Long Division: When Zero Gets in the Way

I was going to move on from arithmetic to algebra, but the discussion of long division led me to think about some of the more ordinary difficulties students have asked about in that area. Here I will show several questions about the process of long division in which zero caused trouble.

Zero in the quotient

When you come across a step in long division where the divisor is larger than the current working dividend, you need to put a zero in the quotient. Students often have trouble here:

Problems with Zero; Writing Remainders

When I do long division I have trouble with some of my answers, where the answer is correct except for one thing in the middle that sometimes contains a zero.

For example, when I did this one:

   38/1294058

I got the answer 3465 r23/28 but when I checked the answer it read 34065 r23/38

Where am I going wrong?

Can you see three different issues here to talk about? The main one will be the zero in the quotient that Lacy missed; but there is also her (or the book’s) use of “r” with a fraction rather than a remainder, and writing 38/1294058 (the format used in doing the division) where we would properly write 1294058/38, or 1294058÷38. In addition, the answer given is not correct! I couldn’t find a likely correction, so I just went with the problem as given. (That was fine, because in doing this I wasn’t giving anything away: Lacy already knew the correct answer, and evidently the problem I did wasn’t the one she had to do, anyway!).

You probably just forgot to write a zero in the quotient when you had to "bring down" an extra digit from the dividend because the remainder was bigger than the divisor. I'll do the problem in a way that avoids that:

       ____34054_r_6
    38 ) 1294058
         114
         ---
          154
          152
          ---
            20
             0   <-- the problem is here
            --
            205
            190
            ---
             158
             152
             ---
               6

(I don't know where one of us copied wrong, but I'll go with this problem anyway!)

What I did here is to follow the standard process carefully, actually writing down the product, 0, where the more efficient method is to skip that step and just write a 0 in the quotient:

       ____34054_r_6
    38 ) 1294058
         114
         ---
          154
          152
          ---
            205  <-- two digits brought down need two digits 
            190      in quotient
            ---
             158
             152
             ---
               6

Writing a little more, taking it slowly, is all that is needed here. Like last time, remember that correctness must be developed before efficiency. (At the end, I did point out the other issues in the problem.)

Zero in the quotient, after a decimal point

A mother sent us this division question in 2006:

Adding Zeros in a Long Division Problem

I was trying to remember the long division rules when dividing a small number such as 6 by a larger number like 351.  I am teaching my daughter and I want to make sure I understand the rules about adding zeros correctly.

Zeros come into play twice here. We have to add zeros to the dividend to add decimal places; and for a number as small as this, we will have zeros at the start of the quotient. Dr. Rick just went slowly through the process, paying special attention to how to organize the work:

I prefer not to pile up too many rules, but to help a student see that what you do in this case is exactly what you always do in division. There are really no new rules; the only thing is to recall that the numbers 6, 6.0, 6.00, 6.000, and so on are all names for the same number.

I start by looking at the division problem
      ___
  351 ) 6

How many times does 351 go into 6?  None--that's 0, so I put 0 in the quotient, above the ones digit of the dividend, multiply 0 by 351, and subtract this from 6.

        0
      ___
  351 ) 6
       -0
       --
        6

Then I'm supposed to bring down the next digit.  There is no next digit ... but there is, if I write 6 as 6.0.  When I add the decimal point to the dividend, I also put one in the quotient directly above it.

        0.
      _____
  351 ) 6.0
       -0
       --
        6 0

Now (since I ignore the decimal point at this stage) I'm trying to divide 60 by 351, and 60 still isn't big enough; 60 divided by 351 is 0 again, so ...

        0.0
      _____
  351 ) 6.0
       -0
       --
        6 0
       -  0
        ---
        6 0

It's time to bring down the next digit of the dividend again, but again there isn't one ... unless I write the dividend as 6.00:

… and so on. Note again the value of not trying to go too fast. Once everything makes sense, then we can look for ways to improve the process:

Once a child understands how this works, just following the basic rules, I can show some secrets that make it easier!  You don't have to write so much, because of what you know about how numbers work. Whenever the quotient digit is zero, you know that when you multiply zero by the divisor, you'll get zero; and you know that subtracting 0 from any number leaves it just the same.  You can do those operations in your head, because they amount to doing nothing!  All you need to write down is this:

        0.0 1 7
      _________
  351 ) 6.0 0 0
       -3 5 1
        -----
        2 4 9 0
       -2 4 5 7
        -------
            3 3

The only danger in leaving out those steps is that you might lose track of where the next digit goes in the quotient.  That's why I emphasize that the digit goes directly above the ones digit of the partial dividend.  If you find that this leaves a blank to the left of the digit you add, it means that you forgot to write down a zero; so put it in.

