Are There Times When Following the Rounding Rules Isn't Best?

Date: 03/14/2007 at 22:22:07
From: Mary
Subject: Rounding either up or down while using a two digit number

With 32 - 15 the actual answer is 17.  However, when rounding to the
nearest 10 I am certain that 32 is rounded to 30.  What way is the
number 15 rounded if it is to be rounded once again to the nearest 10?
My teacher is wanting us to round the numbers first before we solve
the problem to see if our answer is close to the answer of the
estimation to figure out if we got the correct answer.

What I don't understand here is the number 15.  If I round the 15 up 
to 20 for my estimation, then my answer to the estimated part of the 
problem would be 30 - 20 = 10.  But if I round the 15 down to 10 as 
the closest 10 then my estimation is 30 - 10 = 20.  

My Mom says my teacher is trying to show me that by rounding 15 down,
since it is 5 or less in the ones column, then my estimated answer of
20 is only 3 digits away from 17. 

My Dad is telling me that we should round the 15 up to 20.  But that
makes my estimated answer of 10 farther away from 17 since it is 7
digits away.

Who is correct here?  My mom or dad?  Is the rule 5 and down rounds
down?  Or is it 5 and up rounds up?  I'm so confused.  Help!

Mary

Date: 03/15/2007 at 06:13:37
From: Doctor Peterson
Subject: Re: Rounding either up or down while using a two digit number

Hi, Mary.

Both are right in some sense.  Ultimately, the point here is that 
following rules is not the best way to do math; you should instead 
think about what will best accomplish your goal.

The rule usually taught in school is that when the only digit you 
would drop in rounding a number is a 5, you round up.  (There is 
another rule often taught that says you always round to the nearest 
even number.)  If you just follow that rule here, rounding both 
numbers to the nearest 10, you get

  32 - 15 =~ 30 - 20 = 10

(I'm using "=~" to mean "is approximately equal to".)

But our goal is not just to follow rules; it is to get the best 
approximation we can, as easily as possible.  And the round-up rule is 
only a custom: in reality, both 10 and 20 are equally close to 15, so 
neither can be called THE CLOSEST multiple of 10 to 15.  We just pick 
one because we have to.  If there is a better reason to choose one 
direction over the other, then we should follow that reason; and ANY 
reason is better than "because that's the way we do it"!

So we think, instead: We're already rounding 32 down; if you decrease 
one number and increase the other in a subtraction, you are sure to 
change the result significantly.  But if you decrease both by the same 
amount, you don't change the answer at all, and if you decrease them 
by similar amounts you won't change it much.  So when you estimate a 
subtraction, it is best if you can round them both in the same 
direction.  (For an addition, you'd want to round in opposite 
directions, one up and the other down.)

So let's round the 15 down, contrary to the usual rule; we get

  32 - 15 =~ 30 - 10 = 20

As you noted, this is a better estimate than following the usual rule 
for rounding.  The exact answer is 17, which is closer to 20 than to 
10.  That's not surprising, since for our second answer we thought 
carefully about what would work best, and we were right!

Now, I don't know which way your teacher expects you to do it.  She 
may want you to follow the rule, as your father did, and will then 
show you that there is a more accurate way to estimate; or she may 
want you to think for yourself and discover the better answer (as 
you have done and your mother recommends).  Either way, I think if 
you give the answer of 20 and explain why you chose to round down, 
it can't hurt!

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


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