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### Paint Formulas in Base 48

```
Date: 08/20/98 at 14:56:34
From: LINDA BEHYMER
Subject: BASE 48

I work for a paint store where our formulas are based on an ounce
being 48 parts. Example: 33Y42, which is 33 ounces plus 42/48 of an
ounce. All formulas are given for a gallon mix. When we want to do a
quart, we have to convert everything to the 48 base and then divide by
4 to get a quart formula. Would you know how we can make this an
easier solution or even a computer program that would help? I've
searched all over the internet and feel lucky to find someone I could

Thank you,
Linda
```

```
Date: 08/21/98 at 09:45:15
From: Doctor Peterson
Subject: Re: BASE 48

Hi, Linda.

This is a unique question. It was very insightful of you to identify
this as base 48, because that's the key to a neat solution. This is
really very much like the base 60 fractions the Babylonians used. In
their case, they didn't have a separate symbol for each of 60 "digits,"
but wrote a sequence of base-ten numbers, each representing a power of
60, much as we write 23 35' 27" for degrees, minutes, and seconds.

What I would suggest is that you do your calculations in base 48. I'll
demonstrate how I would do your division example. If you have to do
other things like multiplying, my illustration may be enough to help
you get started (since division is by far the hardest thing to do in
another base), or you could write back with other problems.

We want to divide 33Y42 by 4. (What does the Y stand for, by the way?)
I'll write this as a regular long division problem, and make up a new
notation to let us handle the weird base.

__________
4 ) 33 Y 42

First, we just divide the 33 as usual, in base 10:

___8______
4 ) 33 Y 42
32
--
1

Now, we have to cross the "base line", and this is the key to any work
with a base like this, even "base 12" calculations with feet and
inches or "base 60" calculations with time. A 1 in the ones column
translates into 1 x 48 in the 1/48's column (that is, 1 = 48/48), so
I'll add 48 to the 42:

___8______
4 ) 33 Y 42
32
--
1 = 48
--
90

Now I can just divide this 90 by 4 as usual, in base ten:

___8_Y_22_
4 ) 33 Y 42
32
--
1 = 48
--
90
8
-
10
8
--
2

So the answer is 8Y22.5. This is much easier than converting
everything to 48ths before dividing. I tried that, and got the same
answer, but I won't even bother trying to show it.

Alternatively, here's how to do the same thing on a calculator, which
is probably what you would really want to do. Divide the 33 by 4 to
get 8.25; write down 8Y, then work out the remainder by multiplying
8 * 4 and subtracting it from 33 to get 1. (A scientific calculator
would have a key to get the remainder. On my Windows computer's
calculator, it's called MOD, and you'd type 33%4 to get 1.) Now
multiply this by 48 to get 48, add the 42, and divide that by 4 to get
22.5, and write that down after the Y. You're done.

I hope that helps. Is this a standard problem throughout the industry,
and has everyone always been doing it the long way?

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Number Theory