems until you're very good at doing votes [Music] have you practiced doing your two-digit numbers times 1 digit numbers I hope so because otherwise these next problems will be very difficult indeed let's try I'd like to do everything by example let's start off with 628 times 7 once again we do all of our calculating from left to right I'll start with 7 times 600 to give us 4,200 now let me pause for a moment here because we're working from left to right we know already that the answer has to be a little higher than 4200 and that's very important in fact if we were to stop the calculation right now we could say the answer is a little bit over 4200 but if we were working from right to left like we do on paper all we would know is that the answer ended in 6 and that's not very important ok end of lecture let's go back to the calculation 7 times 600 is 4200 seven times 20 is 140 now at this point ladies and gentlemen I stop before I do 40 before I do 7 times 8 to get 56 I stop to add 4200 plus 140 left to right of course giving us 40 340 got it now we do the 7 times 8 to get 56 and that 240 340 and the answer is 40 396 ku it takes a little while before you can hold all these numbers in your mind but once you do once you practice it enough and of course you've got to practice practice practice once you've practiced it enough you'll be able to do these problems in a matter of seconds these are so important I'm going to do a few more examples some three-digit numbers times one digit numbers are easier some are harder let's start off with some of the harder ones let's get those out of the way and then we'll do some easier three digit times one digit problems let's do three hundred and seventy three times four okay left to right four times 300 is 1200 four times 70 is 280 now add 1200 plus 280 and you get listen to yourself 1200 280 gives us 14 hundred and 80 now I do 4 times 3 which is 12 add that to 14 hundred and 80 and we get fourteen hundred and ninety two which was the Year Columbus discovered America right okay here's another historical problem try two hundred and ninety six times six okay six times 200 is 1200 six times 90 is 540 1200 plus 540 is 1740 you see where we're heading now 6 times 6 is 36 1740 plus 36 is 1776 that's the spirit hey I got an idea let's do that same problem a slightly different way let's do 296 times 6 and let's treat 296 as 300 minus 4 so I'll do 6 times 300 and then I'll subtract 6 times 4 6 times 300 is 1800 minus 6 times 4 which is 20 for 1800 minus 24 what's the complement of 24 it's 76 so the answer is 1700 and 76 now let's try some easy ones 805 times 3 I promise this was going to be easy three times 800 is 2400 3 times 5 is 15 2400 plus 15 is you can hear it 20 415 here's another one 573 times 6 doesn't look as easy but it is start with 6 times 500 that's 3000 you can now say the 3,000 you don't have to worry about 73 times 6 because it's going to be a three-digit number of some sort say the 3,000 now do 73 times 6 that's 420 plus 18 is 438 and that's the rest of your answer 3,000 438 here you try this one 649 times five five times 600 is say it verbally say it three thousand good five times forty is two hundred you can say that - I love fives they're so convenient five times nine is forty-five and that's the answer 3245 I want to show you two more three digit times one digit numbers that sort of break the pattern that we've established I mean who's 811 times seven okay I'll start with seven times eight hundred to get 5,600 and now I'll do 11 times seven in one fell swoop because I know that seven times 11 is 77 so why not just do it fifty-six hundred plus seventy seven is fifty-six hundred and seventy seven wasn't that easy let's try four hundred and twenty five times eight as my last example again I'll start with eight times four hundred is thirty-two hundred and now I take advantage of the fact that I know that 25 times eight is two hundred right because if you had eight quarters that would make exactly two dollars right so eight times 25 is two hundred and that to the thirty two hundred and now you have an even 3400 actually we can apply this problem to do problems involving percentages for instance suppose you brought a VCR for four hundred and twenty five dollars and that the sales tax on that VCR was exactly eight percent well we've just done the calculation that you would need to do to determine what the sales tax would be because taking 8 percent of something is just multiplying it by eight and then dividing by a hundred and since four hundred and twenty five times eight is thirty four hundred when you divide it by a hundred you're left with an even thirty four dollars here let's do one more problem for you to try let's say you bought a camera for 240 and at the sales tax was 7% okay let's do 