Cancelling FractionsDate: 03/06/2002 at 13:58:48 From: KJJ Subject: Cancelling Fractions Dear Doctor Math, I have a question about cancelling fractions. How do I cancel this problem: 1/5 x 40/1 x 1/2 ? Can you please help me? Sincerely, KJJ Date: 03/06/2002 at 16:04:21 From: Doctor Ian Subject: Re: Cancelling Fractions The thing you have to keep in mind is that a c a * c - * - = ----- b d b * d Now, usually you use this to go from left to right, to turn multiple fractions into one fraction: 3 5 3 * 5 15 - * - = ----- = -- 4 6 4 * 6 24 But it also works from right to left! 15 3 * 5 3 5 5 -- = ----- = - * - = 1 * - = 5/8 24 3 * 8 3 8 8 Note that each time I can break off a fraction like a/a, it's equal to 1. And since multiplying by 1 is the same as doing nothing, I can just ignore it. So let's look at a case like 1 40 1 - * -- * - 5 1 2 Note that 40 is divisible by 5: 1 5 * 8 1 - * ----- * - 5 1 2 And note that 8 is divisible by 2: 1 5 * 2 * 4 1 - * ---------- * - 5 1 2 So now I can mix the terms up, pair them, and unmix them: 1 5 * 2 * 4 1 - * ---------- * - 5 1 2 1 * 5 * 2 * 4 * 1 = ------------------- 5 * 1 * 2 5 * 2 * 1 * 4 * 1 = ----------------- 5 * 2 * 1 5 2 1 = - * - * - * 4 * 1 5 2 1 = 1 * 1 * 1 * 4 * 1 = 4 Now, this is _why_ you can cancel. But in practice, you would do something like this: 8 x 1 40 1 Note that 40 divided by 5 is 8; - * -- * - Get rid of the 5, and replace 40 with 8. 5 1 2 x Do you see why this works? I'm just saying that if I went to the trouble to expand 40 into 5 * 8, the 5's would cancel, leaving me with an 8. The next step would be 4 x 8 x 1 40 1 Note that 8 divided by 2 is 4; - * -- * - Get rid of the 2, and replace 8 with 4. 5 1 2 x x This is just the same thing again. You keep going until there is nothing else to cancel. In this case, I'm done, and the only thing left is a 4 in the numerator, so the answer is 4. Note that you might have to do the expansions to see the opportunities for cancellation. For example, 1 12 1 -- * ---- * -- 15 1 28 Now, in this case, nothing divides into anything else. But if we break everything into factors, 3*2*2 1 12 1 -- * ---- * -- 15 1 28 3*5 2*2*7 Now I can see the things that should cancel: x x x 3*2*2 1 12 1 1 -- * ---- * -- = ----- = 1/35 15 1 28 5 * 7 3*5 2*2*7 x x x Now, note how much easier this is than multiplying 15 by 28, and then dividing the result by 30. Every pair of operands that you can cancel is a pair of operations that you don't have to do. And _this_ is one of the reasons that everyone makes a big deal out of being able to find the prime factors of numbers. When you've got all the prime factors out in the open, you can just tick-tick-tick them off in pairs, leaving you with only the operations that you can't escape. I hope this helps. Write back if you'd like to talk more about this, or anything else. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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