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Mathematics I (Math 1132)

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Durham College, Mathematics
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Mathematics I (Math 1132)

Simplification of Fractions

A fraction is a mathematical representation of a comparison of two or more quantities. This comparison consists of a numerator, denominator, and a fraction line.

A fractions can also be represented as ratio → 2 : 5 (2 to 5), in words → 2 over 5, and symbolically → 2 ÷ 5 (2 divided by 5). When 2 is divided into 5, the resulting quantity is called a quotient.

Furthermore, any fraction over zero will not result in a quotient, hence you will receive an error message on your calculator every time you do. This is also referred to as a restriction or constraint.

A common fraction is one whose numerator and denominator are both integers. However, if it contains algebraic experiences, it’s referred to as an algebraic fraction. These fractions will be the focus of a unit to come.

Proper and Improper Fractions

A proper fraction is when the numerator is of lesser quantity than the denominator – an improper fraction is when the opposite is true. This can be applied to both common and algebraic fractions. Here are some examples:

Common Fractions

34   proper common fraction119   improper common fraction

Algebraic Fraction

xx2   proper algebraic fractionx3+4x2   improper algebraic fractionx+2x   neither  

To determine if an algebraic fraction is proper or improper, the numerator needs to have a lower degree than the denominator and it must be rational. Notice that in the first example, the numerator x has an exponent of 1, while the denominator is raised to the power of 2; hence, the fraction is improper. The third example is neither because the numerator √(x+2) isn’t a polynomial given the square root. An algebraic fraction is called rational if the numerator and the denominator are both polynomials. As mentioned, a full unit on algebra will be introduced next.

Sometimes you may want to show the sum or difference of an integer with a fraction as a mixed fraction. For example:

5+12=512      or      234=234

A video showing how to convert between mixed and improper fraction is shown below:

Fraction to Decimal (and vice versa)

To change a fraction to an equivalent decimal, simply divide the numerator by the denominator.

1   25      50.42   =0.42   911      110.8181...9   =0.81¯

In the second example, the quotient has a repeating decimal. The repeating portion can be written with a bar over the repeating part called a vinculum.

To change a decimal number to a fraction, write a fraction with the decimal number in the numerator and 1 in the denominator. For example, if we have the decimal number 0.42, decimal 42 is written over 1.

.421

Then you multiply numerator and denominator by a multiple of 10 that will make the numerator a whole number.

.42×101×10=4.2×1010×10=42100

Finally, reduce to lowest terms using whatever method you’re comfortable with (more on this below). Given that 42 and 100 are even, you can divide both by 2.

42÷2100÷2=2250

Question:   Convert the decimal number as a fraction → 5.387.

Answer

5.38715.387×10001×100053871000

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Interestingly, the method is slightly different when the decimal portion is repeating. The video below shows a technique how to write these numbers as fractions.

Reducing Fractions

This part will be important when you learn about ratios later on in this unit. A reduced fraction is one in where the numerator and denominator have no common factors. Take, for instance, the fraction 4/10. Both the numerator and denominator have a factor of 2 in common:

410=2(2)5(2)=2255=25

Notice how 2/5 is an equivalent fraction to 4/10 that can’t be reduce any further without generating decimal numbers in the numerator and denominator. Think about, what other number divides perfectly into 2 and 5 – nothing. Therefore, to reduce a fraction to lowest terms, divide the numerator and denominator by its greatest common factor (GCF), and if you don’t know the GCF, divide the top and bottom by any common factor until it’s completely reduced. This process is also called simplifying the fraction.

Question:   Reduce 81/18, then write as a mixed fraction.

Answer

8118=81÷318÷3=27643643÷36÷3412

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