Dividing Fractions to find DecimalsDate: 03/12/2001 at 19:58:13 From: Erin Subject: Fractions into decimals and WHY I have some students who do not understand dividing the numerator into the denominator to find out the decimal. I cannot give them a reason why you do this except that it works. Why does this operation work to give you the decimal form of a fraction? Date: 03/12/2001 at 23:01:09 From: Doctor Peterson Subject: Re: Fractions into decimals and WHY Hi, Erin. Let's be careful with our terminology, first: you mean dividing the numerator BY the denominator! A lot of kids get that wrong. The basic reason is that division is really what fractions are all about. In a very real sense, we can say that a fraction is simply a division we haven't bothered to perform yet, a division problem frozen in time. We write "3/4" to mean "I want to divide 3 by 4, but I don't want to do the work just yet, so I can simplify the work before I finish up." That's why we use the virgule "/" or the horizontal fraction bar to signify division in more advanced math (and in computer programming languages), rather than the old-fashioned obelus. Let's go back to the basics of fractions to see why I can say this. What are fractions? The essence of a fraction is a division, a breaking into pieces. Take a whole object and divide it into 5 (equal) pieces; each piece is 1/5. We've divided 1 by 5, just as we divide 10 by 5 by dividing a set of 10 things into 5 parts, each of which consists of 2 objects. So 1 divided by 5 is 1/5; and 10 divided by 5 is the fraction 10/5, which simplifies to 2. Likewise, we can divide 2 pies into 5 parts by dividing each pie into 5 parts (fifths) and taking 2 of them at a time: 2 divided by 5 is 2/5. The denominator represents the number of parts we divide each whole into (a divisor); the numerator represents a multiplier, the number of parts we have. Let me repeat that, because it's easy to miss: 2/5 MEANS 2 divided by 5. The fraction IS a division. Now look at the operations on fractions. When we multiply by a fraction, we're really dividing: 1/2 * 10 is half of ten, or ten divided by 2, or 10/2 again. Fractions mean division. So when we want to convert a fraction to an ordinary (decimal) number, all we're really doing is waking up a division problem that has been in suspended animation, and letting it continue: "Where was I? Oh, yeah ... 3 divided by 4 ... that's 0.75." Here's another approach to the whole thing, focusing more on the decimals themselves. A (terminating) decimal can be thought of as a way to represent a fraction whose denominator is a power of ten. So when we convert 3/4 to a decimal, we are looking for an equivalent fraction whose denominator is, say, 100: fraction decimal 3 ? --- = --- 4 100 How do we do that? We have an object divided into 100 parts, and need to know how many of them make 3/4 of the object. To do that, we can multiply 100 by 3/4. But that is 300/4, and we can simplify that by dividing both the numerator and the denominator by 4: 300 300 divided by 4 75 --- = ---------------- = -- = 75 4 4 divided by 4 1 So 3/4 = 75/100, or 0.75 - and we found the 75 by dividing 300 by 4, which is exactly what you are doing when you divide 3 by 4 and put in the decimal point where it belongs, two places from the right. I hope one of these ideas will help. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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