Converting a Mixed Decimal to a Fraction

Date: 11/09/2001 at 13:54:23
From: Kathryn Leist
Subject: Converting mixed decimal to fractions

I am homeschooling an 8th grader. We have not been able to locate how 
to do this problem:

   Write 4.16 2/3 as a mixed number.  

We know the answer is 4 1/6. We just can't figure out how they reached 
that. When I was in school we just did repeating decimals and not 
fractions. Is this a new learning style?   

Thank you!

Date: 11/09/2001 at 17:01:44
From: Doctor Peterson
Subject: Re: Converting mixed decimal to fractions

Hi, Kathryn.

You can read about this notation in our archives:

   Clarifying Fraction Notation
   http://mathforum.org/dr.math/problems/goetzke7.2.97.html   

I wouldn't call it new, just rare. As Dr. Sonya said in that answer, 
this is not considered a standard notation; but it is sometimes seen, 
perhaps in certain fields that we are not familiar with as 
mathematicians and scientists. I think it would probably arise most 
often in converting from a percentage; 83 1/3% could naturally be 
written as 0.83 1/3, meaning (83 1/3)/100. This is confusing, and I 
wouldn't teach it as something to do deliberately, only as something 
one should be able to understand when we come across it. Like certain 
obsolete or vulgar words, it may be worth having as part of your 
"reading vocabulary," but should not be in your normal "speaking 
vocabulary."

In reading this notation, you have to think of the fraction as part of 
the last digit, not taking a decimal place of its own. In your 
example, 4.16 2/3 would mean 4 + 1 tenth + 6 2/3 hundredths, or 
4 + 16 2/3 hundredths, not as 4 + 1 tenth + 6 hundredths + 2/3 
thousandth. The ambiguity as to where the fraction belongs is the 
reason we recommend not using this notation. But again, in a context
where you expect two decimal places, and want to be more precise
(as in percents, or in money situations like 83 1/3 cents), there 
may be some value in mixing fractions with decimals.

To work out your problem, multiply the number by 100 to get rid of the 
decimal point, and then divide by 100 as a fraction to get back to the 
intended value:

               416 2/3   1250/3   1250   125   25
    4.16 2/3 = ------- = ------ = ---- = --- = -- = 4 1/6
                 100       100     300    30    6

You can save some work by taking the 4 off and putting it back on at 
the end:

                      16 2/3       50/3        50        1
    4 + .16 2/3 = 4 + ------ = 4 + ---- = 4 + --- = 4 + --- = 4 1/6
                        100         100       300        6

I'm curious as to which book teaches this; since we do get questions 
on this from time to time, clearly someone teaches it, but probably 
not many texts do so. It seems particularly odd, though, that the text 
itself wouldn't explain it; it must have been taught earlier in the 
series.

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