Monday, April 12, 2021

The Case for Inches and Fractions

Or, Fraction Math Without a Calculator!

Buckle up!

I recently was watching a tutorial video from a well-known and very talented leather maker on layout and design. It was painful for me to watch. Not because he wasn't a good teacher, but because of how he was struggling with his method for measuring distances and dividing those distances in two.

He did what many of us would now first think to do: he measured the distance between two points in inches and fractions, he converted the fractions into decimals using a table he printed out so he could input those fractions into his calculator, made his calculations and then used his table again to re-convert those decimals back into fractions so he could find the new measurements on his ruler.

Those of you who do all of your measuring in the metric system are laughing about now.

There are so many ways to screw up calculations in the above scenario that one really needs to be careful.

First, I suggest don't measure at all, if you can. Use dividers to make your measurements and transfer them directly. No math, no figuring. If you need to divide those distances you've measured, there are easy ways to do it that have been around thousands of years. Others have covered this, and I suggest looking these ways up because they are foolproof and involve no measuring, measurement systems, or math. In other words, it's foolproof: no mistakes.

If you insist on measuring with a tape, ruler, measuring stick, etc. (and I know you do), don't blow off the imperial measuring system just because dealing with 10s is sometimes easier.

What if you need to divide 293 milimeters by two? Likely you can do that in your head, but there is still a tiny bit of guesswork involved.

Inches were born for this. As an example, let's take the above situation with the leather worker. He measured his leather and found it to be four and a quarter (4¼) inches long. So far, so good. 

What he did was convert 4¼ into 4.25 so he could put it in his calculator. He divided this number by two to find the midpoint, and the calculator told him the midpoint was at 2.125 inches.

Out came his trusty piece of paper with a table on it telling him that 0.125 inches is ⅛ inch. Add two and his midpoint is two and one-eighth inches.

For the love of God! Did no one learn fractions in the third grade?

Here's how dividing 4¼ should be done (in your head):

divide the whole number: 4/2=2. This is easy because the number is even, we don't have to do anything else to it.

Look at the fraction - ignore the top number and double the bottom number: ¼ becomes ⅛.

2⅛ inches. Done.

Let's try another measurement from my own leather project: four and seven eighths (4⅞) inches:

Half of the whole number: 4/2=2.

Ignore the top number of the fraction and double the bottom: ⅞ becomes 7/16.

Half of 4⅞ is 2 7/16.

One more example from my current project: seven and three eighths (7⅜) inches.

Half of the whole number: 7/2=2.5. Woops! If the whole number is odd, this doesn't work. What do we do? 

Don't panic.

Take the next lower even number. In this case, six. 6/2=3.

That extra whole number needs to go to the fraction. Add the bottom number of your fraction to the top, and that is your new top number. In our example, ⅜ becomes 11/8 when we add eight to the top number, three.

We're not done yet. We have an inch added to our original fraction. We need to find half of this new fraction. Same as before, just double the bottom number. 8*2=16. Our new fraction is 11/16.

Add it all up, and our midpoint for 7⅜ inches is three and eleven sixteenths (3 11/16).

Here are a few common measurements for you to practice. Write down the answers before you look at the answer key.

Find the midpoint:

  • 11½
  • 5⅝

Here are the answers (no cheating!):

  • four and three eighths
  • five and three quarters
  • two and thirteen sixteenths
Easy, right? Let me know your method.

16 comments:

  1. Just out of curiosity, what was the video you were watching? Can you post a link?

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    1. Hi J.J.! I didn't want to call the artist out, because the content of his video is excellent. But since you asked:

      https://youtu.be/r0CgLSKvAac

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  2. Thanks for sharing. I've been looking at maybe adding leatherwork to my skill set. This looks really good.

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    1. I've gone deep into leather work this year. It's a great skill to add to your arsenal. Go for it!

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  3. Just don't measure.... fold it - LOL
    I was born in metric. We "got" it in Portugal since XVIII century (back then these things took a while). So I had to learn the Imperial measurement of length - Inche and foot because of the woodworking YT videos...
    BUT
    Today if I go to my local hardware store to buy a tap for a hose today it is still in 1/4", 1/2" 3/4" or 1" and that is true for screws on older machines as well due to the XIX century British industrie influence

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  4. In a casual setting friends on the metric system will usually ask "can you even give a single rational reason why the USA stubbornly adheres to the tradition of imperial measurement" (Usually this is asked less articulately with some profanity and judgmental looks).

    My go to answers is that most trades people are doing calculations in their head and that the most common operation is division by 2 to achieve symmetry within a +/- 1/64in precision; and that given those constraints the imperial system is actually easier to use then base 10. (Something most people learn from experience as an journeymen in the trade as they are trusted to start doing layouts)

    Thank you for an example of how bad this can go for the uninitiated.

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  5. Good examples Brian.
    It is why I stick with Imperial, not only the ease of handling fractions but fractions I can visualized easily in my head. 290 mm not so much. But for everyday operations in my shop, I eschew most measurements, letting the tools dictate whatever size 1/4 inch or 1/2 in chisel I'm using happened to be. And try to stick to using same one throughout project. Just because you have a few, it's a bad idea to switch midstream. Saying for a friend :-)
    Setting up machines and tools for precision I used set up blocks, again no measurements.
    That and using go nogo gauge, story stick, matching tenon to mortise and etc.