Here’s another example of the same kind of problem, but the tutor who wrote us had hit a roadblock with the student:

Decimal Multiplication and Division

I'm tutoring a 6th-grade student, and I'm having a hard time explaining to her why you sometimes need to put a zero in the tenths place, in the quotient - as a sort of placeholder, I tell her. 

For example, we were dividing 0.5 into 0.025. She's okay with multiplying both numbers by 10, so that the problem becomes 0.25/5, but when we start writing digits for the quotient, she won't believe that the answer is 0.05. She insists that it must be 0.5.

I've tried showing her that 0.5 x 5 = 2.5, and not 0.25. And I've tried using 25 base 10 cubes to represent 25 one-hundredths, and then having her divide them into five groups, each of five one-hundredths. She still says you cannot put that zero in after the decimal point. 

It seems like something is just missing. For one thing, she has it firmly in mind that her teacher said to never add a zero after the decimal point, so she refuses to believe any of my explanations. (For another, she's still a bit shaky on decimal multiplication.)

Do you have any other way of explaining decimal division when the numbers that are not strictly analogous to regular division - i.e., when you have to put in a zero in the tenths place, or hundredths place, etc.? (I suspect my student is not the only one who could use help in this area.)

I took Susan through several ways to help the student overcome her “phobia about zeros” and see that meaning trumps memorized rules.

It sounds as if the student is stuck on rules and doesn't dare try to understand what's going on. The claim that "you cannot put that zero in after the decimal point" must come from something the teacher said in some other context, and she feels math is all about rules. That probably happens more often than we'd like to think; too many teachers emphasize rules rather than reasons. So let's first see where that rule might come from, and maybe we can untangle it from this problem. All I can think of is just that once you have a number, you can't insert a zero between the decimal point and other digits without changing its value. You'll have to ask her to give an example where her teacher might have used that rule, and then show why it doesn't apply here.

I explain decimal multiplication and division in terms of fractions (assuming a student has an understanding of fractions!). In this case,
 
                  25     5    25    10    25 * 10    10    25
   0.025 / 0.5 = ---- / -- = ---- * -- = -------- = ---- * --
                 1000   10   1000    5   1000 * 5   1000    5

So the answer will be 1/100 of 5, or 0.05.

This will probably be too complicated for your student. Here's an alternative way to show the same thing:

    0.025   1000 * 0.025    25   25    1
    ----- = ------------ = --- = -- * ---
     0.5    1000 * 0.5     500    5   100

Susan said she would try my ideas out, but I never heard back about the results. The most important thing would be to break the dependence on rules. Misremembered rules are worse than no rules at all!

19 thoughts on “Long Division: When Zero Gets in the Way”

        1. It happens that we’ve had two questions about this sort of problem in the last couple days at our Ask a Question page!

          For your example, after you write 4 in the quotient, you multiply by 26 and get 104, then subtract that from 106. The remainder is 2, and you “bring down” the 4, making 24.

          But this is still less than the divisor, 26. So the next digit in the quotient is 24/26, which is 0.

          If you want an integer quotient, you are finished; the remainder is 24.

          If you want a decimal quotient, you now “bring down” the next digit, the 0 after the decimal point, and divide 240 by 26 to find the next digit, 9.

          The important thing is that as soon as you find that 24 is less than 26, you write the 0 in the quotient. I find that the American style of division, as shown in my post, where we write the quotient above the dividend, putting each digit in its own place, helps to make sure that you don’t omit a digit.

          It is possible that you are using a slightly different method for division (such as the scaffold method), where you just wrote the 40 and multiplied to get 1040. In that case, you now subtract and get 24; the fact that that is less than 26 means you can stop. There is no new digit to add to the 40.

  1. Concerning whether or not to put a zero in the quotient of a long division problem: I always ran into the issue (and still do) that sometimes you have to put a zero in the quotient, but then there are sometimes situations when you don’t put a zero in the quotient, even though to me it seems like the same situation as when you are suppose to. I think I remember something about if the divisor is over half of the part of the dividend in question, then you do the “zero-in-the-quotient part” different or something— or maybe I misunderstood the rule. lol. PLEASE HELP!

    1. Hi, Jonathan.

      You’ll have to find a specific example. The only possible rule I can think of involving half the dividend would be when you want to decide whether to round up the last digit of the quotient, which is not about placing a zero there unless you are rounding up from 9.

  2. I want steps regarding the problem
    1010/ 100 .
    I know the answer but I couldn’t get the steps. Will you please help me sir.

    1. For this, there is a very quick solution: Division by 100 just moves the decimal point two places to the left, so 1010. becomes 10.10, and there is the answer.

