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### Clarifying Fraction Notation

```

Date: 07/02/97 at 00:04:39
From: Paul
Subject: HELP

The problem:

83 1/3
.83 1/3 =    ---------------------    =  ?
100

The answer is supposed to be

5
---
6

problem?

```

```

Date: 08/29/97 at 14:57:06
From: Doctor Sonya
Subject: Re: HELP

Hi Paul:

It is hard to tell what this notation means. We don't usually include
a fraction such as 1/3 after a decimal point. It confuses the meaning
of a decimal, a fraction whose denominator is a power of 10, not 10
and 3 as here.

The number of digits after the decimal place is the same as the power
of 10 in the denominator.  For example, 3/10 = 0.3, 40/100 = 0.40,
333/1000 = 0.333, etc. You could rewrite 0.83 1/3 as a series of
decimals by separating the digits like this:

0.83 1/3 = 0.8 + 0.03 + 0.00 1/3

which can be written

8/10 + 3/100 + (1/3)/100.

The sum of these terms will simplify correctly into the answer you
want.

But there is a more direct way to solve your nephew's problem, and
it involves a different interpretation of the notation. The "decimal"
part of the number has two decimal places. Therefore, you can think
of 0.83 1/3 as the compound fraction (83 1/3)/100, where the mixed
fraction 83 1/3 is the numerator, and 100 is the denominator.

Rewrite the mixed fraction in the numerator as the compound fraction
250/3. The expression now looks like (250/3)/100.

Dividing by 100 is the same as multiplying by 1/100. Therefore,
(250/3)/100 is equivalent to (250/3) x 1/100. Multiply the two
fractions: The product is 250/300 which reduces to 25/30 = 5/6.

There's also an algebraic way to solve the problem, if you're
interested. It is the most sophisticated, but also the most
elegant. The number 1/3 as a decimal is 0.3333..., where the ellipsis
after the 3 means that the 3s never stop coming. Then, write
0.83 1/3 as 0.83333.... with lots of trailing 3s implied.

Let   x = 0.833333... Multiply both sides by 10:
10x = 8.3333...

Subtract the first equation from the second and all the repeating 3s
disappear.

10 x = 8.3333...
1 x = 0.8333...
_______________
9 x = 7.5 (Divide both sides by 9.)
x = 7.5/9 = 75/90 = 5/6.

Whew! This was a fun problem!

-Doctors Chita and Sonya,  The Math Forum

http://mathforum.org/dr.math/

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Associated Topics:
Middle School Fractions

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