1. Dividing Decimals #8 8.25 ÷ 3.5 next Taking the Fear out of Math
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2. So far we have discussed the operations of addition, subtraction and
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So far we have discussed the operations of addition, subtraction and
multiplication of decimal fractions, and we saw that once we knew where to
place the decimal point in the answers, these operations were identical to
their counterparts for whole numbers.
In this presentation, we shall see
that division works the same way. That is, any division problem involving decimals can be converted into an equivalent problem that uses only whole numbers.
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Let’s first start with the simplest case in which we divide one whole number by another1.
Specifically, let’s start with an example in which the divisor is a factor of the dividend.
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1 Notice that every whole number can be represented by a decimal fraction. For example, the whole number 6 can be written as 6.0 which is a decimal fraction. This is the decimal equivalent of saying that every whole number can be represented by a common fraction. For example, 6 = 6 ÷ 1 = 6/1 .
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Based on our adjective/noun theme the fact that 300 ÷ 4 = 75 allows us to draw conclusions such as…
300 apples ÷ 4 = 75 apples 7 5
If we wanted to use the usual division algorithm (“recipe”) to arrive at this result we might first have written…
3 0 0 4
2 8 2
2 0
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5. Now add the noun “apples” to both the dividend and the quotient.
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Now add the noun “apples” to both the dividend and the quotient.
3 0 0 4 7
2 8 2
2 0 5
apples
apples
To help get to the main point of this presentation, let’s simply replace the word “apples” above by the word “hundredths. When we do this we obtain…
apples
3 0 0 4 7
2 8 2
2 0 5
hundredths
hundredths
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6. Recalling that in place value notation
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Recalling that in place value notation
75 hundredths = 0.75, and
300 hundredths = 3.00,
we may rewrite…
3 0 0 hundredths 4 7
5 hundredths
3.0 0 4 .7 5
in the form…
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7. In summary, the above algorithm is simply another way of writing
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In summary, the above algorithm is simply another way of writing
300 hundredths ÷ 4 = 75 hundredths,
or equivalently 3.00 ÷ 4 = 0.75.
When we use the division algorithm to divide a decimal by a whole number, we simply place a decimal point in the quotient directly above where it appears in the dividend and then divide exactly as we would have done if there had been no decimal point in the problem.
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For example, suppose we wanted to divide 3.36 by 4. That is, we wanted to write the quotient 3.36 ÷ 4 as a decimal.
In “paper and pencil” format, we might use the following sequence of steps…
Step 1
We would rewrite 3.36 ÷ 4 as…
3.3 6 4
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9. Then perform the division as if there had not been any decimal point.
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Step 2
We would then place a decimal point in the quotient so that it is aligned with the decimal point in the dividend…
3.3 6 4 .
Step 3 . 8 4
3.3 6 4 6
Then perform the division as if there had not been any decimal point.
3 2 1
1 6
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10. In other words, 3.36 ÷ 4 means the same thing as 336 hundredths ÷ 4;
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Notes
What we really did when we used the above algorithm was the same as if we had viewed 3.36 as 336 hundredths.
In other words, 3.36 ÷ 4 means the
same thing as 336 hundredths ÷ 4;
and since 336 ÷ 4 = 84,
336 hundredths ÷ 4 = 84 hundredths.
Thus, the previous division could have been written in the form…
apples
3 3 6 4 7 5
hundredths
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Recall that division is still “unmultplying”. In other words, 3.36 ÷ 4 means the number which when multiplied by 4 yields 3.36 as the product. Thus, we can check our result by observing that 0.84 × 4 = 3.36
The same logic applies even if the dividend is a decimal that is less than 1.
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For example, suppose that we want to express the quotient ÷ 6 as a decimal.
We could avoid the use of decimals completely by observing that represents 12 millionths. Then, we may use the fact that 12 ÷ 6 = 2 to conclude that 12 millionths ÷ 6 = 2 millionths.
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13. Using decimal notation, the above sequence of steps would look like…
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In summary…
÷ 6
= 12 millionths ÷ 6
= (12 ÷ 6) millionths
= 2 millionths =
Using decimal notation, the above sequence of steps would look like… 6 .0 2
1 2
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14. rewrite 0.000012 as 12/1,000,000, whereupon the problem becomes…
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Keep in mind that there are other ways to do this problem. In particular, if you prefer to work with common fractions, simply
rewrite as 12/1,000,000, whereupon the problem becomes…
12/1,000,000 ÷ 6 or
12/1,000,000 × 1/6 or
2/1,000,000
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Notice that even though 2/1,000,000 can be reduced to 1/500,000; there is no need to do this if we want to express the answer as a decimal.