240 times seven seven times 200 is 1400 seven times 40 is 280 add those two get sixteen hundred and eighty all right and now what do you do you take 16 hundred and eighty divided by a hundred to be left with 16 point eighty in other words $16.80 which when you add it to the price of the camera of two hundred and forty dollars would give you a total bill of not too hard to do in your head left to right two hundred and fifty six dollars and eighty cents well that wasn't too bad was it [Music] [Applause] maybe the next time you're asked to divide the check you won't have to resort to this [Music] let's talk about division now if you think about it division is the only operation that we're taught to do in school from left to right already think about it I suppose if they could figure out a way of teaching you division from right to left they might but until that day comes we're all going to be doing it from left to right let's try a simple division problem like 65 divided by 7 you need to know your multiplication tables through 10 pretty thoroughly for you to do this you ask yourself how many 7s can I squeeze into 65 7 times 9 is 63 7 times 10 is 70 well that's too much so 65 divided by 7 will be 9 with a remainder when I subtract 63 from 65 I get 2 so my answer is 9 with a remainder of 2 or 9 and 2/7 ok your turn to try this 147 divided by 8 all right well let's see 8 times 5 is 40 8 times 6 is 48 well that's a little too high so the first number is 5 what will our remainder be 47 minus 40 is 7 and so the answer would be 5 with a remainder of 7 or 5 and 7/8 all right let's go up a level let's try three-digit numbers divided by one digit numbers the answer will always be either a two-digit number or a three-digit number for example let's try 326 divided by 9 now let's see that answer would be 36 and two ninths let me show you how I do that first I try and see how many nines I can squeeze into 326 I see that 9 times 30 is 270 9 times 40 is 360 which would be way too big so the first part of my answer is 30 and I'll say the 30 I'll get it out of my memory and give myself an easier division problem I've now subtract 270 from 326 giving me 56 so my new division problem becomes 56 divided by 9 oh that's easy because 9 times 6 is 54 so the answer is 36 now all I have to do to figure out the remainder is subtract 54 from 56 which is 2 so the answer is 36 with a remainder of 2 or 36 and 2 ninths all right here you try this one 523 divided by 6 okay first of all how many sixes can you squeeze into 523 6 times 70 would be 420 can we go higher 6 times 80 is 480 can we go higher 6 times 90 is 540 up to but too late you went above 523 so 6 times 80 is 480 say the 80 let me hear it 80 right ok now subtract 480 from 523 what do you get well if you're good at subtraction you should get 43 ok 43 divided by 6 is oh that's easy that's 7 with a remainder of 1 so the answer would simply be 87 with a remainder of 1 or 87 and 1/6 ok finally let's do a three-digit number divided by a one digit number that gives us a three-digit answer let's say 655 divided by 3 okay how many threes can I squeeze into 655 well 3 times 100 is too small 3 times 200 is 600 that looks just about right so we'll say 200 now subtract 600 from 655 leaving us with 55 so we've said the 200 and now our problem reduces to 55 divided by 3 well let's see 3 times 10 is 30 3 times 20 is 60 that's too high so I can next say the the 10 okay and subtracting 30 from 55 gives me 25 how many threes go into 25 8 with a remainder of 1 so the answer is 218 and 1/3 let's apply this to the following restaurant situation imagine the bill came to fifty seven dollars and twenty four cents and let's say there were four of you to divide the bill evenly of course well I break this problem down into two parts first I'll divide four into fifty seven dollars and then depending on the remainder I'll divide four into the rest let me illustrate how many times does four go into 57 well 4 times 10 is 40 which when I subtract from 57 gives me 17 how many times does 4 go into 17 4 times so 4 times 14 is fifty-six dollars so you can say 14 dollars right away 54 times 14 is $56 subtract that from $57 and 24 cents leaving us with a dollar twenty four cents so the problem has been reduced to dividing 4 into a dollar 24 4 times 30 cents is a dollar 20 subtract that from a dollar 24 you're left with 4 cents 4 goes into 4 once so you get exact answer of 14 dollars and 31 cents okay let's try another problem let's suppose instead of 4 people at the dinner table we have 6 okay how many times does 6 go into 57 dollars well 6 times nine dollars is $54 so you can say nine dollars subtract that from 57 be left with three dollars and let's not forget the 24 cents how many times the 6 go into three dollars and twenty four cents remember