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Creating Improper Fractions

Date: 02/28/98 at 14:42:26
From: Walter
Subject: improper fractions and mixed numbers

How do you change a mixed number into a improper fraction?

Date: 03/26/98 at 12:45:03
From: Doctor Roberta
Subject: Re: improper fractions and mixed numbers

Dear Walter,

You asked Dr. Math how to change a mixed number into an improper 
fraction. Let's start by investigating what a mixed number is. One 
example is the number 3 1/4. It is mixed because it has a whole 
number, 3, and a "broken" number, 1/4. The whole number and the 
fraction are added together, although people do not usually write the 
+ symbol between them. That is, 3 1/4 means three plus one fourth.  
There are three wholes added to one fourth of a whole.

An example of 3 1/4 is three whole dollars and one quarter. One 
dollar can be broken into four quarters and one quarter is 1/4 of a 
dollar. Then changing 3 1/4 into an improper fraction is like changing 
$3.25 into all quarters. How many quarters would you have? I get 13 
quarters. Each dollar is broken into 4 quarters (3 x 4 = 12) and I 
have one more quarter. I write this amount as 13/4.  

So when one dollar is the whole, 3 1/4 means three wholes and one 
fourth of a dollar or one quarter. 13/4 means only quarters are used 
and there are 13. The numbers 3 1/4 and 13/4 have the same value.

Is there a shortcut to changing a mixed number into an improper 
fraction? Yes, there is. It works like this: to change a mixed number 
into an improper fraction, multiply the whole number by the 
denominator of the fraction and add the numerator to this product to 
get the numerator of the improper fraction. Use the same denominator.  
Here is an example and an explanation of why the shortcut works.

Change 2 3/10 to an improper fraction.

Shortcut:  Take 2 x 10.  The answer is 20.  Add the numerator, 3, to 
20. 20 + 3 = 23.  We keep the denominator of 10, so the improper 
fraction is 23/10.

This shortcut works because multiplying 2 x 10 is like breaking the 
two wholes up into 10 parts each. Then we add the three parts we 
already have for a total of 23 parts which are each a tenth of the 

Since one dollar can be broken into 10 dimes, we can think of 2 3/10 
as two whole dollars and 3 dimes. If this amount is changed completely 
into dimes, there would be 2 x 10 = 20 dimes in the two whole dollars 
and three more dimes. There are 23 dimes or 23/10 in 2 3/10 dollars.

-Doctor Roberta,  The Math Forum
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Associated Topics:
Elementary Fractions

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