Exploring Estimation

Date: 08/29/2003 at 01:48:56
From: Lisa
Subject: Estimating

If I pay $13.20 for a 0.44 oz can of a rare truffle, about how much
would I have to pay for a whole pound?

I tried this:  0.44 divided into 16 = 36.36.  Round to 36.  
36 x $13.20 = $475.00

Date: 08/29/2003 at 12:42:38
From: Doctor Peterson
Subject: Re: Estimating

Hi, Lisa.

Your work is correct, as far as actually solving the problem is 
concerned; and that's really the hard part of the problem. If we 
wanted an exact answer, we would divide 16 oz/lb by 0.44 oz/can to 
find the number of cans per pound, and then multiply that (without 
rounding) by $13.40/can to find the price per pound. The answer then 
is 16/0.44 * 13.40 = $487.27/lb. (Note how I'm keeping the units with 
the numbers in order to make sure they mean the right thing.)

Now, assuming that the problem says to estimate the answer rather 
than find the exact price, what is the best way to estimate? In a 
relatively complex problem like this, it will not surprise me if 
there are several good ways to estimate that give slightly different 
answers with the same amount of work. (We judge the value of an 
estimate by both how close it is likely to be to the exact answer, 
and how much work is saved.)

How we estimate may depend on the order in which we do the 
multiplication and the division. Let's try a couple ways.

First, if we divide first as you did, then we might either look for a 
compatible number and divide 16 by .4 to get 40, or round 0.44 to 0.5 
and get 32. Both are easy; the former happens to be a little closer. 
Then we would multiply one of these answers by a rounded price per 
can, say $13. The result will be either 520 or 416. The 
multiplication of 32 times 13 is a little hard to do in your head, so 
you might want to re-round, rounding 32 down to 30 and 13 up to 15, 
giving an answer of 450. So that gives us three different reasonable 
estimates.

What if we reasoned differently and first divided $13.40/can by 0.44 
oz/can to get $30.45/oz, and then multiplied by 16 oz/lb to get 
$487.27/lb? The exact answer, of course, is the same; but we might 
want to round differently. I might round both 13.40 and .44 down and 
divide 12 by .4, giving 30, and then multiply that by 16 to get 
$480/lb.

And if we recognized that the final answer of 16/0.44 * 13.40 could 
be calculated by multiplying first (as 13.40*16/0.44), we would first 
estimate 13.40*16 as, say, 10*20 = 200, and then divide by .4 to get 
$500.

So I've found five different, equally reasonable estimates:

  520  416  450  480  500

Of these, it turns out that the most accurate is 480; but I can't 
really say that any one of them is the best estimate.

This is probably more than what you wanted; your answer was correct 
in terms of the calculation, but not quite an estimate in the sense 
of rounding wherever it would save work. If you're going to calculate 
any part with three or four digits of accuracy, you might as well not 
round at all.

My main point here is not whether you were right or not, but that 
there are many valid ways to estimate, something that often is 
forgotten in school due to the need to grade everything!

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

Date: 08/30/2003 at 14:26:26
From: Lisa
Subject: Thank you (Estimating)

Wow!  I am so impressed.  Thank you so much for such an in-depth
answer and explanation.