you've said the nine dollars already 6 times 50 is three dollars 300 so the next amount will be 50 and your oil that you're left with when you subtract three dollars from 324 is twenty-four cents so and six goes into 24 evenly that was convenient leaving us with six times four is 24 so the answer is exactly nine dollars and fifty four cents finally let's pretend instead of having four people at the dinner table we had five I saved five for last because that's actually the most fun that's the easiest one to divide a number by five it's usually a lot easier to multiply the original number by two that is double it and then divide that new number by ten see what I'm saying if I have fifty seven dollars and twenty four cents and I double it I get a hundred and fourteen dollars and forty eight cents now dividing by ten is a piece of cake to slide the decimal point over by one and you're left with eleven point four four eight which is roughly eleven dollars and forty-five cents per person now go to the workbook practice more division exercises and soon you'll find yourself conquering that great divide yourself whether you're in the laboratory in the kitchen or ask them all you have to know about fractions [Music] I hear from so many people that they understood math just fine until they encountered that dreaded F word fractions well I think that F should stand for fun because once you understand how to do simple arithmetic with whole numbers it's just as easy to do them with fractions let's start by multiplying fractions together that's actually the easiest operation with fractions let's say you wanted to do 3/4 times 5/7 ok all you do is you multiply the top and you multiply the bottom what could be simpler 15 divided by 28 fifteen twenty eighths is the answer I won't even do any more examples of that because that's so easy we've got more examples in the workbook anyway dividing fractions is practically as easy if you think about it division is the reverse of multiplication and dividing fractions is kind of like multiplying in reverse let me give you an example suppose the problem was 3/4 divided by 5/7 we take the second fraction 5/7 and we reverse it by that I mean we turn it upside down so it becomes seven fifths seven over five and now you multiply those two fractions so 3/4 times seven fifths is 21 over 20 or 21 twentieths okay here you try this one try 2/3 divided by 3 11 okay what do you do you take three 11s and you reverse it so it becomes 11 thirds multiply 2/3 times 11 thirds and you get 22 over 9 22 ninths exactly now the answer we have there 22 ninths can be simplified whenever your the number on top we call that number on top by the way the numerator hats a fancy term we use is bigger than the number on the bottom called the denominator then the answer can be simplified we have 22 over ninths we can think of that as a division problem 22 over 9 that's 2 with a remainder of 4 or we can just say our answer as 2 and 4/9 was I clear on that one so let's do another example 38 fifths the numerator is 38 the denominator is 5 since the numerator is bigger than the denominator we think of that as a division problem how many times does 5 go into 38 right 7 times with a remainder of 3 so the answer would be 7 and 3/5 is a simpler way of expressing 38 fifths the next thing I want to talk about is we can simplify another way if I took a number and multiplied it by 1 that wouldn't change the number right you know that so if I took the number one half and I multiplied it by 1 it would still be a half right of course right okay now suppose I replace one with something that's just the same thing as 1 let's write 2 over 2 2 over 2 is the same as 1 right if I take 1/2 and I multiply that by 2 over 2 that gives me what 2 over 4 and since 2 over 2 was just the same as 1 we must have the 2 over 4 is the same as 1 over 2 that is 2/4 is the same as 1/2 similarly if I took one half and I multiplied it by 1 but now i disguised 1 as 3 over 3 I should get the same thing and 1/2 times 3 over 3 is 3 over 6 so that means that 3 sixths is also equal to 1/2 do you see that that those two fractions are the same thing they represent the same quantity okay by the same logic if we can divide the top and the bottom by the same number we'll still have the same for action because really what we're doing is we're dividing by one and dividing by one gives you the same number for instance let's say you were looking at the fraction 27 over 36 now what number what's the biggest number you can think of that divides into the top and divides into the bottom nine right 9 times 3 is 27 9 times 4 is 36 if I divide both the top and the bottom by 9 all I'm left with is 3 over 4 so 27 over 36 simplifies to the fraction 3/4 now for a practical application let's suppose you had