Estimating and Rounding

Date: 03/04/2002 at 15:49:48
From: Amy 
Subject: Division - one answer problems

I am doing my math homework and we have to divide a two-digit number 
into another number and get a one-digit quotient. For example, 192  
divided by 86 = . Can you please help me?

Date: 03/04/2002 at 22:56:59
From: Doctor Peterson
Subject: Re: Division - one answer problems

Hi, Amy.

You can find some discussions of the tricks you need in our archives:

   Division by Estimation
   http://mathforum.org/dr.math/problems/humphrey11.4.96.html   

   Long division, Egyptian Division, Guessing
   http://mathforum.org/dr.math/problems/kiran.06.13.01.html   

   Compatible Number Estimating
   http://mathforum.org/dr.math/problems/matt.01.31.02.html   

There are a couple of important things to remember: you have to 
estimate (because you don't have a multiplication table that goes up 
to the 86's), and when you estimate you expect not to be exact. 
Therefore, you will be learning not only to make the best guess you 
can, but to correct the guess WHEN (not if) it turns out to be wrong. 
That's just part of the process, and doesn't mean you've made a 
mistake.

So let's look at your example. The first thing I usually do is to 
round both numbers, generally so that there is one non-zero digit left 
in the divisor and two in the dividend (though that's not always true 
- you'll get used to how to make this decision with a little 
practice). In this case, 86 rounds up to 90, and 192 rounds down to 
190. It won't matter here, but I like to round both numbers in the 
same direction, because that's more likely to give a good estimate. 
So I'd round both numbers up here (giving preference to the divisor), 
making it
       ______
    90 ) 200

Now, we can divide both numbers by 10 and it won't change the 
quotient; so ignore the zeroes on the end:
      _____
    9 ) 20

Now we've got something we can do: the answer is 2, since 9 * 2 = 18.

That's our estimate; but is it the right answer for the real problem 
we're doing? All we can do is check it by multiplying. We're hoping 
that
       ____2_
    86 ) 192

To check that (and also to find the remainder), we multiply the 
divisor by the quotient: 2 * 86 = 172. This is good: it's less than 
192, but not so much less that we could fit another whole 86 into it. 
That is, we can subtract to get a remainder, and the remainder is less 
than the divisor:

       ____2_
    86 } 192
         172
         ---
          20

So the remainder is 20.

At leat one of the links I gave you goes into how to correct an 
estimate if the check doesn't work out; briefly, you subtract one from 
the quotient if it's too big (so that the product was too big to 
subtract at all), and you add one if the quotient is too small (so 
that the remainder is too big).

Let me know if you need more help. You might want to send a sample 
problem worked out, so I can see where you might be going wrong or 
getting stuck.

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