Subtleties of Rounding

Date: 09/24/2003 at 13:05:43
From: John
Subject: Rounding decimals

There are 480 freshmen at Washington High.  If 36% of the freshmen
play sports, how many play sports?

Is the correct answer 172 or 173?

The argument for 172 is that you cannot have .8 of a person so you
round down to 172.  The argument for 173 is that you should round to
the nearest whole number (and that 173 is closer to 36% of 480 than
172 is).

Date: 09/24/2003 at 14:55:18
From: Doctor Peterson
Subject: Re: Rounding decimals

Hi, John.

I think rounding to 173 is best if you need a specific answer; but in 
reality we don't quite know.

If someone who knew the actual number calculated the percentage to 
the nearest percentage point, and told us that it is 36%, then the 
exact percentage might be anywhere from 35.5% to 36.5%, corresponding 
to numbers from 170.4 to 175.2. That means that the original number 
might have been 171, 172, 173, 174, or 175! In each of those cases, 
the percentage would come to 36% after rounding; so all are equally 
possible.

If we take the given percentage as 36.0%, accurate to the nearest 
tenth of a percentage point, then there have to be 173 playing 
sports, since percentages of 35.95% and 36.05% correspond to 172.56 
and 173.04. You can see why rounding to the nearest whole number 
makes sense.

If we take the percentage as being _exactly_ 36%, of course, then 
somebody on the team is missing a few body parts. That's why we HAVE 
to make some assumption about the precision of the data.

I love these little problems that turn out to be a lot more 
complicated than they seemed on the surface!

If you have any further questions, feel free to write back.

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