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Mathematics LibreTexts

6.4: Division and Decimals

  • Page ID
    9859
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    When you studied fractions, you had lots of different ways to think about them. But the first way, and the one we keep coming back to, is to think of a fraction as the answer to a division problem.

    Example \(\PageIndex{1}\):

    Suppose 6 pies are to be shared equally among 3 children. This yields 2 pies per child. We write:

    $$\frac{6}{3} = 2 \ldotp$$

    6-pies-3-children.png

    The fraction \(\frac{6}{3}\) is equivalent to the answer to the division problem \(6 \div 3 = 2\). It represents the number of pies one whole child receives.

    In the same way…

    sharing 10 pies among 2 kids yields \(\frac{10}{2} = 5\) pies per kid,

    sharing 8 pies among 2 kids yields \(\frac{8}{2} = 4\) pies per kid,

    sharing 5 pies among 5 kids yields \(\frac{5}{5} = 1\) pies per kid, and

    the answer to sharing 1 pies among 2 children is \(\frac{1}{2}\), which we call “one-half.”

    We associate the number “\(\frac{1}{2}\)” to the picture onehalf.png.

    In the same way, the picture onethird.png represents “one third,” that is, \(\frac{1}{5}\).

    (This is the amount of pie an individual child would receive if one pie is shared among three children.)

    The picture onefifth.png is called “one fifth” and is indeed \(\frac{1}{5}\), the amount of pie an individual child receives when one pie is shared by five kids.

    And the picture threefifths.png is called “three fifths” to represent \(\frac{3}{5}\), the amount of pie an individual receives if three pies are shared by five kids.

    We know how to do division in our “Dots & Boxes” model.

    Example: 3906 ÷ 3

    Suppose you are asked to compute \(3906 \div 3\). One way to interpret this question (there are others) is:

    “How many groups of 3 fit into 3906?”

    In our “Dots & Boxes” model, the dividend 3906 looks like this:

    divide1-300x74.png

    and three dots looks like this: divide2.png

    So we are really asking:

    “How many groups of divide2.png fit into the picture of 3906?”

    divide3-300x153.png

    Notice what we have in the picture:

    • One group of 3 in the thousands box.
    • Three groups of 3 in the hundreds box.
    • Zero groups of 3 in the tens box.
    • Two groups of 3 in the ones box.

    This shows that 3 goes into 3906 one thousand, three hundreds and two ones times. That is,

    $$3906 \div 3 = 1302 \ldotp$$

    Of course, not every division problem works out evenly! Here’s a different example.

    Example: 1024 ÷ 3

    Suppose you are asked to compute \(1024 \div 3\). One way to interpret this question is:

    “How many groups of 3 fit into 1024?”

    So we’re looking for groups of three dots in this picture:

    1024div3a.png

    One group of three is easy to spot:

    1024div3b.png

    To find more groups of three dots, we must “unexplode” a dot:

    1024div3c.png

    We need to unexplode again:

    1024div3d.png

    This leaves one stubborn dot remaining in the ones box and no more group of three. So we conclude:

    $$1024 \div 3 = 341\; \text{R} 1,\; meaning\; 1024 = 341 \cdot 3 + 1 \ldotp$$

    In words: 1024 gives 341 groups of 3, plus one extra dot.

    We can put these two ideas together — fractions as the answer to a division problem and what we know about division in the “Dots & Boxes” model — to help us think more about the connection between fractions and decimals.

    Example: 1/8

    The fraction \(\frac{1}{8}\) is the result of dividing 1 by 8. Let’s actually compute \(1 \div 8\) in a “Dots & Boxes” model, making use of decimals. We want to find groups of eight in the following picture:

    18a.png

    Clearly none are to be found, so let’s unexplode:

    18b.png

    (We’re being lazy and not drawing all the dots. As you follow along, you might want to draw the dots rather than the number of dots, if it helps you keep track.)

    Now there is one group of 8, leaving two behind. We write a tick-mark on top, to keep track of the number of groups of 8, and leave two dots behind in the box.

    18c.png

    We can unexplode the two dots in the \(\frac{1}{10}\) box:

    18d.png

    This gives two groups of 8 leaving four behind. Remember: the two tick marks represent two groups of 8. And there are four dots left in the \(\frac{1}{100}\) box.

    18e.png

    Unexploding those four remaining dots:

    18f.png

    Now we have five groups of 8 and no remainder.

