Fractions |
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A lot of people have trouble with fractions, I think we have all become rather dependant upon the old calculator, and along the way we forget what we don't use. Then of course there is today's education system, which only seems happy when teaching the young how to use a calculator and not the basics that you need in life. Many years ago I had an apprentice straight out of school, and he didn't have a clue what a fraction was, it took me a week in spare moments, just to give him the basics that he should have been taught, very sad, and this lad had qualifications too. Well I am not going into any depth here, this is just for those of you having fun with change wheels, if you want more get a book. So lets look at simple fractions and what we can do with them to our advantage. What is a fraction? just the decimal equivalent, for example lets take half of something, that is 1/2 and that is a proper fraction, to convert it into a decimal carry out the obvious arithmetic 1 divided by 2 = 0.5 But before we go too far lets understand the terminology of numerator and denominator, these you will see referred to in books a lot. Its easy stuff, think of currency and what denomination is that, pounds dollars etc. So your fraction above was in halves ie 0.5's of which we had one of them, that is 1 over 2, or one numerator of a denomination of 2. Or half a dollar, or half a pound or 50 pence if you wish. Another fraction is an improper fraction, all this is is one where the top is greater than the bottom, in other words there are more of them than make a whole number. It makes no difference if we are talking fractions or decimals, they are only part of a whole. So if we had the vulgar fraction 3/2 what do we have, well divide it yourself and its 1.5, a number greater than a whole, hence a vulgar fraction. Or a dollar and a half etc. But how to adjust them this is the fun part, because we wont have change wheels with two teeth on them will we...Not unless you had an accident anyway. So 1/2 = 1 part of 2 equal pieces and two of those equal pieces make a whole, you can do anything you like to this fraction provided that you do the same to both the top and the bottom and the result will be the same, for example lets multiply both by 30 and we get 30/60 now divide it to make a decimal and the answer is 0.5 still, if that's no good we can do it again lets divide it by 10 = 3/6 and divide it it's still 0.5. Okay I'm sure that is clear. In our examples for change wheels we made up a larger fraction into two seperate ones, we did this in order to have 4 change wheels instead of two, because we could not make the ratio using only 2 wheels. Lets look at that again, this time we will use the fraction 250/40, so how do we do this. Take each number one at a time and find 2 numbers that when multiplied together equal it, for the 250 two obvious ones are 10 and 25, move on to the 40 and lets go for 5 and 8. Now prove to yourself that this is the same 10/5 x 25/8 = 6.26 which is the same as 250/40. We now have two different fractions to play with, but now you can do all sorts of things and as I can't do this in this format please look at the picture below....
So the basic rules are what ever you do to the top of one fraction you must do the same to the bottom, ie divide or multiply, or as shown above the top of one fraction to the bottom of the other, all you need do at the end is check the final result with the ratio you started with and you have got it, play with the numbers until you get the ones you have gears to match.
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