Operations with
Mixed Numbers and Improper Fractions
Mr. Martin
Definitions. An improper fraction is simply a fraction with a numerator (top) larger than the denominator (bottom). In other words, it is a fraction with a value over 1. There is really nothing improper about improper fractions and in algebra we usually want to keep fractions as improper fractions especially if we will be performing further operations. An improper fraction can be expressed as a whole number and a fraction under 1, however. This is called a mixed number. It is often easier for us to quickly understand the amount of a mixed number and therefore teachers in the past have often told you to convert to a mixed number. It is important to realize what a mixed number is. It is really a whole number plus a fraction under 1. For example, the mixed number 3 ¾ means 3 + ¾.
Example.
An example of an improper fraction is 
.
4 will go into 15, 3 times, leaving ¾ remaining. Therefore, =
3 ¾. Conversely, we can take the mixed number 3 ¾ and convert it into an
improper fraction. In 3 wholes, there are 12 fourths. (4 multiplied by 3 is
12.) With 3 additional fourths we have a total of .
Addition and Subtraction. In adding and subtracting mixed numbers you can add and subtract the whole numbers and fractions separately.
For example, the
sum of 3 ½ + 4 ¾ is equal to 3 + 4 or 7, plus ½ + ¾ or 
,
or a total of 7 ,
which can be expressed as 8 ¼. This also works for subtraction, but sometimes
you must borrow. For example, the problem 4 ¼ - 2 ½ can be solved as follows.
Since ¼ is less than ½ we must first change 4 ¼ to 3 .
With this change and a change to a common denominator, the new problem is
hence 3 -
2 
.
3-2 is 1. -
=
¾. 1 + ¾ = 1 ¾. You can also solve such problems using integer operations
without borrowing. 4-2 = 2. ¼ - ½ = - ¼. 2 + - ¼ = 1 ¾. (Remember, a mixed
number adds a fraction and a whole number. For example, 4 ¼ means 4 + ¼.)
Finally, you can, but do not have to, use improper fractions in adding and
subtracting mixed numbers. For example, 4 ¼ - 2 ½ becomes 
Multiplication
and Division. With multiplication and division you must convert to an
improper fraction first. For example, 3 ¾ x 4 ½ = 
If
you multiplied the whole numbers and fractions separately you get 12 
,
which is not the right answer. Try it out with decimals. 3.75 x
4.5 = 16.875 = 16 
.
To divide, convert to improper fractions and multiply by the reciprocal of the second
number. For example, 3 ¾ ÷ 4 ½ = 
.
You cannot divide the whole numbers and fractions separately. You
will get the wrong answer.
Calculators. You can perform all of these operations with fractions on most scientific calculators with two line displays such as the TI-30X IIS®. The Ab/c key allows you to enter mixed numbers and improper fractions. The 2nd key with the Ab/c key gives the Ab/c-d/e function which converts fractions to mixed numbers and mixed numbers to fractions. The 2nd F-D key converts a decimal to a fraction and a fraction to a decimal. We will review these operations further in class. Further information is available in chapter 8 of the publication, TI-30X IIS: A Guide for Teachers, at the Texas Instruments web site, http://education.ti.com/us/product/tech/30xiis/guide/30xiisguideus.html
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