Math Mojo - Making Math Meaningful

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(Information about division of decimals is further down on this page)
The bottom of the page includes new information about using factoring to make division simpler, and using a very simple method to check your answers in division.

This was the question:

One of my tutoring students who is studying for the SAT doesn't know how to do long division. I guess she got through all these years by using a calculator. I tried to show her how but she acted like I was speaking a foreign language.

Do you have a simple explanation that I could show her that would break through her fear?

Professor Homunculus' answer:

The way long division is usually taught, is one of the greatest "Weapons of Math Destruction" in the public school's arsenal.

The method I am about to describe is simpler and makes more sense than the typical way. Unfortunately, the description is long. The process is not really long, though, and once you understand it, it will take less time, and you will make fewer mistakes than with what you were originally taught.

While reading this, keep in mind (as usual with Math Mojo methods) that the way you originally learned was drilled in over a few years of elementary school. You were shown and shown again (rather than actually taught) and tested and tested again. And still most students have problems with it. So don't expect to get the Math Mojo method automatically. Give it a chance to sink in.

I promise, that once you have understood it and practiced 5 or 6 problems with it, it will be easier than the other way. It may take a day or two to sink in. That sure beats years, doesn't it?

Let's get right to an example. How about 36,550 / 86?

• First, make a table of the first 10 multiples of the divisor (86) by multiplying 86 by 1, 2, 3 ...10.
  

The easiest way to do it is by repeatedly adding 86 to the number you get. *

The actual writing you would do would look something like this table:

Unless you didn't need to write the "+86" each time. In that case the table would look like the one below:

The numbers in those tables are multiples of 86.

Here is the special way to compare the above table to the dividend:

• Look at the original divisor (86). Now, in your mind, imagine zeros behind it. First imagine one zero behind it (making it 860). Compare that to the dividend. Could 860 go into 36,550? Yes. So far so good.

• Now imagine another zero behind it (making it 8,600). Could that number go into 36,550 safely? Yes, it also could. So far two zeros work.

• Keep trying till it won't work. Will three zeros behind 86 (86,000) go into 36,550? No, not this time. Three zeros won't work. Well, if 86,000 won't go into 36,550, neither will any number higher than it. That means that there will be no thousands in the answer. (In other words, no number with three zeros behind it, or higher.)
  

The above method has determined that the highest number which we can multiply 86 by, without getting over 36,550, will be in the hundreds.
Now, if we multiplied all the multiples of 86 in the table above by 100, we would get the following numbers:

The zeros in gray are imagined.
Of course, you could actually write these numbers somewhere, but getting good at math means improving your imagination, so try to do it without writing. The less writing you have to do in math, the more you will be increasing your brain-power. (Obviously that goes even more for avoiding the accused calculator!) Face it - writing those numbers down would be a waste of time.

• Now simply determine which of the above numbers is the highest one which goes into your dividend. In this case it would be 34,400. So the first digit of the answer is a 4 (in the hundreds column).

Because you have recorded the 4 in the hundreds column of the answer, you can take 400 x 86 (which is 34,400) away from the dividend safely, before you continue on with the problem. How do you do that? Subtract it. 400 x 86 = 34,400, as we have already shown. (Just to remind you how we showed it: 4 x 86 is 344, which was shown in our table. Then, by adding two zeros at the end, we multiplied 344 by 100. And you can easily see that 4 * 86 *100 is the same as 86 * 400).

• Therefore, the next step is to subtract 34,400 from 36,550.
  36,550 - 34,400 = 2,150.

• If you repeat the exact same procedure with 2,150 which you did with 36,550, you will get the next digit of the answer (which will be in the tens-column, obviously). We'll do this step together after the following "nutshell" explanation.

1. Make a table of the first ten multiples of the divisor.
   
   
2. Imagine zeros behind the divisor. Determine how many zeros would be the most that could be there, and have that number fit in the dividend.
   
   
3. Imagine that many zeros behind all the multiples in the table you have already made.
   
   
4. Determine which of those number is the highest which will fit into the dividend (or the new dividend, if you have already done these steps once).
   
   
5. Notice which multiple of the divisor that number (before you imagined the zeros behind it) was next to in the table. Write that number in the quotient.
   
   
6. Subtract that number (the one which you had determined was the highest that would fit in the dividend) from the dividend.
   
   
7. Repeat with the number you get after that subtraction (we'll call it the new dividend).

Okay, let's apply those steps to the rest of our problem, which was 36,550 / 86.

