We have looked at how we add, subtract, and multiply decimals. Now we’ll conclude with division: what we do, why we do it, and how we don’t really need to do it that way.
This was the subject of a recent question, from Edite:
I teach at a community college, and when explaining how to divide decimals, I emphasize the importance of making the divisor a whole number. While I often mention that it simplifies the process, I believe there is more to the reasoning, and I want to provide my students with a deeper understanding.
Could you please offer some insight into the underlying principles behind this method, so I can offer a more comprehensive explanation to my students?
This is just what we like. I answered:
Hi, Edite.
I’m always happy to help a fellow community college instructor. I’ve never taught a class at this level, but often work with such students in our tutoring center, both “experienced” adult learners who have forgotten, and some who never learned these things well.
This is a topic I have long considered making a post about; I’ll primarily quote from a pair of old answers from Ask Dr. Math that may form the core of a future post.
Those two answers are the first two below. As we’ll see, Edite’s explanation (that it makes the work easier) is actually a central part of the answer, but there’s more to say.
How we do it
First, we have a question from 1998:
How to Divide Decimals by Decimals I don't understand dividing decimals. Can you send me any information so I can understand it more? Thank you! Tiffany Wright
Doctor Lim answered with a detailed explanation of the process:
Dear Tiffany,
Don't worry. Doing mathematics problems is just like eating a piece of cake. Eating a piece of cake gets difficult when we try to eat a big piece of cake all in one gulp. Let's break it into smaller pieces, and we can eat it easier.
Start with dividing a one-place decimal into another one-place decimal:
0.2 / 0.5
It actually means
_____
0.5 )0.2
The first form is the way we say “two tenths divided by five tenths” (also written as \(0.2\div0.5\)); the second is how we set up the work, with the divisor (which we’re dividing by) on the left, and the dividend (which is to be divided) “on the operating table” to be worked on.
We cannot have a divisor that is not a whole number, so we must change 0.5 into a whole number. To do this, we have to multiply by 10. This is because 0.5 is in the tenths place. We have to do this for both the divisor and the dividend in order to balance them.
0.2 x 10 2
---------- = ---
0.5 x 10 5
So, we have
_____
5 )2.0
As we’ll see below, it is not quite right to say we can’t divide by a non-whole number; it’s just easier to keep track of things this way. It’s necessary in order to follow this method. Multiplying by 10 (that is, moving the decimal point one place to the right) changes the divisor into a whole number; we have to do the same to both numbers to keep the same quotient.
We need to show where the decimal point is in the dividend. The next thing we need to do is to fix the decimal point of the answer. What you do is just to move another decimal point straight up from the position in the dividend.
___.__
5 ) 2.0
So, now you do normal division. Does 5 go into 2? No, so you can either leave it blank, or put a zero there to remind yourself there's nothing there (and it will ensure that you keep your decimal point in the right place). Does 5 go into 20? Yes, so we put a 4 above the zero. Our final answer is .4.
In many cultures the quotient is placed to the right rather than above the dividend; I don’t know how they teach to place the decimal point there. I find this style very straightforward.
Here’s what the work looks like:
0.4 ------ 5 ) 2.0 2 0 ----- 0
Putting everything directly above the digit you’re working on helps keep things straight. We explained these details in Long Division: When Zero Gets in the Way, and again in Long Division with Zero, Revisited.
Now let's go through a more complicated problem:
_______
0.14 )0.448
As above, we check out the divisor. The divisor is 0.14, which is 14 hundredths. So we have to multiply by 100.
0.448 x 100 44.8
------------- = ------
0.14 x 100 14
The first thing we do is take care of the decimal point of the divisor (multiply by 100). So, we move the decimal of the dividend over two places. Now we have
______
14 )44.8
Position the decimal point,
___.__
14 )44.8
xx.x
------
xx.x
Once you have positioned the decimal point, divide the numbers as usual. You should have no problem, because it is the same as division of a big number, 448, divided by a smaller number, 14. The only difference is that in your answer, you have to remember the position of the decimal point.
Here is the complete work:
3.2
------
14 )44.8
42
------
2 8
2 8
-----
0
We’ll do more complicated examples later, and talk about why we “move the decimal point straight up”..
In conclusion:
Take things easy. Once you practise more often, you will find that it gets easier and easier. Maths then will become a breeze.
Why we do it that way
But we’d like to understand more deeply how that process works, and why. Here is a question from 2001:
Why Decimal Division Works For my college math education class I have to write a paper on why (not how) the procedure of division with decimals works. The question is, "All elementary school students learn how to divide with decimals such as in the problem 551.2/1.06. Explain as if talking to 5th and 6th graders why this procedure works." If you could help me it would help very much! Thanks, Marc
I answered:
Hi, Marc. If you'd like to look over our shoulders as we actually explain division of decimals to a 5th or 6th grader, visit our archives: Fractions and Decimals - Elementary School Level http://mathforum.org/dr.math/tocs/fractions.elem.html Division - Elementary School Level http://mathforum.org/dr.math/tocs/division.elem.html Here's a sample: How to Divide Decimals by Decimals http://mathforum.org/dr.math/problems/wright3.7.98.html That should give you some good ideas for your own explanation.
