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.