Explaining Proportions and Unit Prices

Date: 09/21/2006 at 11:47:01
From: Ta
Subject: 5 lbs. divided by 10 lbs. = .5 lbs. How??

You need 5 lbs of flour to make 10 lbs of batter.  How many pounds of
flour do you need for 1 lb of batter?

I know how to do percents, and I know how to get these answers.  But I
can't grasp this concept: Why does dividing (or getting the %) the 5
lbs by the whole (10 lbs) give you the amount for a SINGLE pound?

Date: 09/21/2006 at 16:00:32
From: Doctor Ian
Subject: Re: 5 lbs. divided by 10 lbs. = .5 lbs. How??

Hi Ta,

Really, what's going on here is a proportion.  It takes 5 lbs of flour
to make 10 lbs of batter:

    5 lbs flour
   -------------
   10 lbs batter

Now, the main thing about a proportion is, you can scale it by n/n,
where n is any number.  For example, if we want to double the recipe,
we'd have

    5 lbs flour    2   10 lbs flour
   ------------- * - = --------------
   10 lbs batter   2   20 lbs batter

If we wanted 40 lbs of batter, we'd have to quadruple it:

    5 lbs flour    4   20 lbs flour
   ------------- * - = --------------
   10 lbs batter   4   40 lbs batter

Does this make sense?  If so, we can ask:  By what would we have to
scale 10 lbs of batter to end up with 1 lb of batter?  That would be
1/10, right?  So 

    5 lbs flour    1/10   5/10 lbs flour
   ------------- * ---- = --------------
   10 lbs batter   1/10      1 lbs batter

The principle doesn't change just because the scale factor isn't an
integer. 

Note that this is all that's going on when we compute unit fractions,
e.g., 

   189 cents    1/12    (189 * 1/12) cents
   ---------- * ---- =  ------------------
    12 ounces   1/12               1 ounce

or speeds,

    242 miles   1/1.5   (242 * 1/1.5) miles
    --------- * ----- = -------------------
    1.5 hours   1/1.5               1 hour

or even percentages--except in this case, we want a "unit" of 100,
since a percentage is a fraction with 100 in the denominator:

    36   1/129   100   (36 / 129) * 100 
   --- * ----- * --- = ---------------- = [(36 / 129) * 100]%
   129   1/129   100         100

In the other cases, once we end up with 1 in the denominator, we just
express it using "per":

   26.3 miles
   ---------- = 26.3 miles per hour
      1 hour 

So you see, it's really just the same idea--proportions--used again
and again, in slightly different ways.  So once you've played around
with proportions enough to really understand them, everything else
will click into place, too. 

Does that make sense?  Let me know if you need more help. 

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