      In terms of long division, dividing 101 by 100 gives a quotient of 1 (which goes in the tens place); subtracting 100 leaves a remainder of 1. Bringing down the zero in the ones place, we have to divide 10 by 100; the quotient this time is 0, so we write a 0 in the ones place of the quotient, and our remainder is still 10. Bringing down another zero (this one from the tenths place), we have to divide 100 by 100, which gives us another 1 to put in the tenths place of the quotient. So the quotient is, again, 10.1.

  3. I came across a similar situation in long division where I had trouble with understanding zeros. In the problem 245/8 I went about solving it the usual way and got the answer as 3.625 which was wrong when I checked it with my calculator, the correct answer was 30.625. The thing I would like to know is what is the meaning of the zero before the decimal point.

    1. Hi, Gamer.

      The key idea is that each time you divide by the divisor, you must put the quotient digit in the answer, even if it is a zero.

      In your example, you first divide 24 by 8, and write the quotient, 3, in the answer (specifically, in the tens place, as 24 is in the tens place of the dividend – that is, it means 24 tens, not 24 ones). After multiplying and subtracting, the remainder is 0.

      You then bring down the next digit, 5, and append it to the remainder to get 05. When you divide this by 8, you get 0 (because 5 < 8), and you must write this 0 in the answer. It is needed because we need a ones place in the answer, corresponding to 5 being in the ones place of the dividend.

      After that, you can continue. Since the next digit you bring down is a 0 from the tenths place, the quotient you get, 6, must go in the tenths place of the answer.

      So, what is the meaning of the zero? It can be considered a place holder, to ensure that the 3 you got represents 3 tens. That is because when you divided 24 by 8, you were really dividing 240 by 8, and the quotient is 30, not just 3.

      We can express the work algebraically. To divide 245/8, we treat it as (240 + 5)/8 = 240/8 + 5/8 = 30 + 5/8. This corresponds exactly to what I just described.

      Compare a similar problem, 236/8. Here we treat it as (160 + 76)/8 = 20 + 76/8; then we treat 76/8 as (72 + 4)/8 = 72/8 + 4/8 = 9 + 0.5. So the answer is (160 + 72 + 4)/8 = 160/8 + 72/8 + 4/8 = 20 + 9 + 0.5 = 29.5. The difference between the two problems is merely that there is a 0 in the ones place in the first example.

      Does that help at all?

      1. Yes it did help me understand the meaning of zero in the quotient. Thank you for helping me out on this.

  4. Pingback: Long Division with Zero, Revisited – The Math Doctors

    1. I presume you mean to divide 8200 by 5, which is properly written as 8200/5, but is written as you did when you start the work the traditional way.

      The only hard part, I think, is in the final digit. I would divide 8 by 5, putting 1 in the quotient and subtracting to leave a remainder of 3; then bring down the 2 to make a partial dividend of 32. I would then divide 32 by 5, putting 6 in the quotient and subtracting to leave a remainder of 2; then bring down the first 0 to make a partial dividend of 20. I would then divide 20 by 5, putting 4 in the quotient and subtracting to leave a remainder of 0.

      At this point, I still have another 0 to bring down; that gives me a partial dividend of 00. I divide 00 by 5, putting 0 in the quotient (which I suspect is what you may not have done), and subtracting to leave a remainder of 0. So the final quotient is 1640; and when we check the answer by multiplying, we find that 5*1640 = 8200.

      You will probably find it helpful to read my recent supplementary post, Long Division with Zero, Revisited, which gives more examples much like yours. If you need additional help, please use Ask a Question, where we can discuss details more easily.

  5. Thanks for the detailed explanation. I am having tough time explaining to my son why 653 / 13 is 50.2 and not 5.2… Could you provide an explanation please. Thanks.

    1. I think you, like others, will benefit from reading the follow-up post, Long Division with Zero, Revisited, which starts with an example very similar to yours.

      Another approach to showing that 5.2 is wrong is just to do the multiplication that corresponds to the division. If you say that 653 / 13 = 5.2, then you are saying that 653 = 13 * 5.2. But 13 * 5.2 = 67.6, not 653. This is how we check divisions; and teaching your son to do that regularly can help him catch this sort of mistake, as well as better understand what division is.

      Of course, since your answer is rounded, not exact, if you check 653 / 13 = 50.2, you find that 13 * 50.2 = 652.6, not exactly 653; but clearly it’s close.

      It would also be good to teach him how to check by estimation. 653 / 13 should be close to, say, 600/10 = 60, and 5.2 is very far from that.

      But ultimately, the explanation for why his work is wrong is, as I say in the newer post, “Never ignore a quotient.” You’ll see what I mean when you read it.

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