To recap: So far we have determined that 4 will be the first digit of the answer, and that we still have to deal with 2,150.

1. We can still use the same table. Once you have made it for any problem, it never changes. (The divisor always remains the same, therefore, so will the multiples of it in the table).

2. Same as before - Imagine zeros behind the divisor. Determine how many zeros would be the most that could be there, and still ha ve that number fit in the dividend. The only difference is that this time the dividend is 2,150. So we imagine one zero behind 86, giving us 860. That would fit into 2,150. Would 8,600 fit? No, it's larger than 2,150.

3. Therefore, we imagine only one zero behind the numbers in our table.

4. Determine which of those number is the highest which will fit into thenew dividend (2,150). Here's the table again (remember, the zeros in grey are just imaginary):

86 times
1 =	860
2 =	1720
3 =	2580
4 =	3440
5 =	4300
6 =	5160
7 =	6020
8 =	6880
9 =	7740
10 =	8600

 

1,720 would be the highest which would fit into 2,150.

5. Notice which multiple of the divisor that number was next to in the table. Write that number in the quotient. In this case, it was 2, so write the 2 in the tens-column of your answer, next to the 4, which is already in the hundred-column.

Here's how your problem might look so far:
36,550	/	86	= 42
-34,400
2,150

6. Subtract that number (the one which you had determined was the highest that would fit in the dividend, which in this case is 1,720) from the dividend (which in this case is 2,150).

2,150 - 1,720 = 430.

Here's how your problem might look so far:
36,550	/	86	= 42
-34,400
2,150
1,720
430

7. Repeat with the number you get after that subtraction (we'll call it the new dividend). In this case it is 430.

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We have come to the final step.

Hopefully, you can see that we only have to determine which digit will have to go in the ones-column of the answer.

To do this, you really only need to do step 5 in the above instructions. Simply look at the table, and determine which multiple of the divisor the remaining number (in this case, 430) was next to in the table. Write that number in the quotient. In this case, it was 5, so write the 5 in the ones-column of your answer.

Here's how your problem might look , notw that it's done.
36,550	/	86	= 425
-34,400
2,150
1,720
430

 

You have written a 4 in the hundreds-column, a 2 in the tens-column, and a 5 in the ones-column. That gives you 425. That means the answer to 36,550 / 86 should be 425.

Now check it.

No, really, check it. What's the use of doing something without being sure that you at least have a chance of being right? Especially when you are doing a new method.

Step away from the calculator. I repeat, step away from the calculator! Don't use that sleazy thing to check your answer. Any idiot can do that (and most do!)

Multipy the divisor (86) by the quotient (425) and see if you get 36,550.

The next step in Math Mojo is to check each of the above steps and the answer, to make sure you make no mistakes along the way. There is a simple way to do this, although, like most of Math Mojo, it takes effort to explain than it does to learn. And once you learn it and practice it a bit, it takes almost no effort to use. This method of checking will be in a forthcoming booklet. The booklet will cover an amazing way to check all your arithmetic (addition, subtraction, multiplication, division and exponents) using one simple system (and it is NOT using the reverse of the operation you originally used!) It has nothing to do with the way you learned in school, and it is quicker than using a calculator, It is the best thing I could possibly teach you for use on standardized math tests (although that should never be the main incentive). I hope to have the booklet available by the spring of 2003. If you really are curious, or you need it for upcoming tests, you can send me an e-mail to hurry me along.

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---

Alright, we both need a break after doing that. There will be another example of this up soon, so check again if you don't get it so far.

You worked hard, you need a break. Click here for a very weird mathematical anagram, which might give you a smile.

If you need practice doing long-division, it is easy to make up your own problems. Take a phone book, look at some random numbers in it, and multiply two of them together. Use two-digit numbers first. Take the answer and use it as the dividend. Use either of the numbers you multiplied together as the divisor. When you do the division, you should get the other number (of the two you multiplied together) as the quotient (the answer).

If you don't, it doesn't mean that you can't divide well. It might mean you can't multiply! That's why you should use this method to get practice questions. It helps you learn to divide and to multiply!

