Fraction Multiplication and DivisionDate: 02/10/2001 at 22:21:50 From: (anonymous) Subject: Fractions are killing me I don't understand problems like 2 1/10 * 7 5/8 or 27/30 divided by 75/100. I try to work the problem by dividing the bottom by the LCD and just multiplying or dividing the top. Date: 02/12/2001 at 12:28:48 From: Doctor Ian Subject: Re: Fractions are killing me Hi Tiffany, Maybe we should start at the beginning. When we have a fraction like 3/5, that is really shorthand for a pair of operations - 'cut something into 5 pieces, and keep 3 of them.' So, let's say that we want to take 3/5 of six candy bars. We cut each of the bars into five pieces: +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ | | | | | | | | | | | | | | | | | | +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ | | | | | | | | | | | | | | | | | | +-+-+-+-+-+ +-+-+-+-+-+ +-+-+-+-+-+ Then we keep three from every bar: +-+-+-+ +-+-+-+ +-+-+-+ | | | | | | | | | | | | +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ | | | | | | | | | | | | +-+-+-+ +-+-+-+ +-+-+-+ So we have 18 pieces, any five of which could be put together to make up a whole candy bar. In other words, we have 18 'fifths' of a candy bar...or 18/5 of a candy bar. If we try to combine the pieces into whole bars, +-+-+-+ +-+-+-+ +-+-+-+ |1|1|1| |1|1|2| |2|2|2| +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ +-+-+-+ |2|3|3| |3|3|3| | | | | +-+-+-+ +-+-+-+ +-+-+-+ we get three whole bars, plus three extra pieces. In other words, 18/5 = 3 3/5 This is what we're doing when we say that 3 18 3 6 * - = -- = 3 - 5 5 5 Okay, so what if we have something like 2/3 * 3/4? Well, imagine that we have something like a cake. Can we take 2/3 of the cake? Sure - cut it into three pieces, and keep two of them: +----------------+ +----------------+ | | | | +----------------+ +----------------+ | | => | | +----------------+ +----------------+ | | +----------------+ So, can we take 3/4 of the remaining part? Sure - cut each of those pieces into four smaller pieces, and keep three out of each four: +----------------+ +---+---+---+---+ +---+---+---+ | | | | | | | | | | | +----------------+ => +---+---+---+---+ => +---+---+---+ | | | | | | | | | | | +----------------+ +---+---+---+---+ +---+---+---+ So we end up with six small pieces. But how big are these pieces? In fact, 12 of these pieces would make up one whole cake. So each piece is 1/12 of a cake. And we have six of them, so we have 6/12 of a cake. Note that this happens to be the same thing as having 1/2 of a cake. Now, can we get this same answer without drawing pictures? In fact, we can, by multiplying the numerators and multiplying the denominators to get a new fraction: 2 3 2 * 3 6 - * - = ----- = -- 3 4 3 * 4 12 We don't really have to worry about things like LCDs when we're multiplying or dividing by fractions. We just multiply the numerators and denominators. So what about something like 2 1/10 * 7 5/8? Well, there are a couple of things that we need to keep in mind. The first is that 2 1/10 is the same thing as 2 + 1/10... the '+' is implied. (Mathematicians are always making up notations that let them cut down on the amount of writing they have to do. So they write '4*5' instead of '5+5+5+5' or '4+4+4+4+4,' and they write '2^6' instead of '2*2*2*2*2*2,' and so on.) The second thing we need to keep in mind is that 2 is the same as 20/10 because 20 divided by 10 is 2. And 7 is the same as 56/8 because 56 divided by 8 is 7. So we can rewrite the problem this way: 2 1/10 * 7 5/8 = (2 + 1/10) * (7 + 5/8) = (20/10 + 1/10) * (56/8 + 5/8) = 21/10 * 61/8 And now we just have a regular fraction multiplication problem, which works the same way as 2/3 * 3/4. Multiply the numerators, and multiply the denominators: 21 * 61 = ------- 10 * 8 So, what about dividing by fractions? There is a trick to dividing fractions, which is this: To divide by a fraction a/b, you multiply by the fraction b/a. So, for example, 2/3 --- = (2/3) * (4/3) = (2*4)/(3*3) = 8/9 3/4 Why does this work? It's not easy to come up with a picture to explain this, but the reason it works is that you can always multiply something by 1 without changing it, and '1' comes in many disguises! 2/3 2/3 4/3 --- * 1 = --- * --- 3/4 3/4 4/3 2/3 * 4/3 = --------- 3/4 * 4/3 2/3 * 4/3 = ------------- (3*4) / (4*3) 2/3 * 4/3 = --------- 12 / 12 2/3 * 4/3 = --------- 1 = 2/3 * 4/3 You could go through these steps every time you wanted to divide by a fraction, e.g., 5/6 5/6 * 17/9 ---- = ----------- = 5/6 * 17/9 9/17 9/17 * 17/9 but in the end, it's easier just to remember the trick, and go right to the final step. If you understand _why_ the trick works, then you can always figure it out again if you forget. This is one reason that you should never be content just to memorize something without understanding it. If you don't use the things you've memorized, they fade in your memory, and then when you can't remember them, you're just stuck. I hope this has been helpful. Write back if any of this wasn't clear, or if you're still stuck. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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