a recipe that would show you how to make a 3 dozen chocolate chip cookies and the recipe called for 3/4 of a cup of chocolate chip cookie dough but you don't want three dozen cookies you only want two dozen cookies so the question is how much cookie dough are you going to need well since you don't want three dozen you only want two dozen that means you're gonna have to make 2/3 the amount of cookie dough that's 2/3 the amount of cookie dough is how much you're gonna have to use so we take 3/4 of a cup times 2/3 multiply those together and we get 6 twelfths which of course can be simplified to 1/2 a cup of cookie dough is how much you'd need to make 2 dozen cookies before we go any further now would be a good time to go back to the workbook and practice practice practice that's what you need to be able to get very good at doing fractions okay now we come to the next step which is adding fractions which is actually a little bit trickier than multiplying and dividing the easiest problems to do are when the denominators that is the numbers on the bottom are the same for example let's suppose you wanted to add three tenths plus 4 tenths okay add that all you have to do is add the numerators and keep the denominators the same and you get 7/10 you see let's do one more example for you suppose you wanted to add 9/16 plus 3/16 add them together what do you get you should get 12 sixteenths which can be simplified to 3/4 because the top and the bottom are divided divisible by 4 what if the bottom numbers the denominators are not the same let's suppose you wanted to add Oh 3/16 plus 1/4 okay now the denominators are different but we can make them the same and the way we do that as we say hmm 4 divides into 16 so I can write 1/4 as four sixteenths if I multiply the top and the bottom by 4 1/4 becomes four sixteenths and now the problem becomes adding 3/16 plus the same number four sixteenths which of course is 7/16 was that a little confusing here let's do one more example and you'll catch on try adding 1/5 plus 3/10 ok we're lucky here because I can multiply 2 times 5 and get 10 so let's multiply the top and the bottom by 2 so 1/5 is the same as 2/10 I'm now going to add that to 3/10 and we get what 5/10 exactly and 5/10 simplifies to 1/2 ok now suppose the denominators are different and you can't just multiply one by something and get the other like let's say 3/4 plus 5/7 ok well I'm going to make the denominators the same by multiplying the first fraction by 7 over 7 and the second fraction by 4 over 4 so this makes the first fraction equal to 21 over 28 the second fraction equal to 20 over 28 add 21 over 28 plus 20 over 28 now the denominators are the same and we get 41 over 28 which by the way if you wanted to simplify that fraction well since 41 is bigger than 28 we can simplify that to 1 and 13 28 okay we'll see one more example of this difficult problem kind of problem let's add one-half plus five ninths okay so multiply top and bottom of the first fraction by nine gives me 9/18 I chose nine because that was the other denominator I multiply five ninths by two over two to give me 10 18 we add 9/18 plus 10 18 and that gives us 1918 which if you want to simplify is one and one eighteenth finally subtractions a lot like addition if the problem was say four tenths minus 3/10 then just go ahead and subtract the numerators and you get 110 okay so here if you wanted to subtract nine 16 minus three sixteenths what do you get 6/16 of course which simplifies by the way to 3/8 I like simplifying things I like to keep life as simple as possible ok let's try now the denominators are different let's try 1/4 minus 3/16 what do we do we want to get both denominators to be the same let's multiply the first fraction by 4 over 4 so we have four sixteenths minus three sixteenths and now it's easy the answer simply 1/16 okay here you try it try eight ninths minus two thirds okay what are you gonna do let's convert two thirds into a fraction that'll have 9 on the bottom ok so I'll multiply top and bottom by 3 so 2/3 becomes six ninths right so eight ninths - six ninths is simply two nights that wasn't so hard okay one last this will be the toughest of them all let's try subtracting seven tenths minus 2/9 okay seven tenths minus 2/9 well nine and ten well neither of those divided into each other so I'm going to multiply the fractions seven tenths by nine over nine and the fraction two ninths by ten over ten so my first fraction becomes 63 over 90 my second fraction becomes what 20 over 90 63 over 90 minus 20 over 90 is 43 over 90 which would is the answer to the problem seven tenths minus 2/9 okay before we go any further I insist you take some time and practice using problems from the work booklet and then when we get back we'll do some more interesting stuff