Another Warning! This is not the end. There are simplifications of this method, but none will make sense to you unless you learn this method well first. Math Mojo is not just a bunch of math tricks and shortcuts. It is built on the principle that if you undertstand something deeply (not just "know how to do it"), that will lead you on to a deeper understanding of the next step, too. You can't get there from here without work, thought and experimenting. The simplifications of this method will lead back to something like the way you were taught in school. But by then, you will be armed with methods of how do to all the steps in that method quicker, more accurately, and with a deeper understanding of how and why they work. As often is the case, it is not the material which you learned in school which stinks, it is was the method you were taught, and the meaningless stress you were put under. I'd like to put you under more meaningful stress, here. The stress of learning for your own sake and enjoyment, not for the sake of some stupid grade, and some poorly-thought-out, artificial "standards". Make your own standards. But set them high. 'nuff said. Professor Homunculus

Some curious reader sent this in April, 2005:

I was just wondering how your deviant method for long division might work if a decimal was thrown in? Example:
17 / .04138

Professor Homunculus replies:

Good question!
Whenever you are dividing with decimals involved, it is good to "get rid of the decimals." To do that, you take the number with the largest amount of digits behind the decimal point (in this case this would be .04138, because it is the ONLY number with digits behind the decimal point) and move the decimal point to the end of that number.
When you do that, you are moving the decimal point 5 places to the right. So you have to do that to the other number (the 17), also.
Now the problem becomes 1,700,000 /4,138, and solve this by exactly the same method that is shown on this web page.

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The same curious reader wrote back:

Alright professor,
Your reply to my decimal division was extremly
helpful, UP TO THE POINT WHERE I GET TO THE REMAINDER.
How do I divide a smaller number by a bigger number?
My original division was 1700000/4138.
I have a remainder of 3420.
How do I divide 3420 into 4138 pieces?
I've tried other websites to try and decipher the concept of solving improper fractions...i'm lost.

Thanks again for your time!

Professor Homunculus replies:

And thanks for your question! Readers like you are the ones that keep this website up to date, (and fun to maintain!)

I think I get what you are asking. The concept is a bit screwy, but it is not hard. The best way to expain it is to simplify it, so:
Imagine you have to break 6 into 45 pieces. (Same kind of problem, right?)
What you do is turn that 6 into 6.000...(with as many zeros as you want behind the decimal point). After all, 6 is the same as 6.000000000, isn't it?
Now, you divide just like you would 45 into 60000000... (keep going until either you get no remainder, or until the answer starts repeating).
In the answer, you put a decimal point at the place where you started having to to put a decimal point in the dividend.

Let's do 4138 into 3420:
Here's the list for the multiplications of 4138:
4138 x   1 =   4138
4138 x   2 =   8276
4138 x   3 = 12414
4138 x   4 = 16552
4138 x   5 = 20690
4138 x   6 = 24828
4138 x   7 = 28966
4138 x   8 = 33104
4138 x   9 = 37242
4138 x 10 = 41380

Now, 3420 becomes 34200
Just continue the regular way, but remember that everything you get from now on goes behind the decimal point of the quotient (the answer).

The largest multiple of 4138 that goes into 34200 is 33104, so write an 8 after the decimal point in your answer.

When you subtract 33104 from 34200, you get 1096.
We know that we can put an unlimited amount of zeros behind the dividend because they are all behind the decimal point. So go ahead and put a zero behind the 1096 and you will get 10960.

The largest multiple of 4138 that goes into 10960 is 8276, so put a 2 at the end of your answer so far.
Now we have to subtract 8276 from 10960, and we'll get 1230.
Put zero at the end of that, and keep dividing that by multiples of 4138 like we have been doing up till now.

I think you will find that you will need to keep doing this quite a few more times until you get a number with no remainder. If you have lots of patience keep going and let me know what you get. I am getting tired of typing now.
I wish you lots of luck. It was fun going over this with you. Thanks for the inspiration!

Yours truly,
Professor Homunculus

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The same curious reader wrote back:
I'm sort of stuck again. After trying to divide
76,013 into 23 pieces I hit a roadblock. Here goes...

23x1=23
23x2=46
23x3=69
23x4=92
23x5=115
23x6=138
23x7=161
23x8=184
23x9=207
23x10=230

I realize that adding one zero to 23 could work adding
2 zeros could work, adding 3 zeros could work, but 4
would not.

Adding 3 zeros to 69 (23x3) (69,000) would be the
closest number to subract from 76,013.
So I put 3 in the hundreds position and subracted
69,000 from 76,013 and got 7013

Due to the decending nature of your zero-adding method, I naturaly added 2 zeros to the closest number that could be subtracted from 7013, which again was the product of 23x3= 69.