    18g.png

    Remember: the tick marks kept track of how many groups of eight there were in each box. We have

    • One group of 8 dots in the \(\frac{1}{10}\) box
    • Two groups of 8 dots in the \(\frac{1}{100}\) box.
    • Five groups of 8 dots in the \(\frac{1}{1000}\) box.

    So we conclude that:

    $$\frac{1}{8} = 1 \div 8 = 0.125 \ldotp$$

    Of course, it’s a good habit to check our answer:

    $$0.125 = \frac{125}{1000} = \frac{5 \cdot 25}{5 \cdot 200} = \frac{5 \cdot 5}{5 \cdot 40} = \frac{5 \cdot 1}{5 \cdot 8} = \frac{1}{8} \ldotp$$

    On Your Own

    Work on the following exercises on your own or with a partner. Be sure to show your work.

    1. Perform the division in a “Dots & Boxes” model to show that \(\frac{1}{4}\), as a decimal, is \(0.25\).
    2. Perform the division in a “Dots & Boxes” model to show that \(\frac{1}{2}\), as a decimal, is \(0.5\).
    3. Perform the division in a “Dots & Boxes” model to show that \(\frac{3}{5}\), as a decimal, is \(0.6\).
    4. Perform the division in a “Dots & Boxes” model to show that \(\frac{3}{6}\), as a decimal, is \(0.1875\).
    5. In simplest terms, what fraction is represented by each of these decimals? $$0.75, \qquad 0.625, \qquad 0.16, \qquad 0.85, \qquad 0.0625 \ldotp$$

    Repeating Decimals

    Not all fractions lead to simple decimal representations.

    Example: 1/3

    Consider the fraction \(\frac{1}{3}\). We seek groups of three in the following picture:

    13a.png

    Unexploding requires us to look for groups of 3 in:

    13b-1.png

    Here there are three groups of 3 leaving one behind:

    13c.png

    Unexploding gives:

    13d.png

    We find another three groups of 3 leaving one behind:

    13f.png

    Unexploding gives:

    13g1.png

    13g2.png

    And we seem to be caught in an infinitely repeating cycle.

    We are now in a philosophically interesting position. As human beings, we cannot conduct this, or any, activity an infinite number of times. But it seems very tempting to write:

    $$\frac{1}{3} = 0.33333 \ldots,$$

    with the ellipsis “\dots” meaning “keep going forever with this pattern.” We can imagine what this means, but we cannot actually write down those infinitely many 3’s represented by the \dots

    notation

    Many people make use of a vinculum (horizontal bar) to represent infinitely long repeating decimals. For example, \(0. \bar{3}\) means “repeat the 3 forever”:

    $$0. \bar{3} = 0.33333 \ldots,$$

    and \(0.296 \overline{412}\) means “repeat the 412 forever”:

    $$0.296 \overline{412} = 0.296412412412412 \ldots$$

    Now we’re in a position to give a perhaps more satisfying answer to the question \(1024 \div 3\). In the example above, we found the answer to be

    $$1024 \div 3 = 341\; \text{R} 1 \ldotp$$

    But now we know we can keep dividing that last stubborn dot by 3. Remember, that represents a single dot in the ones place, so if we keep dividing by three it really represents \(\frac{1}{3}\). So we have:

    $$1024 \div 3 = 341\; \text{R} 1 = 341 \frac{1}{3} = 341.3333333 \ldots = 341. \bar{3} \ldotp$$

    Example: 6/7

    As another (more complicated) example, here is the work that converts the fraction \(\frac{6}{7}\) to an infinitely long repeating decimal. Make sure to understand the steps one line to the next.

    67a.png

    67b.png

    67c.png

    67d.png

    67e.png

    67f.png

    67g.png

    67h.png

    67i.png

    67j.png

    67l.png

    67m.png

    67o.png

    With this 6 in the final right-most box, we have returned to the very beginning of the problem. (Do you see why? Remember, we started with a six in the ones box!)

    This means that we will simply repeat the work we have done and obtain the same sequence of answers: \(857142\). And then again, and then again, and then again. We have:

    $$\begin{split} \frac{6}{7} &= 0.857142857142857142857142 \ldots \\ &= 0. \overline{857142} \ldotp \end{split}$$

    On Your Own

    Work on the following exercises on your own or with a partner. Be sure to show your work.

    1. Compute \(\frac{4}{7}\) as an infinitely long repeating decimal.
    2. Compute \(\frac{1}{9}\) as an infinitely long repeating decimal.
    3. Use a “Dots & Boxes” model to compute \(133 \div 6\). Write the answer as a decimal.
    4. Use a “Dots & Boxes” model to compute \(255 \div 11\). Write the answer as a decimal.