dividing decimals © math as a second language all rights reserved next #8 taking the fear out of...

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© Math As A Second Language All Rights Reserved

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#8

Taking the Fearout of Math

8.25 ÷ 3.5

So far we have discussed the operations of addition, subtraction andmultiplication of decimal fractions, and

we saw that once we knew where toplace the decimal point in the answers,

these operations were identical totheir counterparts for whole numbers.


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In this presentation, we shall seethat 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.

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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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Specifically, let’s start with an example in which the divisor is a factor of the

dividend.


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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

If we wanted to use the usual division algorithm (“recipe”) to

arrive at this result we might first have written…

3 0 04

7

2 82

0

2 0

5

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next Now add the noun “apples”

to both the dividend and the quotient.

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…

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apples

apples3 0 04

7

2 82 02 0

5

apples

apples3 0 04

7

2 82 02 0

5 hundredths

hundredths

Recalling that in place value notation75 hundredths = 0.75, and

300 hundredths = 3.00,


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we may rewrite…

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3 0 0 hundredths4

7 5 hundredths

in the form…3.0 04

.7 5

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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.


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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.

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.


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In “paper and pencil” format, we might use the following sequence of steps…

Step 1

We would rewrite 3.36 ÷ 4 as…

3.3 64

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nextnextStep 2

Then perform the division as if there had not been

any decimal point.

3.3 64

8

3 21

6

1 6

4

We would then place a decimal point in the quotient so that it is aligned with the

decimal point in the dividend…

3.3 64 .

Step 3 .

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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.


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Notes

Thus, the previous division could have been

written in the form…apples

apples3 3 64

7 5 hundredths

hundredths

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


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Notes

The same logic applies even if the dividend is a decimal that is less than 1.

For example, suppose that we want to express the quotient 0.000012 ÷ 6

as a decimal.


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We could avoid the use of decimals completely by observing that 0.000012

represents 12 millionths. Then, we may use the fact that 12 ÷ 6 = 2 to conclude

that 12 millionths ÷ 6 = 2 millionths.

In summary…


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Using decimal notation, the above sequence of steps would look like…

0.000012 ÷ 6

= 0.000002

= (12 ÷ 6) millionths

= 2 millionths

= 12 millionths ÷ 6

0.0 0 0 0 1 26

.00 0 0 0 0 2

1 2

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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, simplyrewrite 0.000012 as 12/1,000,000, whereupon the

problem becomes…


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Notes

12/1,000,000 ÷ 6 or

12/1,000,000 × 1/6 or

2/1,000,000

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.


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Notes

In other words, in decimal notation, the “denominators” must be powers of 10.

Let’s now generalize how we divide a decimal by a decimal.


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In the present context, we now know how to divide a decimal fraction by a

whole number.

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.

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Let’s suppose we were thenasked the following question.


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Express the quotient 0.000012 ÷ 0.006 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.

With this in mind we notice that to convert 0.006 into a whole number wemust 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 rightmeans that we have multiplied the

decimal by 1,000. Hence, to keep thequotient the same, we must also multiply the dividend (that is, 0.000012) by 1,000.


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In summary, then we may move the decimal point 3 places to the right

in both 0.000012 and 0.006 to obtain the equivalent problem 0.012 ÷ 6;

which we already know how to solve.

0.012 ÷ 6 = 12 thousandths ÷ 6 =

2 thousandths = 0.002

Notice that 0.000012 ÷ 0.006 does not look like 0.012 ÷ 6, but the two quotients

express the same ratio.2


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Notes

One way to check this is to do bothcalculations 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…


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Notes

0.000012 = 12/1,000,000, and 0.006 = 6/1,000

Hence…

0.000012 ÷ 0.006 = 12/1,000,000 ÷ 6/1,000


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Notes

0.000012 = 12/1,000,000, and 0.006 = 6/1,000

0.000012 ÷ 0.006 = 12/1,000,000 ÷ 6/1,000

= 12/1,000,000 × 1,000/6

= 2/1,000

= 0.002

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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Notes

0 0 0 6 0 0 0 0 0 1 2

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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Notes

0 0 0 6 0 0 0 0 0 1 2

We then had to move the decimal point the same number of places in the divisor

to obtain…


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Notes

0 0 0 6 0 0 0 0 0 1 2

We were then told to carry out the division in the usual way…

0 0 2

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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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Notes

However, the magic vanishes and the true understanding occurs when we see

that 0.006 0.000012 is just another way of writing 0.000012 ÷ 0.006 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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Notes

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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Notes

To help make sure that you are becoming more comfortable with thisnotion of relative size let’s see what

happens when we compute thequotient 0.02 ÷ 0.0004.


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To convert 0.0004 to a whole number, we have to move the decimal point

4 places to the right. This means we are multiplying 0.0004 by 10,000.

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And if we multiply 0.0004 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 0.0004 and 0.02 to obtain the equivalent ratio 200 ÷ 4, which is 50.


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As a check we notice that…

50 × 0.0004 = 0.02

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What the answer means is that as small as 0.02 is it is still 50 times

as great as 0.0004.


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Notes

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 be20,000 ÷ 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 0.02/0.0004 after which we could multiply numerator and denominator by

10,000 to obtain…


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0.02 × 10,000

0.0004 × 10,000=

2004

= 50

Up until now, our trip through the world of decimal fractions may not have

seemed overly exciting.


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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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However, as we shall discuss in our next

presentation, there is more to the division of

decimals than what first meets the eye.


Fractions to Decimals

dividing decimals © math as a second language all rights reserved next #8 taking the fear out of...

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