I placed a 3 in the TENS position and continued with
solving the problem. So far my answer was 33.
7013
-6900
113

This is where I was confounded, the closest logical
answer from the 23 times table would be
23x4=92...113-92=21...I'm sorry to say i had to divert
to my calculator::prostates head in shame::
Through my insurection i dicoverd that the answer was
3304.9130435...3304!? Where did the ZERO come from!?

Professor Homunculus replies:
Remember, step 2 was:
"Imagine zeros behind the divisor. Determine how many zeros would be the most that could be there,and have that number fit in the dividend."

You have to imagine at least one zero behind the 23 in this step.
If you can’t, it obviously goes in no times. That means you have to put a zero in the quotient, and go on to the next number. I hadn’t made this perfectly clear in the instructions, so it is not your fault.

So now you would have 330 in the quotient so far.

Simply add a zero behind the 113 in the dividend now to continue. That gives you 1130.

From our table, we see that 230 is the highest that could go in there, and the highest multiple of that would be 920. So put another 4 in the quotient, (making it 3304 so far) and subtract 920 from 1130 to get 210.

We are into the decimals, now, and as you know from our last session (the one about decimals) you have to start adding zeros to the dividend (because everything in the quotient from now on is going behind the decimal point). So the 210 becomes 2100.
230 goes into that 9 times, so but a nine in the quotient, giving us 3304.9 so far.

From here on in you’re on your own until you come up across anything else you think I missed. Please let me know if this solves your problem.

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Another astute reader wrote in:

I really enjoyed your "easier way to divide" lesson. I had never
thought of making tables of the ten first multiples of a number.
I'm not sure it's very convenient to have to do that for every
computation, but as you say it's a good exercise and it made me
understand what dividing is about better than ever before.

Still, take a simple problem like 684/6.

Once you've written all the multiples of six up to ten, you only get 60
(plus an imaginary gray "0"), which is not enough to go into 684. It is
66 that fits (6 * 11). What if the right multiple was further down the
list.... like in the 23s or 28s times 6?

I think you intend that method only for long division as I realize it's
easier to do 684/6 digit by digit.

Thanks again for the trouble you take to make maths enjoyable.

Professor Homunculus replies:

If you look at 60, with an imaginary 0 behind it, you get 600, which goes into 684 one time. So the first digit of the quotient would be a 1 in the hundreds place.

That leaves you with the rest of the quotient (86) do deal with. 60 is the highest that goes in there, so a 1 goes in the tens place, leaving you with just 24 left in the quotient. 24 is the highest that goes in there, giving you a 4 in the ones place, and as the Brits used to say, "Bob's your uncle!" You're done.

You said, "I think you intend that method only for long division as I realize it's easier to do 684/6 digit by digit"

Yes, I do, actually. But some people have a really hard time doing multiplications "on the fly." Even the easy ones. For those people, I think the above method is a good way to learn even simple division. It doesn't take a long time to get used to it, and then the mind says, "Hey, I don't have to write that simple stuff down anymore!" Obviously you are past that point. But I still use this simple way to teach even simple division, because it makes so much sense. Once people learn it, they really do get the "Idea" of how division works, which takes away the frustration, and lets them get on to frying bigger fish.
Well, that's my theory, anyway!

There is a great trick which I haven't mentioned on the site yet. It involves factoring. It works great for your example: 684/6.
You can immediately see that 2 is a factor of both numbers (this is always the case when both the dividend and divisor are even). So you can safely divide each number in two, without affecting the outcome of the problem.

Thus, 684/6 becomes 342/3. Now the problem is so much easier, and you haven't really expended any significant energy.

Once Professor Homunculus gets around to writing the booklet on factoring, it will contain a way to immediately identify 342 as divisible by 3 evenly, and how to do it without much effort at all. All of which would enable one to simply look at 684/6 and come up with 114 without doing any "classical" division.

By the way, to check the answer, you don't need to do a large multiplication. In the See-Say-Write book about addition a method is explained to check addition answers. The same method of crunching to check can be used for division. Crunch the quotient (it crunches to 6), crunch the divisor (it also crunches to 6). Multiply them. 6 * 6 = 36, which crunches to 0. If the dividend crunches to to anything but 6, it is wrong. 684 does crunch to 0, which doesn't make it definitely right, but it is a good bet.

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