The last link is to the question above. But I wanted to say more about the “why”.
The basic idea is that we first learn a method for dividing whole numbers. When we introduce decimals, we find that the work can be divided into two parts, just as in multiplication: we can first ignore the decimal point entirely, and get the DIGITS of the answer using the methods we've already learned; then we can determine where in that answer to put the decimal point. This is because, for example, we can rewrite a division of decimals as a division of whole numbers times a division of powers of ten: 1.95 195 * 0.01 195 0.01 195 ---- = ---------- = --- * ---- = --- * 0.01 * 10 = 3 * 0.01 * 10 6.5 65 * 0.1 65 0.1 65 This says that, after dividing 195 by 65, we put the decimal point in the same place it was in the dividend, 1.95 (multiplying 3 by 0.01 to get 0.03), and then shift it one place to the right (multiplying by 10, which is the same as dividing by 1/10, to get 0.3).
Here, rather than just multiply numerator and denominator by 100 as we did before, I’ve instead analyzed the two numbers, breaking each into a whole number times a power of ten, and then handling the two parts separately, much as we handled multiplication last time. It isn’t quite the traditional way to do this (at least as taught in America), but we could do just what this says: write the division without any decimals,
3
-----
65 )195
195
---
0
and then move the decimal point two places from the right, as in the dividend, and then one place to the right from there, undoing the decimal in the divisor:
3 --> .03 --> .3 ----- 65 )195 195 --- 0
In fact, this might be a good way to do it if you write the answer to the right, as we see in other countries:
6.5 ) 1.9 5( .3
1.9 5 ^
---- |
0 +-- make 2 - 1 = 1 decimal places
But it’s easier to remember if we first move the decimal point in the dividend itself:
The traditional way to make this memorable is to first move the decimal points in the divisor and dividend until the divisor is whole, and then put the decimal point in the quotient directly above the new position in the dividend. This effectively shifts the decimal point in the quotient first to the left, accounting for the decimal in the dividend, and then the to right, accounting for the divisor.
This makes the work look like this:
.3 ----- 65 )19.5 19 5 ---- 0
There’s a lesson here:
So you see, it's not that we MUST divide only by a whole number; rather, this is just one easily remembered way to decide where to put the decimal point in the quotient. It's not much different from what we do in multiplication of decimals, where we first ignore the decimal point, and then add the number of decimal places in the two numbers. In division, we are really SUBTRACTING the number of decimal places in the two numbers; but explaining it in terms of moving the decimal point ensures that we won't accidentally subtract in the wrong order.
Routine reduces risk — if the routine is remembered right.
What about the remainder?
Division has a feature the other operations don’t have: the remainder. How does that affect what we do? Here is a question from 2001:
Dividing Decimals with a Remainder Dear Dr. Math, I am a student in an elementary school and just got accepted to be in accelerated math next year in Jr. High. I have a question about dividing decimals. I know how to divide decimals, but when there is a remainder my teacher tells me I have to add zeros to the inside number. He says you can either add 2 or 3. I usually add 2 zeros, but if you add 2 you will get a totally different answer than if you add 3 zeros. Also, sometimes when you have a remainder and you bring it out 2 or 3 places (or however many it tells you to), you still have a remainder. My teacher says to just leave the remainder and round the answer, but I don't understand. If you could explain to me what to do when the directions only say DIVIDE: and you get a remainder, and also if the directions say ROUND TO THE HUNDRETHS PLACE: what to do when there is still a remainder, I would be very happy. Thank you very much, Genna
This is a wonderful question! I answered it:
Hi, Genna.
Let's take an example, so we have something specific to talk about:
___1.9_
23 ) 45.6
23
--
22 6
20 7
----
1 9
The original problem might have been \(4.56\div2.3\); we’ve already moved the decimal and done the work. But the remainder tells us that the quotient is not exact. Now what?
Okay, we have a remainder, and we want to round our answer to the hundredths place, so we'll add two zeros to the dividend and continue. (I'll deal with how many zeros to add in a moment.)
___1.982_
23 ) 45.600
23
--
22 6
20 7
----
1 90
1 84
----
60
46
--
14
Now what do we do? Since we were told to round to two decimal places, we just look at the 2 in the quotient, drop it, and leave the rest as it is. The answer is 1.98.
