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Mar 31 2008

Understanding Fractions, The Key to High Accuracy When Measuring.

Published by at 4:23 pm under Metrology,Skill Development

I want to have that little talk with you about, Fractions. Yeah. But the plan is, if all goes well, that it won’t hurt – as much as it did last time. Working in sub inch territory usually involves the use of little buggers. The problem many people have when working with fractions, is that they relate the use of the common fraction to their math education experience when they were in school as children.

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Our school systems scared the bejeezus out of everyone by forcing us all to learn a series of mathematical exercises, which evolved around fractions that we would never use again in our entire lifetimes. For many, this often created mental blocks to the entire notion of fractions, even the simple useful ones, because after that harrowing experience, it seemed that nothing pleasant could possibly come from the manipulation of fractions at all. In fact, when people are faced with dealing with fractions, they generally feel some panic along with it. A panic that rates up there with the sound of high speed dental drills and root canals, and it is most likely from their harrowing experience in math class. Folks remember what all the wonky practice of solving mismatched fractions was really like, and relate that it was way, way too similar, and maybe even the diabolical preparation, for diagramming English sentences later on during their high school education.

I hope I can help make this a lot more user friendly!

Fractional arithmetic for measurements within the sub inch territory is much simpler than the nightmare a lot of people conceive it to be, because there is a fixed set of fractions that are in use, and memory tricks which you can learn to make well practiced calculations that you can do in your head. This is the reason why the trades have been slow to discard the use of the inch. The fractions are just too handy for halving, quartering and doubling.

There are just some basic constructs we need to know about fractions. A quick few definitions of terms, and memory tricks for using them quickly, and we can be off and running with this for most linear measurement purposes.

Basically put, a fraction is a division problem, which is meant to deal with components of a unit that are smaller than the unit of one. Within the realm of linear measurement, the constructs used are expressing the halving of the unit from the whole on one end, to what is commonly considered usable with common tools on the other. The tools are made to fit the fractional intervals.

Commonly, the scale by incremental division looks like this:
1, 1/2. 1/4, 1/8, 1/16, 1/32, 1/64, 1/128.

In descending order you have a process of halving, and in ascending order you have a process of doubling. Divide or multiply by 2. So there is a memory trick. Get used to doubling and halving units less than one.

Though the numbers in the fractional divisions seem to become larger, it does not indicate the size of the unit, but rather, the quantity of units, which will fit within an inch, so it is best to think of it inversely. The larger the number gets, the smaller the unit is.

The parts of a fraction are worth touching on for a moment.

1 (the numerator) It is the first number or the above number as written.
— Then there is this dividing line, which signifies the division problem.
2 (the denominator) It is the second number or the below number as written.

The dividing line has a name and the name is different depending on how the fraction is written, but the names of the separator do not help you work with fractions easier. So don’t worry about what to call it. For us, “Slash” and “Dash” are fine.

The denominator signifies the sizes that the divisions of one (1 inch in our discussion) are. The numerator signifies the quantity of those denominated divisions we have.

For most of our purposes in linear measurements, these fractions when used strictly as fractions of an inch, will only be added and subtracted to and from each other.

It is important to accept that 1-inch can be expressed as a fraction. No matter what the denominator may say, if you have any quantity of numerators equal to the denominator, the quantity is equal also to one. Such as 4/4ths, 8/8ths, 32/32nds.

For expressing results when two fractions added together create a numerator, whose quantity exceeds the value set by the denominator, then a whole number, such as 1, 2, or what have you, is generated, and the fractional remainder is then expressed with the whole number. This expression is called a mixed number, and is the final expression of your result. As an example you would express 3/4 + 3/8 as 1-1/8, instead of 9/8.

When expressing fractions, they are best expressed in the reduced or simplest form possible. If the numerator is able to be added to another numerator which will derive an even number, the denominator level you are working with may not need to be referred to as a smaller unit, in fact, when numerators added together resulting in an even number, the denominator can likely be expressed as a reduced or simpler unit. As an example: 1/4 + 1/4 = 2/4 = 1/2, and 3/8 + 3/8 = 6/8 = 3/4. It is considered best practice to express the fraction in its simplest form.