In other words, in decimal notation, the “denominators” must be powers of 10.
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16. Let’s now generalize how we divide
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Let’s now generalize how we divide
a decimal by a decimal.
One of the nice things about mathematics from a purely logical point of view is that it gives us practice in reducing new problems to equivalent problems which we were able to solve previously.
In the present context, we now know how to divide a decimal fraction by a whole number.
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17. Let’s suppose we were then asked the following question.
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Let’s suppose we were then
asked the following question.
Express the quotient ÷ as a decimal fraction.
Here’s one approach for solving this problem. We already know how to divide
a decimal by a whole number. We also know that if we multiply both numbers in a division problem by the same (non-zero) number, we obtain an equivalent ratio.
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18. must move the decimal point (at least)
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With this in mind we notice that to convert into a whole number we
must move the decimal point (at least)
3 places to the right. As we noted in
our previous presentation, shifting the decimal point 3 places to the right
means that we have multiplied the decimal by 1,000. Hence, to keep the
quotient the same, we must also multiply the dividend (that is, ) by 1,000.
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19. In summary, then we may move the decimal point 3 places to the right
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In summary, then we may move the decimal point 3 places to the right
in both and to obtain the equivalent problem ÷ 6;
which we already know how to solve.
0.012 ÷ 6 =
12 thousandths ÷ 6 =
2 thousandths =
0.002
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20. One way to check this is to do both
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Notes
Notice that ÷ does not look like ÷ 6, but the two quotients express the same ratio.2
One way to check this is to do both
calculations on a calculator and see that the answers are the same.
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2 It’s similar to saying that 30 ÷ 10 doesn't look like 6 ÷ 2, but they both name the same rate. For example, at a rate of 6 for $2 you would get 30 for $10.
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We could also do this problem by rewriting each decimal fraction as an equivalent common fraction. More specifically…
= 12/1,000,000, and = 6/1,000
Hence…
÷ = 12/1,000,000 ÷ 6/1,000
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= 12/1,000,000, and = 6/1,000
÷ = 12/1,000,000 ÷ 6/1,000
= 12/1,000,000 × 1,000/6
= 2/1,000 =
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What we have just shown is the equivalent of the way most of us were taught to do this problem by a rather rote method. Specifically, given a problem such as…
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We first moved the decimal point just enough places to convert the divisor into a whole number (in this example it is 3 places to the right) to obtain…
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25. We were then told to carry out the division in the usual way…
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We then had to move the decimal point the same number of places in the divisor to obtain…
0 0 2
We were then told to carry out the division in the usual way…
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Not only was this sheer rote, but it was an eyesore trying to keep track of where the decimal points were originally and where they were after they were moved. In essence, it was like a “magic trick” with little meaning to most students.
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However, the magic vanishes and the true understanding occurs when we see
that is just another way of writing ÷ and that moving the decimal point 3 places to the right in both the divisor and the dividend was simply another way of saying that we were multiplying both numbers by 1,000, thus maintaining the same ratio.
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Moreover, if we were willing to work with greater numbers and wanted to wait until the very end before we had to deal with decimals, nothing prevents us from moving the decimal points in a way that converts both decimals into whole numbers.
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29. To help make sure that you are becoming more comfortable with this
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To help make sure that you are becoming more comfortable with this
notion of relative size let’s see what happens when we compute the
quotient 0.02 ÷
To convert to a whole number, we have to move the decimal point
4 places to the right. This means we are multiplying by 10,000.
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30. As a check we notice that…
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And if we multiply by 10,000 we also have to multiply 0.02 by 10,000 in order not to change the ratio. In other words, we move the decimal point
4 places in both and 0.02 to obtain the equivalent ratio 200 ÷ 4, which is 50.
As a check we notice that…
50 × = 0.02
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31. What the answer means is that as small as 0.02 is it is still 50 times
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What the answer means is that as small as 0.02 is it is still 50 times
as great as
We could move the decimal point even more places to the right but it wouldn’t affect the answer. For example, if we move the decimal point 6 places to the right in both numbers the ratio would be
20,000 ÷ 400, or still 50.
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If we wanted to use the fraction format for division, we could have rewritten the problem as 0.02/ after which we could multiply numerator and denominator by 10,000 to obtain…
0.02 × 10,000
× 10,000
200 4 = = 50
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33. seemed overly exciting.
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Up until now, our trip through the world of decimal fractions may not have
seemed overly exciting.
When all is said and done, it seems that except for learning where to put the decimal point and how many places to move it, we do decimal arithmetic the same way that we do whole number arithmetic.
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34. next
Fractions to Decimals
However, as we shall discuss in our next presentation, there is more to the division of decimals than what first meets the eye.
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