Deciding How to Express the Answer to a Division Problem

Date: 07/22/2006 at 06:52:09
From: Donald
Subject: How do I know when to express a remainder?

There are 3 possible answers when I divide 3 by 2:

  3 divide by 2 = 1 and 1 left over
  3 divide by 2 = 1 and 1/2
  3 divide by 2 = 1.5

How do I know which form I should use?

Date: 07/22/2006 at 11:53:16
From: Doctor Rick
Subject: Re: How do I know when to express a remainder?

Hi, Donald.

That's a good question!

If you are dividing 3 by 2 because you were given a problem that said,
"divide 3 by 2", then you may have been also instructed whether to
write the result as quotient and remainder, mixed number, or decimal.
If not, it's probably best to give the result as a mixed number.  I
say that because, on the one hand, in the future you will be needing a
quotient and remainder less often than you'll need a fraction or
decimal.  On the other hand, decimals have the problem that, if you
want an *exact* answer, it can take a lot of work to get to the point
where the decimal either terminates or repeats.  Have you learned
about that?  For instance, if I divided 281 by 34, the exact answer in
decimal form (with a repeating decimal) is
     ________________
  8.26470588235294117

That's a mess, and a lot of work, compared to the mixed-number 
answer, which is 8 9/34.

If you are dividing 3 by 2 because you're solving a word problem and 
you decided that dividing by 2 would help you find the answer, then 
you have to decide for yourself which form is best.

For most problems, an answer with a quotient and remainder will not 
make sense.  For example, suppose you have this problem: "The 24 
children in a class sold a total of 421 candy bars for a fund-raiser.
What is the average number of candy bars each child sold?"  You find
the answer by dividing the total, 421, by the number of children, 24.
If you wrote the answer as "17, remainder 13", someone who didn't know
what you did to get that number (that you divided by 24) wouldn't know
what that meant.  When you say instead, "17 13/24", your answer
includes the information about the divisor.  Without it, the answer is
incomplete. (An answer of 17.54 is also complete; it is less accurate,
but probably more useful.  More on that in a moment.)

There are some special types of problems where the remainder is 
important.  For instance: "Nine boys have 75 carrot sticks to share.
If they share them equally, how many will be left over?"  That's a
remainder problem.  If you divide 75 by 9 to get 8 1/3, you won't see
the answer.  If you get 8.33333..., that won't help either.  But "8,
remainder 3" tells you right off that each boy gets 8 carrot sticks
and 3 are left over.

If the problem calls for an exact answer, then it's best to get a 
mixed number as a result of the division, because as I said, an exact
decimal can be a real pain to calculate and to work with.  It's easier
to be exact with fractions.

If it's a "real-world" type problem involving measurements of things,
then you don't need an exact answer.  A girl's height, for example,
isn't going to be *exactly* 134 12/41 centimeters.  Usually a decimal
answer will be best (especially if you're working in the metric
system).  You can decide to round 134.292682926829268292682 to 134.3
centimeters, accurate to the nearest millimeter--that's probably good 
enough.  (And you can stop calculating when you reach 134.2 with a 
remainder; comparing the remainder to the divisor is enough to decide 
whether to round up or down.)

Even with problems in which you don't need an exact answer, if the 
division comes in the middle of a problem, it may be a good idea to 
keep the result in mixed-number form.  That way you don't have to 
round the intermediate result; you can save rounding until the very 
end, when you have a better idea of how accurate the answer has to 
be.

I hope that helps a bit!  I'm sure I haven't covered all the 
possibilities.  If you have a specific instance in which you aren't 
sure what kind of answer to give, show me the problem and we can 
discuss it.

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

Date: 07/23/2006 at 03:42:22
From: Donald
Subject: Thank you (How do I know when to express a remainder?)

Thanks heaps for answering my question.  It was very much appreciated.
You are the best!