with fractions that will amaze your friends [Music] the last thing I want to talk about is how to convert a fraction into a decimal if the denominator is 11 or smaller and that process is very very easy and it's explained in the workbook okay the only denominator I want to talk about give special attention to now is when the denominator is 7 because it's it's so amazing it's so magical that I got to show it to you I don't want to just let the book show it to you okay let's look at the fraction 1/7 okay if you were to expand it into decimal notation you would get point one four two eight five seven one four two eight five seven one four two eight five seven and so on and so on repeated forever okay all you have to remember is those numbers one four two eight five seven next let's look at the fraction two sevenths two sevenths is also a repeating fraction but it goes point two eight five seven one four two eight five seven one for repeated but do you see that that's the same numbers as we had before just shifted over by a little bit right we have the two eight five seven one four similarly three sevenths is point four two eight five seven one repeated it's the same six numbers except they start with the number four this time point four two eight five seven one four two eight five seven one and so on okay four sevenths is 0.5714 to eight repeated five seven one four to eight and so on okay five sevenths is point seven one four two eight five repeated and 6/7 is point eight five seven one four two repeated so once you memorize that original cycle one four to eight seven then you've got all of them memorized okay now if ever you were given a division problem where you were asked to divide something by seven let's say the problem was 45 divided by seven you could say the answer is forty five sevenths but that's not really saying very much or you could say the answer is six and three sevenths that sounds nice but if you really want to sound like a genius you'll say six point four two eight five seven one four two eight five seven one four [Music] there are times when all you need is a good mental estimate such as when you're reading the stock page or verifying the numbers that your calculator spits out [Music] one of the great advantages of working from left to right is that you work with the most significant digits from the start it often times that's all you really need to get a good estimate of your final answer for example suppose we wanted to add three thousand four hundred and seventeen plus 6195 now gosh that's a lot of numbers to hold in memory and you don't even maybe want the exact answer let's just approximate the two numbers with let's say the first number is approximately 3,400 and the second number is approximately 6200 add those numbers together and we get 9600 which is very close to the exact answer of 9600 and 12 ok let's try let's try a larger problem but it's just as easy if you want an estimate let's say you wanted 23456 plus 56789 okay let's just round to the two most significant digits 23456 that's approximately 23,000 and the second number is closer to 57,000 than the 56,000 so we'll call it 57,000 now we add 23,000 Plus 57,000 to get 80,000 which is almost the exact answer you probably couldn't even hold on to that entire problem in your head unless you use some of the advanced memory devices that we teach in the advanced video but if you just want an approximation this will do just fine subtraction just as easy let's say we wanted to do 6195 and this time instead of adding it will subtract 3417 the first number is about 6200 the second number is about 3400 subtract those numbers and you'll get about 2800 and it's as easy as that now I don't actually compute my grocery bill to the exact penny when I go shopping you remember Todd did that on our television show and let's see what he did exactly and then I'll show you how you could have done the same thing approximately okay let's have a look at it can we give you a little test take your best shot Oh were he pretty confident also join us on today's show is Tony from Tony's deli welcome back to shopping at tony's deli we have some items there ready to get checked out and Tony I want you to check them out using your cash register go ahead no problem Mike listen 325 the first item amazing brand good stuff I got for 80 cents I got unspecial for you Mike comeall $5.99 Mike Rowe wave popcorn and at 435 if you're a couch potato like my cousin Vinnie eat these all night chuck full of nuts don't push the total button wait Tony cuz Todd's gonna tell us how much we spent so far Tony's deli how much we spent she spent $14 and 39 cents all right all right now you see what he's done here 325 plus 80 cents adding that left to right that's not a problem right it's four dollars and