Numbers Containing Whole Number, Decimal, and Fraction at Once

Date: 10/19/2004 at 15:32:57
From: nesha
Subject: solving a problem that has both a decimal and a fraction 

When dealing with a problem with a whole number, decimal, and a 
fraction all in one number, what kind of formula would I use?  
Example: 3.366 2/3.  The problem asks to rename the following decimals 
as proper fractions in lowest terms.

I know how to find a fraction from a decimal, and I know how to find 
a decimal from a fraction.  The formulas are different, so when the 
problem has both I don't know what kind of formula to use.

Date: 10/19/2004 at 18:47:49
From: Doctor Ian
Subject: Re: solving a problem that has both a decimal and a fraction 

Hi Nesha,

First of all, I wouldn't worry too much about this, because you're
never, ever going to come across something like 

  3.366 2/3

in real life.  There is absolutely no point in writing a number this
way.  

However, when in doubt, start from definitions.  When we write a
decimal number, like 3.366, it's really a sum:

       3    6      6
  3 + -- + --- + ----
      10   100   1000

So what would 3.366 2/3 mean?  It would mean

       3    6     6 2/3
  3 + -- + --- + ------- 
      10   100    1000   

which is to say, 

       3    6    20/3      
  3 + -- + --- + ---- 
      10   100   1000   

or

       3    6     20      
  3 + -- + --- + ---- 
      10   100   3000   

Does that make sense?  Now you need to convert everything to a common
denominator, add, and simplify:
                                        
  9000    900    180     20   10100    
  ---- + ---- + ---- + ---- = -----  
  3000   3000   3000   3000    3000    

                              /   /           /   / 
                              2 * 2         * 5 * 5       * 101
                            = ---------------------------------
                              2 * 2 * 2 * 3 * 5 * 5 * 5
                              /   /           /   /

                                 101
                            = ---------
                              2 * 3 * 5

                              101
                            = ---
                               30

To check, divide 101 by 30:

            3.3 6 6
       ____________
   30 ) 1 0 1.0 0 0
          9 0
         ----
          1 1 0
            9 0
          -----
            2 0 0
            1 8 0
            -----
              2 0 0
              1 8 0
              -----
                2 0

Now, we could keep going; or we could stop here, and think about what
to do with the numerator.  

If you think back to when you were first learning division, you worked
on problems like 

       5
     ___
  3 ) 17             17 divided by 3 = 5 remainder 2
      15
      --
       2

Later on, you learned to interpret the remainder as a fraction, with
the denominator equal to the divisor:

       5
     ___
  3 ) 17             17 divided by 3 = 5 2/3
      15
      --                           
       2

It's the same thing here:


            3.3 6 6
       ____________
   30 ) 1 0 1.0 0 0         101 divided by 30 = 3.366 20/30
          9 0
         ----
          1 1 0
            9 0
          -----
            2 0 0
            1 8 0
            -----
              2 0 0
              1 8 0
              -----
                2 0


where 20/30 is the same as 2/3, and it means 2/3 of the last PLACE
VALUE for which you obtained part of the quotient.  In the case of 17
divided by 3, the last place is the units place, so it's 

  5 + (2/3 of 1) 

In the case of 101 divided by 30, the last place is the thousands
place, so it's

  3.366 + (20/30 of 1/1000)

which is what we would mean by 

  3.366 2/3

if we ever actually wrote it that way.  Which we don't.  :^D

Does this make sense? 
 
- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 10/20/2004 at 10:28:51
From: nesha
Subject: Thank you (solving a problem that has both a decimal and a
fraction )

Thank you so much for the answer to my question.  My math book did not
explain in as much detail as you did.  Now I understand how they were
coming up with their answers, whereas before I was completely lost.