Young's Formula

Date: 08/21/99 at 20:33:53
From: Eric Smith
Subject: Problems with units

I'm having trouble finding information on how to solve problems with 
units, such as physics or chemistry problems. Specifically, I don't 
know how to solve "Young's formula" for calculating a child's dose of 
medicine based on the adult dose.

     C = A( g / g + 12) 

where

     A = adult dose in mg
     g = child's age

For example, if adult dose is 600 mg and child's age is 3 years, what 
is C? This sounds so easy, but it's driving me crazy.

     C = 600 mg (3 yr / 3 yr + 12)
     C = 1800 mg * yr / 3 yr + 12

Now what?

Thanks.

Date: 08/22/99 at 23:27:16
From: Doctor Peterson
Subject: Re: Problems with units

Hi, Eric. Surprisingly, this is a fascinating question, because I can 
see how you can get hung up on this even though it really is a simple 
problem. You're being too conscientious, in a sense, and you have hit 
upon a conflict between two ways of presenting formulas that I hadn't 
thought about before.

It looks like you've learned to carry the units along in the 
calculation, which is a great method that I like to use. You just have 
to be careful that the formula is set up right before you use this 
method, something I don't think I've ever heard taught.

First, we have to be clear what the formula is. Parentheses would have 
been helpful. Here's what I think it should be:

               g
     C = A * ------
             g + 12

Second, because there is an unlabeled constant in the formula, before 
we can replace the variables with labeled values, we need to label the 
constant. The formula was written with a different convention in mind, 
namely that each variable is expected to be a number representing a 
_fixed unit_; I would write it carefully like this:

               g
     C = A * ------
             g + 12

where

    A = adult dose in mg
    g = child's age in years
    C = child's dose in mg

To work the problem this way, as intended, you simply plug in the 
numbers and apply the specified unit at the end:

For A = 600 and g = 3:

                 3             3
     C = 600 * ------ = 600 * -- = 120 mg
               3 + 12         15

To work this problem with units attached to the numbers, we have to 
recognize that since g is assumed to be in years, 12 must be in years:

                 g
     C = A * ---------
             g + 12 yr

where

     A = adult dose
     g = child's age
     C = child's dose

Written this way, we could use any units for A and g, and will get an 
answer for C that is correct in whatever units it ends up with, 
including any conversions you want to do:

For A = 600 mg and g = 3 yr:

                      3 yr                3 yr
     C = 600 mg * ------------ = 600 mg * ----- = 120 mg
                  3 yr + 12 yr            15 yr

If g were in days, you would find it necessary to convert it to years 
before adding; if A were in pounds, you would get C in pounds.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/