A useful property of numbers, which creates a memory trick that we can use, is that if one of two numbers to be added together is an odd number, it will always result in the sum of the numbers being added together to be an odd number. Interestingly, even numbers when added together will result in an even number, and when any two odd numbers are added together they will also result in an even number. Learning to notice this trick will alert you to when the numerator will result in an odd number. When the fractions numerator is an odd number, the denominator cannot be expressed more simply than the finest size being used amongst the mixed fractions, and you will not likely be able to simplify it beyond that resultant fraction. As an example: 7/32 + 1/8 = 11/32. The trick here is that 1/8 has to be converted to its 32nd equivalent, 4/32, and then you can easily add it to the 7/32. The result is as simple as it can be made, because the numerator is an odd number, and cannot be reduced.

Halving all fractions, which are not part of a mixed number is a pretty simple process. Halve the denominator, (multiply the denominator by 2 as this doubles the fractional division making them smaller by half) the numerator remains the same. The result is always half. For example: 3/4 halved is 3/8. 5/16 halved is 5/32. 7/8 halved is 7/16, and so on.

For mixed numbers it is almost as easy. Convert the mixed number into a pure fraction and multiply the denominator by 2. When finished, convert the fraction back to its simplest form, which includes reverting back to a mixed number if that is the simplest form. For example: 1-7/8 = 15/8ths. 15/8 halved is 15/16ths. This is its simplest expression. 2-9/16 = 41/16ths. 41/16 halved is 41/32nds, which is not a simple fraction, so converted back to a mixed number it becomes 1-9/32nds. Please note again, the numerator numbers in simplest expression form did not change from the original expression. Remember when you see the numerator, the trick is that it will still remain the same but the denominator changes by half, leaving little to think about once you remember the trick.

Fractions by the 128th, from 1/128th to 1/4th, as expressed in simplest form. Observe the patterns: 1/128, 1/64, 3/128, 1/32, 5/128, 3/64, 7/128, 1/16, 9/128, 5/64, 11/128, 3/32, 13/128, 7/64, 15/128, 1/8, 17/128, 9/64, 19/128, 5/32, 21/128, 11/64, 23/128, 3/16, 25/128, 13/64, 27/128, 7/32, 29/128, 15/64, 31/128, 1/4. See the interchangeability of denominators?

And finally the last biggie is that fractions only hit specific decimal locations, so sometimes, in order to get to an increment near the fraction you have, you need to convert to a decimal and work it from there. It is simple. Divide the denominator into the numerator for the decimal equivalent. For instance 7/64ths would be converted by dividing 7 by 64, and the result would be .1094, 3/4 would be .750, and 3/32nds would be .0937. How fine is markup to the 128th of an inch? It is .0078 of an inch. Call that about the width of two whiskers.

Why is the fractional to decimal relationship important? Say you are working on adding a shelf to a cabinet project in 3/4 Baltic Birch. This is nice plywood, commonly available, but the thickness is actually metric. 3/4 is close to 18mm but there is a catch. The decimal equivalents are not exact. 3/4 = .750 and 18mm = .709. This is a 41 thousandth of an inch difference, which will need compensation. Usually the compensation is made by working to the fraction nearest to the metric equivalent, which in this case is 23/32nds, but there is a .010 of an inch remainder. This can be an unacceptable gap in some work, so this too is good to know.

Another workaround which creates a better looking fit, could be to disregard the metric size for joinery altogether, and create a 1/2 inch tongue and groove, but in order to center the .500’s of an inch tongue on the .708 inch thick board, you are going to have to subtract .500 from .708, and halve that result of .208 to make it .104, so you know what the shoulders for the tongue will need to measure, in order to center it on the metric board. The nearest fractional increment to the proper size of this shoulder is 7/64 and as you see by dividing 64 into 7 that .109 will make the shoulder too large. The shoulder too large will make the tongue too small. If you cut to the nearest fractional increment here, the tongue will be centered but only .490 wide, and this is a sloppy fit in a .500 groove for very fine work. In fact the same quantity of slop you had working 18mm into 23/32nds. Now you can dial in the necessary compensation.

Knowing how to manipulate the fractions and knowing where the fractions lie amongst the decimals will help you build higher quality into your fine woodworking or machine work, where fit and finish is everything.

Hopefully these memory tricks and conventions will help you to work with fractions faster, easier, and more proficiently with more confidence. The understanding of fractions for use in linear measurements is conquerable, and for the most part is kept pretty simple and doable by the constructs involved with them. One could only hope that much of the rest of fractional manipulation could work as easily.

Going forward, feel free to practice these memory tricks, and if you like, add both a fractional and decimal dial caliper to your metrology tool arsenal, they will help you a ton. For further reference, feel free to use the Decimal Equivilents chart I have provided, as well as the other tools available in the Woodworks Reference Library. They are all printable and ready for use in the shop.

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