five cents okay what was the next item $5.99 Mike Rowe wave popcorn okay $5.99 okay well let's add $6 that gives us ten dollars and five cents but we got to subtract the penny giving us ten dollars and four cents well that's an easy number let's now add four dollars and 35 cents to ten dollars and four cents to give us fourteen dollars and thirty nine cents which was his answer nice going Todd but if you just wanted to get an approximate answer here's what I do as the numbers are being crawled out I will always round the numbers either up or down to the nearest 50 cents if I'm exactly between a dollar even dollar amount and a 50-cent amount like if I had 25 cents or 75 cents I'll just go to the even dollar amount because that's easier to work with all right so let's run that again and this time I'll see how we could have done that just using approximation 325 the first item amazing brand good stuff I got for 80 cents I got unspecial for you my co meal $5.99 Mike Rowe way popcorn and at 435 if you're a couch potato like my cousin Vinnie eat these all night Chuck full of nuts okay the first item was three dollars and 25 cents take your pick you want three you want 350 I'll call it three dollars because that's easier okay the next item was what 80 cents well that's closer to a dollar than the 50 cents so I'll add a buck so now the answer is four dollars least that's my approximate answer the next item was $5.99 oh that's easy just add six dollars even so we have ten dollars and finally four dollars and 35 cents let's round that up to 450 because it's closer to 450 than it is to four dollars and so my approximate answer would be $14.50 which is only off by eleven cents if you do this even for very large lists your answer will usually be within about a dollar or so from the actual answer which for me is quite good enough [Applause] [Music] one of my favorite calculations is squaring that is taking a number and multiplying it by itself for instance five squared is 25 now let's watch part of my show at the Magic Castle and afterwards I'll show you how I do it I'm going to try and do now is to square four two-digit numbers in my head faster than they can do on the calculators even using the shortcut method what I'll ask is for the four of you here to each quad a single digit a single two-digit number I'm sorry and if you would square the first the second the third and the fourth I will try and race you to the answer so quickly a two-digit number please 84 123 57 79 would you cross your answers please 7050 6 5 29 3249 thank you very much one of my favorite calculations to do is squaring numbers that is taking a number and multiplying it by itself for instance five squared is 25 let's try squaring some two-digit numbers these are very easy to learn and once you've learned them you'll look like a genius without really trying we'll start off small and work our way up let's start with say 13 squared 13 is not an easy number to multiply by but what number close to 13 is 10 right now you have to go down 3 to get to 10 now whatever comes up must come down or whatever comes down must come up is what I think if I go down 3 to 10 I must go up 3 to 16 so the first part of my calculation is I multiply 10 times 16 that's easy that's 160 we're almost done all we have to do is to add the square of the number we went up and down we went up and down 3 3 squared is 9 and so the answer is 160 plus 9 which is 169 and that's all there is to it now the reason why this works is explained in the booklet but let's go on let's try 28 squared ok now 28 this time I'll go up to 230 if I go up to 230 I must go down to 226 now what's 30 times 26 now don't panic that's just 3 times 26 with a friendly 0 attached 3 times 26 is 78 so 30 times 26 is 780 almost done all we have to do is add the square of what number 2 good 2 squared is 4 and that's the answer 784 here you try this one this one's even easier 75 squared this time we'll go down five to seventy and up five to eighty now what seventy times eighty piece of cake right it's fifty six hundred almost done all we have to add to that is the square of the number we went up and down five squared is 25 5,600 plus 25 is 50 625 you can actually hear the answer all right let's do one that's a little bit tougher but not too much let's try 59 squared okay we go up one two sixty down one two 58 60 times 58 while six times 58 is 348 so 60 times 58 is three thousand four hundred and eighty add to that the square of 1 which is 1 three thousand four hundred and eighty plus one is three thousand four hundred and eighty-one by the way if that 60 times 58 problem gave you some difficulty you should review the two-digit numbers times one digit numbers covered in the multiplication module okay quiz time try and do this one in your head all