Do Fractions Have to be Equal?Date: 05/09/2003 at 11:13:16 From: Phil Collins Subject: Fractions My grandson's textbook defines fractions as "one or more of the equal parts of a whole." Is this true? Do the parts of the whole have to be equal? Date: 05/09/2003 at 15:07:46 From: Doctor Peterson Subject: Re: Fractions Hi, Phil. The book is correct, though there can be some misunderstandings about what it is saying. When we name a fraction such as 1/3, we mean that you COULD obtain this amount by cutting three EQUAL pieces and taking one of them. We don't necessarily mean that you DID cut it just that way. But we do mean that if you cut a pie into three DIFFERENT pieces, they can't all be called thirds. Here is a pie cut into thirds (as well as I can do so in text!): ooooooooo oooooo.........oooooo ooo.................../ ooo oo...................../ oo oo....................../ oo o......................./ o o......................./ o o......................./ o o......................./ o o....................../ o o....................../ o o---------------------. o o \ o o \ o o \ o o \ o o \ o o \ o oo \ oo oo \ oo ooo \ ooo oooooo oooooo ooooooooo All three pieces are equal, and I have chosen one of them, so that piece is 1/3 of the pie. Here is a pie cut into three pieces that are not equal, but the chosen piece is still 1/3, because I COULD have made it the same way as before: ooooooooo oooooo.........oooooo ooo.................../ ooo oo...................../ oo oo....................../ oo o......................./ o o......................./ o o......................./ o o......................./ o o....................../ o o....................../ o o---------------------+---------------------o o o o o o o o o o o o o oo oo oo oo ooo ooo oooooo oooooo ooooooooo Only the shaded piece is actually 1/3; the others happen to be 1/6 and 1/2. Because they are not equal, we can't call them thirds. But the shaded piece is the same size as the third I made before, so it can be called 1/3. Here is another pie, which I cut into six pieces, but I chose two of them; the shaded area is still 1/3, because again I could have cut it into three equal pieces to get the same amount: ooooooooo oooooo.........oooooo ooo.\................./ ooo oo.....\.............../ oo oo........\............./ oo o...........\.........../ o o.............\........./ o o...............\......./ o o.................\...../ o o..................\.../ o o....................\./ o o---------------------+---------------------o o / \ o o / \ o o / \ o o / \ o o / \ o o / \ o oo / \ oo oo / \ oo ooo / \ ooo oooooo oooooo ooooooooo (Of course, this can also be called 2/6, which is equivalent to 1/3.) And here is a pie that I cut into three unequal pieces, none of which is a third: ooooooooo oooooo....| oooooo ooo..........| ooo oo.............| oo oo...............| oo o.................| o o..................| o o...................| o o....................| o o....................| o o.....................| o o---------------------* o o \ o o \ o o \ o o \ o o \ o o \ o oo \ oo oo \ oo ooo \ ooo oooooo oooooo ooooooooo The point is that a third is not defined based on how you make it, but based on how big it is. If three identical pieces put together make up a whole, then those pieces are thirds. Three different pieces, or three pieces that don't add up to a whole, are not necessarily thirds. Note also that this defines "fraction" in this specific mathematical sense. In everyday language we can use the word in other ways, such as "only a fraction of the population understands math well." We can even talk about one child getting the "bigger half" of a pie. But when we are writing a fraction using a numerator and denominator, the denominator has to indicate a number of equal pieces into which a whole can be cut, and the numerator is the number of those pieces chosen. Does that help to clarify things? - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 05/09/2003 at 20:54:04 From: Doctor Dotty Subject: Re: Fractions Hi Phil, Thanks for the question. The textbook definition is correct, for this reason: Here is a pie split up into bits of different sizes: * | * * | 1 * | - * | 4 * 11 |_ _ _ _ _ * -- \ 1 * 20 \ - * \ 5 * \ * * * * Each piece is a fraction of the whole. 11/20 is "11 equal parts of a whole" 1/5 is "1 part of a whole" 1/4 is "1 part of a whole" Together, if you add them all up, you do indeed get a whole. The definition was slightly confusing though, I agree, as it could be taken (as I suspect you did) as meaning that all fractions regarding the same whole have to have the same denominator - rather than you can't have two different denominators in the same fraction. A better way of defining a fraction could be: "A number written in the form a/b where both a and b are integers." Does that make sense? If I can help any more with this problem or any other, please write back. - Doctor Dotty, The Math Forum http://mathforum.org/dr.math/ |
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