Fractions and Adding ZerosDate: 08/26/2001 at 16:11:01 From: cassandra Subject: Fractions I am having problems with doing fractions on the calculator. Is there a way to do it? I can't do long division, like 23/34, and how to get the least common denominator. Date: 09/04/2001 at 11:35:08 From: Doctor Ian Subject: Re: Fractions Hi Cassandra, Let's look at the problem you mentioned, which is how to divide 23 by 34 (or, to put it another way, how to divide 23 into 34 parts): ____ 34 ) 23 How many times does 34 go into 23? Well, it doesn't go any times. However, it goes into 230 some number of times: ? _____ 34 ) 230 How many times? Well, the only way to find out is to guess. If you've memorized your times tables, you know that 3*8 is 24, so 30*8 would be 240, so 8 is probably too high. What about 7? 7 * 34 = 7 * (30 + 4) = 7*30 + 7*4 = 210 + 28 = 238 which is still too big. How about 6? 6 * 34 = 6*30 + 6*4 = 180 + 24 = 204 So 34 goes into 230 at least 6 times: 6 _____ 34 ) 230 204 --- 26 230/34 = 6 + 26/34 Now we're back in the same boat, trying to find out what 26/34 is. 34 doesn't go into 26, but it will go into 260. We already saw that 7*34 is 238, so we know that it goes at least 7 times: 67 ______ 34 ) 2300 204 --- 260 238 --- 22 We can keep going on like this, and eventually, one of two things will happen. Either we'll end up with no remainder, 375 ______ 8 ) 3000 24 -- 60 56 -- 40 40 -- 0 Done! or we'll end up with a remainder that we've already seen, 39 _______ 33 ) 1300 99 --- 310 297 --- 13 We've already seen this! in which case we can stop, because we know that we're just going to keep repeating the same calculations over and over again: 39393... (the sequence '39' repeats forever) _________ 33 ) 1300000 99 --- 310 297 --- 130 99 --- 310 297 --- 130 99 Okay, but what's really going on here? Why can we just keep adding zeros? Don't we need a decimal point in here somewhere? In fact, we do, but it certainly makes life easier if we forget about it until the end. Let's look at what happened with 3/8. Instead of taking 8 into 3, we changed the 3 to 3000. And we found that 3000 / 8 = 375 How does this help? Well, we can multiply both sides of an equation by the same thing without changing the truth of the equation, right? Let's try multiplying both sides of _this_ equation by 1/10. The easy way to do that is to move the decimal point to the left by one place - for example, 375 multiplied by 1/10 is 37.5. So 3000 ---- = 375 8 300.0 ----- = 37.5 Multiply everything by 1/10. 8 30.00 ----- = 3.75 Again... 8 3.000 ----- = 0.375 Again... 8 Now we're back to our original problem, and we've found the answer. Now, there is a quicker way to do this, which is to leave the decimal point where it is, which gives us the answer including the decimal point without having to go through this final step: 0.375 _______ 8 ) 3.000 24 -- 60 56 -- 40 40 -- 0 But if you don't see _why_ this 'adding more zeros' stuff works, then this looks pretty much like some kind of magic spell. And the _reason_ that it works is that we're using the same multiplication trick to try to change our original problem into one that we can solve using only integers: 3 30 1 - = -- * -- 8 8 10 300 1 = --- * --- 8 100 3000 1 = ---- * ---- 8 1000 ^ ^_____________ | | We can solve | this problem with integers... and then move the decimal point to get the final answer. = 375 * (1/1000) = 0.375 In other words, we use multiplication on both sides of the equation to turn one messy problem into two simpler problems, whose solutions we can easily combine to get the solution to the original problem. As you take more mathematics, you'll see this pattern come up over and over again, because mathematicians love to solve hard problems by turning them into easy ones. In fact, when you get to algebra, _most_ of what you'll be learning is how to do this. Does this help? Write back if you'd like to talk about this some more, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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