right and use that pause button when you think you have the answer you can push the pause button again 96 squared now hit that pause button okay I'm about to say the answer the answer is nine thousand two hundred and sixteen is that what the answer you got if not go ahead and rewind and try and do it again and see if you can get that answer okay here's how we got nine thousand two hundred and sixteen we went up four to one hundred down four to ninety to ninety two times a hundred is ninety two hundred add to that the square of four four squared was sixteen and that's the answer ninety two hundred and sixteen if you got that right within the first two tries congratulate yourself as being a budding Mathemagician let's try a problem now of the sort that you might see on the SAT exam this isn't really a calculation problem but it's one that requires understanding where squares come into play the problem might be as follows which of the two squares on the screen has the larger area the square whose sides are all of length 6 or the square that lies on that right triangle whose sides are of length 3 & 4 the square on the left will have an area of 6 squared which is 36 the square on the right well here's where good old Pythagoras comes into play Pythagoras says that if I have a right triangle with sides a B and hypotenuse C then the a squared plus B squared will add together to give me C squared and if I had a square lying on that hypotenuse its area would be C squared now what is C squared here it's 3 squared plus 4 squared that's 9 plus 16 which is 25 so the area of the second square is 25 which is less than the area of the first square by the way it also tells us that the hypotenuse have to have a length of 5 now let's go on and do one last calculation let's say a three-digit square this is a sort of problem that we do of in much more detail on the advanced video but I've included this one here for sentimental reasons because it's the first three digit square that I ever calculated suppose we wanted to square the number 108 this time we'll go down 8 to 100 we'll go up eight to 116 now 116 times 100 that's just 11600 almost done all we have to add to that is the square of 8 the square of 8 is 64 and there's your answer eleven thousand six hundred and sixty-four now do some problems in the workbook and you'll be impressing your friends in no time [Music] wouldn't you say that to be a mathematical wizard you need to learn algebra perhaps the most important subject to learn when taking the SAT is algebra and in order to master this most serious of all subjects I've decided to introduce it with something that's very fun magic tricks let's let's start off with a trick that you can help me with at home I want you to think of a number between 1 and 10 you can think of a larger number but for this first experiment just keep it between 1 and 10 it'll be easier that way now take that number and double it all right now add 10 to the number that you're thinking of now take what you're thinking of and divide that number by 2 ok got it now subtract the number that you originally started with and concentrate on that answer I think you're thinking of the number 5 is that right good and now let me explain how to do that using algebra I did not know what number you thought of admittedly a lot of people think of 3 or 7 to begin with but it could have been any number let's call that original number X that's the most famous number in algebra X now that stands for something we don't know let's now what was the first thing I asked you to do to take that number and double it right so if we started with X then now we have 2 times X now we won't write a time symbol that would be confusing it would look like 2x X and we wouldn't want that so we'll just call that 2 X next what did we do we added 10 to that number okay so now we have the number in our mind to X plus 10 next you took that number to X plus 10 and divided it by 2 well let's divide each term by 2 2x over 2 is X 10 over 2 is 5 and so now you're holding on to the number X plus 5 finally you were asked to take that number and subtract the number you originally started with what was the number you originally started with it was X so X plus 5 minus X leaves you with 5 you have to end with fine and that's the best example of algebra that I can think of for getting us started for my next trick go and bring a calculator to the television set if you don't have one with you go ahead press the pause button I'll wait okay you have a calculator now I'd like you to enter a three-digit number on the calculator with the only requirements being that the first digit has to be larger than the last digit okay so for instance I'll choose the number 781 because 7 is bigger than 1 but don't let me influence your decision go ahead