Estimating Fractions

Date: 12/02/2001 at 17:36:41
From: Justin
Subject: Estimating Fractions

Dr. Math,

I am completely clueless about the process of estimating fractions. 
Here is a simple example of an estimation question using fractions:
There is a plate of assorted cookies. 1/6 of these cookies are 
chocolate chip, while 1/5 of them are peanut butter. The rest of the 
cookies on the plate are regular. What fraction do the regular 
cookies make?

Sincerely,
Justin

Date: 12/03/2001 at 15:31:50
From: Doctor Ian
Subject: Re: Estimating Fractions

Hi Justin,

In this case, I think what you're supposed to recognize is that 1/5 
and 1/6 are very close to the same fraction, with 1/5 being a little 
larger. 

So one way you might proceed would be to say that 

  1/5 + 1/6 ~ 1/5 + 1/5 = 2/5

which would leave 3/5 of the cookies as regular. (The '~' is 
pronounced 'approximately equals'. It's this kind of approximation 
that you're doing when you estimate something.)

On the other hand, you might say that

  1/5 + 1/6 ~ 1/6 + 1/6 = 2/6 = 1/3

which would leave 2/3 of the cookies as regular. So you could say that 
'between 1/3 and 2/5 of the cookies are regular'. 

The idea is that this is much easier (and often much neater) than 
finding the _actual_ answer:

  1/5 + 1/6 = 6/30 + 5/30

            = 11/30

Yuck!  Note that 

    1/3 = 0.3333...

  11/30 = 0.3666...

    2/5 = 0.40

so the real answer is in between the two estimates... which is exactly 
what you'd expect if you made one estimate too large and the other too 
small. 

The basic idea with estimation is always the same: you agree to accept 
a little less accuracy in exchange for doing a lot less work. 
Sometimes that's a good trade, and sometimes it's not. Part of 
learning to estimate things is learning when estimation is a good 
idea, and when it's a bad idea. 

I hope this helps.  Write back if you'd like to talk more
about this, or anything else.

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