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 ? means 3 + ?.
????????? Example. An example of an improper fraction is _{}.? 4 will go into 15, 3 times, leaving ? remaining. Therefore, _{}?= 3 ?.? Conversely, we can take the mixed number 3 ? 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 ? + 4 ? is equal to 3 + 4 or 7, plus ? + ? or _{}, or a total of 7 _{}, which can be expressed as 8 ?.? This also works for subtraction, but sometimes you must borrow. For example, the problem 4 ?  2 ? can be solved as follows. Since ? is less than ? we must first change 4 ? to 3 _{}.? With this change and a change to a common denominator, the new problem? is hence 3 _{}? 2 _{}.? 32 is 1. _{}? _{}?= ?. 1 + ? = 1 ?.? You can also solve such problems using integer operations without borrowing. 42 = 2.??? ?  ? =  ?.? 2 +  ? = 1 ?.? (Remember, a mixed number adds a fraction and a whole number. For example, 4 ? means 4 + ?.) Finally, you can, but do not have to, use improper fractions in adding and subtracting mixed numbers. For example, 4 ?  2 ? becomes _{}
????????? Multiplication and Division. With multiplication and division you must convert to an improper fraction first.? For example, 3 ? x 4 ? = _{}??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 ? ? 4 ? = _{}. 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 TI30X IIS?.? The Ab/c key allows you to enter mixed numbers and improper fractions. The 2^{nd} key with the Ab/c key gives the Ab/cd/e function which converts fractions to mixed numbers and mixed numbers to fractions. The 2^{nd} FD 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, TI30X 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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