Improper Fractions

Date: 04/15/99 at 20:32:29
From: Priyana Kumar
Subject: Fractions (Mixed number.....)

Dear Dr. Math,

I really need help on how to make an improper fraction into a mixed 
number.

Date: 04/16/99 at 16:44:05
From: Doctor Peterson
Subject: Re: Fractions (Mixed number.....)

Hi, Priyana. Dr. Math to the rescue!

Let's look at an improper fraction this way: I'll give you 53 blocks 
of wood, and tell you that each piece is 1/6 of a board. How many 
boards did I use to make them?

    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 1 | | 7 | | 13| | 19| | 25| | 31| | 37| | 43| | 49|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 2 | | 8 | | 14| | 20| | 26| | 32| | 38| | 44| | 50|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 3 | | 9 | | 15| | 21| | 27| | 33| | 39| | 45| | 51|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 4 | | 10| | 16| | 22| | 28| | 34| | 40| | 46| | 52|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 5 | | 11| | 17| | 23| | 29| | 35| | 41| | 47| | 53|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    | 6 | | 12| | 18| | 24| | 30| | 36| | 42| | 48|
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+

I can divide 53 by 6 to find out how many boards I can make by putting 
the blocks back together in groups of 6:

      ___8_
    6 ) 53
        48
        --
         5

This says I can make 8 whole boards, using 8x6 = 48 of the pieces, and 
I'll have 5 blocks left. Since each block is 1/6 of a board, that 
means I've made 8 5/6 boards:

    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    |   | |   | |   | |   | |   | |   | |   | |   |
    +---+ +---+ +---+ +---+ +---+ +---+ +---+ +---+
    \_____________________________________________/ \___/
                   8 boards = 48/6                   5/6

The rule for changing an improper fraction into a mixed number is to 
divide the numerator by the denominator; the whole-number quotient is 
the whole number part of the answer, and the remainder is the 
numerator of the fraction part:

    +------------------------+
    |                        |
    |               5 <--+   |
    | ___8_----> 8 ---   |   |
    6 ) 53          6 <--|---+
        48               |
        --               |
         5 --------------+

Another way to say it is that we find the largest multiple of 6 that 
is less than 53 (that's what dividing means), and change that part 
into a whole number:

    53   48+5   48    5       5
    -- = ---- = -- + --- = 8 ---
     6     6     6    6       6

I hope that helps you make sense of mixed numbers. Here are some other 
explanations in our archives if you need more help:

   Fractions, Improper Fractions, Mixed Numbers
   http://mathforum.org/library/drmath/view/58069.html 

   Converting Fractions to/from Mixed Numbers
   http://mathforum.org/library/drmath/view/58875.html

   Improper Fractions and Mixed Numbers
   http://mathforum.org/library/drmath/view/58936.html   

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