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### Converting a Mixed Decimal to a Fraction

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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.

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
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/
```
Associated Topics:
Middle School Fractions

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