    Bob, the staunch Imperialists' when it comes to measurements. Not because of precision or ease of use, but because I can visualized easily a given size without relying on memory. A 1/2 of something is just that, so is a 1/4 and etc. Easy Peasy for my pre-wired Imperial brain :-)

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  6. Nederland, Belgium and Luxemburg (which were united at that time) were the first nations to adopt definitively the metric system. France came after because Bonaparte made back track under people's pressure. So France had to adopt the metric system a second time later.

    If you visit the BIPM web site, you will see that now the SI (System International) is linked to a few natural constant that are valid anywhere in the universe and that it is not anymore linked to artifact kept in Paris. https://www.bipm.org/en/measurement-units/si-defining-constants .

    I am not convinced by the divide by two story.
    When I want to hang a picture, I try to place it with space above and below more or less according to the golden ratio.
    I remember asking to somebody to hang a cabinet on a wall. He was very proud to have placed it "schoon in de middel" [in Flemish]) wasting the space on each side which was now too small to hang something else.


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  7. I used to teach woodworking at a local community college, you get some really smart students and some that struggled in high school. In dealing with fractions I would see panic in some students faces so when dividing up some measurement I showed them how to use a straight edge as a divider, just pick an easily divisible measurement longer than the width you're trying to divide, using a consistent edge of the rule lay the end on one edge, your easily divisible number on the other and pick the middle, only very simple math involved.

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  8. This hackneyed "fractional argument" is really little more than low-rent fodder for confirmation bias. At least you mention that you shouldn't measure whenever possible. And when it comes to measuring or design or whatever, dividing really ought to be a last resort (at least not without actual dividers). Want to "divide" something? Multiply instead. For example, how about dividing four inches into five parts? It's easier to start with 20mm, multiply by five, and you get 100mm, which is functionally equivalent to four inches for almost anything that doesn't require mechanical precision.

    I will digress on how much easier it is to deal with incremental and go/no-go measurements in SI.

    Anyways, in the US, SI encounters a barrier of unwarranted perceived difficulty, and that's coming from an American who lives in the US. We're familiar with US customary units because those are the labels on our tools. Almost everything at the home center is meted out in those units. Why change if there's no mandate, and the customers are just going to moan anyway? And it's not like an individual can or should make any difference. I can imagine the happy welcome looks I'd get at the lumberyard if I went asking for 25mm roughsawn stock.

    The thing is, though, that converting to SI doesn't mean tearing up everything that already exists, and doesn't involve banning customary units. If you go to a home center or whatever in Japan, Taiwan, the UK, or wherever, you'll see things with divisions their own customary units as well as stuff with SI units, because the customary stuff has never gone away. And there's nothing particularly wrong with that.

    What you won't see there, though, are goods in units that were awkward or just plain stupid to begin with. That stuff went away really quickly once people realized how much easier SI is in practice when you really do need to measure and convert. Yeah, I'm talking about you, pounds and ounces. Fluid ounces, cups, gallons. (Don't even get me started on distances and temperature.)

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  9. Multiply the denominator by the whole number and add the numerator. You can now divide The distance into any number of segments (not just half)by multiplying the denominator by the number of segments.reduce the fraction back to whole number and fraction by simple division. If you wanted to divide a length into twelve segments, it still works. One formula for all possibilities.

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  10. Most of my measuring tools include both metric and imperial digits. A lot of us can walk a mile by counting paces. How did 18th century explorers get so close to satellite measurement. It works and different units are mental stimulation.

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  11. As a scientist, I use metrics all day and am fine. Having grown up in the USA, the two things I can't shake are temperature (as in the local weather) and measurements. I can see out to about 10 cm in mm minds eye but not beyond that. As such, I use imperial measuments and I'm fine with that. In reality, as you mentoined, there is so little actual measuring involved in woodworking. Like you, I often use dividers, use a knife to scribe a distance. If I need to divide something, often dividiers or put a ruler on a angle and demark the divisions (the other ruler trick).

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  12. As an anonymous pointed here above, the intercept theorem is the easy way to divide any length whose measurement is not simple in any system.

    Now, for mortise and tenon, divide by three is more common than divide by two. Although, I usually grab the chisel which is approximately 1/3 of the thickness.

    Using inches and foot might be handy for woodworking, but once you need other units it isn't anymore.

    video:
    https://www.youtube.com/watch?v=r7x-RGfd0Yk

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  13. If you know how, you can divide a width 7-57/64 into 5 equal spaces in seconds without any math.

    Take a rule and put the 0 on one edge and the 10 on the far edge. Mark off at 2, 4, 6, and 8. Done. Those marks are 1/5 the width away from each other.

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  14. Ok, and then there's feet and yards and chains, furlongs and miles. And the difference between US and real pints (and quarts and gallons).

    I can sort of visualise both inches (up to maybe 60cm, that'd be like two feet) and metric. But as you wrote one rarely needs to measure. Just mark off the corresponding part. The final size of the cupboard or shelf will need measuring (very carefully), should the object fit into a certain spot. For that anything works.

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