November 24, 2008

Dividing decimals

I feel students need to get grounded conceptually in this topic. So many times, all they learn about decimal division are the rules of how to go about decimal division when using long division, and it becomes an "empty" skill - a skill that lacks the conceptual foundation.

So for starters, we can do two different kinds of mental math division problems.

  1. Division by a whole number - using mental math

    Here it is easy to think, "So much is divided between so many persons".

    0.9 ÷ 3 is like "You have nine tenths and you divide it between three people. How much does each one get?" The answer is quite easy; each person "gets" 0.3 or three tenths.

    And... remember ALWAYS that you can check division problems by multiplication. Since 3 × 0.3 = 0.9, we know the answer was right.

    0.4 ÷ 100 turns out to be an easy problem if you write 0.4 as 0.400:
    0.400 ÷ 100 is like "You have 400 thousandths and you divide it between 100 people; how much does each one get?" The answer is of course 4 thousandths, or 0.004. Check: 100 × 0.004 which is 100 × 4/1000 = 400/1000 or 0.400 = 0.4.

    Here are some more similar ones:

    0.27 ÷ 9

    0.505 ÷ 5

    0.99 ÷ 11
    ...and you can make more, just think of the multiplication tables.


  2. Division where the quotient (answer) is a whole number

    This time it helps to think, "How many times does the divisor go into the dividend?" In these types of mental math problems, the answer ends up being a whole number. (Of course the teacher has to plan these problems just right.)

    For example, 0.4 ÷ 0.2. Ask, "How many times does 0.2 fit into 0.4?" The answer is, 2 times. So 0.4 ÷ 0.2 = 2. Again, we can check it by multiplying: 2 × 0.2 = 0.4.

    Other similar division problems to solve mentally:

    1 ÷ 0.5

    3 ÷ 0.5

    0.09 ÷ 0.03

    0.9 ÷ 0.1

    2 ÷ 0.4

    1 ÷ 0.01

    ...and so on.


This decimal division lesson taken from my Decimals 2 book illustrates these two kinds of mental division problems.


Towards the general case

After the student is familiar with the two special cases above, we can go forward and study decimal division problems in general. Even here, we will divide the problems into two classes, depending on whether the divisor is a whole number or not.

  1. The divisor is a whole number.

    For example, 3.589 ÷ 4 or 0.1938 ÷ 83. These can simply be solved by long division as they are. Just put the decimal point in the same place in the quotient as where it is in the dividend.

    The "stumbling block" may come when the division is not even (this also leads into the study of repeating decimals). Generally, you can continue the division indefinitely by tagging zeros to the dividend, such as making 3.589 to be 3.589000. Then when you've continued the division as long as you wish (or as long as the book tells you to do it), cut the decimal off at a desired accuracy and round it.

    Typical problem in a textbook would say, "Do 2.494 ÷ 3 and give your answer with 3 decimal digits." For this, you need to do the long division until the fourth decimal digit - so as to be able to round to 3 decimal digits. Since 2.494 does not have four decimal digits, you tag a zero to it to make it have so (2.4940).

    Fortunately, this process is not generally difficult. It's the second case that's more of a problem.



  2. The divisor is not a whole number.

    Here, we do something quite special before dividing, and turn the problem into one where the divisor is a whole number. Then, the actual division is done like explained above.

    I say this is special, because this special thing that we do is based on a very important general principle of arithmetic:

    If you multiply both the dividend and the divisor by some same number, the quotient won't change.

    Let's see it in action with some easy numbers:

    1000 ÷ 200 = 5

    100 ÷ 20 = 5

    10 ÷ 2 = 5

    Each time both the dividend and the divisor change by a factor of ten, but the quotient does not change.

    We can also try it using a factor of 3 (or any other number):

    8 ÷ 2 = 4
    24 ÷ 6 = 4
    72 ÷ 18 = 4

    Let's try one more time, with a factor of 2:

    30 ÷ 6 = 5

    15 ÷ 3 = 5

    7.5 ÷ 1.5 = 5

    3.75 ÷ 0.75 = 5

    H hopefully by now you have convinced the student(s) of this principle. Now we can apply it to those pesky decimal division problems.


    decimal division

    This image shows how the decimal division problem 0.644 ÷ 0.023 can be changed into another problem, with a whole number divisor, and with the same answer.

    In each step, we multiply both the dividend and the divisor by 10. This, of course, is the same process as moving the decimal point.

    Many textbooks only show the student the "trick" of moving the decimal point... but don't show him what that idea is based on.

    An example

    To solve 13.29 ÷ 5.19, we need to first change the problem so that the divisor 5.19 is a whole number. We multiply both the dividend and the divisor by 10 as many times as needful to accomplish that:

    13.29 ÷ 5.19
    = 132.9 ÷ 51.9
    = 1329 ÷ 519, and now off you go to do long division... I'm not saying it's the easiest long division problem in the world, since the divisor is 519. Let's try an easier one.


    2,916 ÷ 0.02
    = 29,160 ÷ 0.2
    = 291,600 ÷ 2 and now you can do the long division.

    Of course, in reality you can also multiply by 100 instead of taking two steps of multiplying by 10. But students can start out by multiplying by 10 as many times as needed.


Please also see the lesson on dividing decimals by decimals, from my Math Mammoth Decimals 2 book.
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