Because we added two zeros, we got a third decimal place in the answer, which lets us know that we don’t have to increase the hundredths place when we round. If the third digit had been 5 or more, we would have rounded up to 1.99.
Now, there’s a shortcut:
I didn't really have to add that last zero. I could have stopped with the remainder 6, and seen that the next digit of the quotient would be less than 5, because 6 is less than half of 23. See if you can see why that is true.
If we’d only added one zero, our work would look like this:
___1.98_
23 ) 45.60
23
--
22 6
20 7
----
1 90
1 84
----
6
Without actually doing the next digit, we could see that \(2\times6=12<23\), so \(6<\frac{23}{2}\), and we are “less than half past” the 8, and don’t need to round up. So this remainder is worth looking at. But the remainder in the first version had no effect on the final answer because of rounding.
(I don't know what you mean when you say that you get a totally different answer if you add two zeros or three. I just see one extra decimal place in the answer, which is not very different. Can you give me an example of what you mean?)
Adding another zero, the work becomes
___1.9826_
23 ) 45.6000
23
--
22 6
20 7
----
1 90
1 84
----
60
46
--
140
138
---
2
This will still round to 1.98.
It is possible that the “totally different answer” she saw was the different remainder; or she was referring to the value before rounding, with more digits. It looks different, but doesn’t affect the final answer.
Now let's get back to the big question: What is this all about? Why should you just ignore the remainder? The problem with decimals is that most division problems never end. You can keep adding more zeros to the dividend, and you'll just get more digits in the quotient. If you've learned about repeating decimals, you know why that is: when you change a fraction to a decimal, you get a terminating decimal only if the denominator (in lowest terms) has only 2 and 5 as prime factors, so that it can be converted to a fraction with a power of ten in the denominator. So when we work with decimals, we expect to have to approximate. The reason we can approximate without losing anything important is that each decimal place we get is worth a tenth of the previous one, so eventually they get small enough not to affect whatever we are going to do with them. At that point we can just drop the remainder (and therefore all the rest of the digits), and round if we wish.
If we continued this division farther, we would get $$45.6\div23=1.9826086956521739130434782608696\dots$$ Most of those extra digits add almost nothing to the answer! If we only need two decimal places, we can round to 1.98, and those extra digits won’t change it. So dropping the remainder does no harm.
An important topic related to this is "significant digits." You may want to search our archives for this phrase and read about it; it explains how we can decide how many digits we need, and why digits after a certain point don't matter and can be ignored. When we need to be precise, we use fractions rather than decimals, because we never have to drop anything. When we use decimals, we KNOW that we are going to be rounding, so it doesn't bother us.
This is discussed in a series starting with Significant Digits: Introduction. The difference between fractions and decimals is covered in Fractions vs. Decimals: Pros and Cons.
You don’t really have to do it that way
We’ll close with a question from 2009:
Moving the Decimal Point to Divide In the problem 100 divided by .4 I know you move the decimal over to make it 4 to solve the the problem. My question is why do we move the decimal. I just know it is the rule of thumb (if you will) to do this, I just don't know the reason why it has to be moved.
Doctor Douglas answered:
Hi Patty, It's not a rule of thumb--it is just one of the many possible choices of ways to evaluate this fraction. 100 / .4 = (100 * 10)/(.4 * 10) = 1000 / 4 which is a form that is a little more familiar. Multiplying top and bottom by the same number doesn't change the value of the fraction, so we choose a convenient number (10) to multiply by, and this is equivalent to moving the decimal point over for both top and bottom.
As we’ve seen above, thinking of the division as a fraction explains the process. But it also opens us to other possibilities:
We could have multiplied or divided by whatever number happens to be convenient (but 10 is often a good choice because the multiplications are easy). Here's an example where we might choose something different: 111/.25 We could multiply top and bottom by 100, leading to 11100/25, and then we have a division operation still left to do. But instead, we could have multiplied the top and bottom by 4, and this is nice because 4*(.25) = 1.0, so this choice leads to 444/(1.0) = 444.
For the original problem, I might notice that multiplying 0.4 by 5 makes it a whole number, and do this: $$\frac{100}{0.4}=\frac{100\times5}{0.4\times5}=\frac{500}{20}$$ And then I might notice that I can divide both numbers by 10: $$\frac{500}{20}=\frac{500\div10}{20\div10}=\frac{50}{2}=25$$
Similarly, Doctor Douglas’s example, in detail, would look like this: $$\frac{111}{0.25}=\frac{111\times4}{0.25\times4}=\frac{444}{1}=444$$ I often use tricks like this for mental arithmetic.
But using tens routinely makes almost everything work well, even if not always the fastest possible. And when we are using decimals, being able to think without fractions keeps our mind in one place.
But keeping your mind free to consider alternatives is a good idea too; and that’s why knowing why helps!