pick a different three-digit number okay now reverse your three-digit number and subtract that from your original number so for instance if I reverse 781 I get 187 when I subtract that from 781 I get 594 okay now you should do the same with your numbers all right next if you have if your answer is a three-digit number then leave it alone for the moment okay if it's a two-digit number like let's say your answer was 57 then attach a 0 to the front of your number so now you'd have 0 57 okay all right so now whether you if you had a two-digit number you now have a three-digit number because begins with a zero if you had a three-digit number as your answer it's still a three-digit number either way I'd like you to now take that answer and reverse the digits okay so if I take my 594 and reverse the digits I get 495 you should do the same now add those two numbers together so if I add 594 plus 495 I get 1089 what do you get if you followed my directions properly you should also get 1089 and if you want to understand why that'll always work to see the algebraic explanation consult the workbook [Music] for my last trick I'm going to need a volunteer from the audience but since we don't have an audience here the closest person near me is the camera operator Suzanne Suzanne would you be willing to help me out on this one great now I have a chart here and I want you to help me out by thinking of a number again let's keep it fairly small think of a number between 1 and 10 and enter that number in Row 1 but don't let me see it okay now what do you think of another number between 1 and 10 make it a different number and enter that in row 2 okay next I want you to add Row 1 plus Row 2 and write the answer in Row 3 okay next add Row two to Row 3 and write the answer in row 4 then add Row 3 to Row 4 and write the answer in Row 5 and keep on doing that Row 4 + 5 - what gives you row 6 row 5 plus 6 gives you row 7 and so on until you have numbers in all 10 rows ok next one thank you next what I'm going to do is I'm gonna try and add up those 10 numbers faster than you can now I haven't seen your numbers yet I'm gonna peek at them right now hmm I think the numbers add up to 561 is that what you get ha ha good I'll explain how that works using algebra okay the I did not know what number you started with so let's call that number X again X stands for the unknown number you wrote another number in Row 2 and I didn't know what that one was but it was a different number so let's call it a different letter let's call that Y next we add X plus y when we added Row 1 to Row 2 we put that in Row 3 so Row 3 will have X plus y next we added Row two to Row 3 and where it gets a little tricky by add Y to X plus y I get as an answer X plus 2y okay next if we take X plus y and add that to X plus 2y we get 2x plus 3y if you understood how I did that then the rest should be smooth sailing okay we take X plus 2y and that's a 2x plus y that gives us 3x plus 5y okay if we add 2x + 3 y - 3 X + 5 y we get 5x + 8 Y and so on so what you can fill in the rest of that chart right ok so the last number there was 21 X + 34 Y good now I want you to add up all of those letters okay if we add up all the X's we've got an X than another ax another ax of 2 X 3 X 5 X 18 X 13 X 21 X add all those together and then we get 55 X and if we add up all the Y's we get 88 y so all those numbers will add up to 55 X + 88 Y now how did I figure out your answer so quickly it's based on what's written in box number seven What's in box number seven it says 5 X + 8 Y if you take 5 X + 8 Y and multiply it by 11 then by the distributive law we're gonna have 55 X + 88 Y which is the grand total so what I did was I looked at what was written in box number seven and I multiplied it by 11 let's see your original numbers again okay now in box number seven you have 51 which I multiplied by 11 and quickly got the answer 561 and that was your total okay now I've got one more trick for you I want you to now take the number that you have written in box number ten bring out a calculator for this divide it by the number that's written in box number nine okay so in your case that's 215 divided by 133 before I began this last trick I wrote down a prediction on the back of this card I predicted what would happen if you were to take the number in row 10 and divide it by the number in row 9 go ahead with the numbers that you have there which are 215 divided by 133 okay give me the first three digits of your answer okay I get do you get a hundred and ninety one oh no what do you get one point six one oh I see that's the what I had I just had the card written upside down well if you want to see why that works consult the booklet as well I hope you enjoyed your first exposure to mathematics thanks for joining [Music] [Music]
