Understanding Fractions: Addition, Subtraction, Types

Study Notes

Fractions: Breaking Down Numbers into Parts

Fractions are a fundamental part of mathematics, helping us represent parts of a whole more precisely than integers alone can. Let's delve into different types of fractions—adding them, subtracting them, and understanding their nomenclature.

Addition and Subtraction of Fractions

When dealing with fractions, addition and subtraction entail finding equivalent fractions with a common denominator before performing arithmetic operations. For instance, if we want to find ( \frac{2}{5} + \frac{3}{8}, ) we first need to make the denominators equal by finding the lowest common multiple (LCM):

[ \text{LCM}(5,8) = 40 ]

Now, multiply both numerator and denominator of each fraction by the necessary numbers to obtain new fractions that have the LCM as their denominator:

[ \left( \frac{2}{5} \times \frac{8}{8} \right) + \left(\frac{3}{8} \times \frac{5}{5}\right) = \frac{16}{40} + \frac{15}{40} = \frac{31}{40} ]

Similarly, let's consider subtraction: To calculate ( \frac{7}{9} - \frac{3}{11}, ) follow these steps:

[ \text{LCM}(9, 11) = 99 ]

Multiplying both numerator and denominator of each fraction by the necessary factors to get the LCM results in:

[ \left(\frac{7}{9} \times \frac{99}{99}\right) - \left(\frac{3}{11} \times \frac{9}{9}\right) = \frac{693}{99} -\frac{27}{99} = \frac{666}{99} = \frac{133}{11} ]

Like and Unlike Fractions

Like fractions share a common denominator. In our previous example regarding addition and subtraction, both ( \frac{2}{5} ) and ( \frac{3}{8} ) had an LCM of 40. Conversely, unlike fractions do not share a common denominator; ( \frac{2}{5} ) and ( \frac{3}{11} ) from the subtraction problem are examples of unlike fractions because they don't have a shared LCM.

Proper and Improper Fractions

A proper fraction is one whose absolute value is less than 1; i.e., it has a numerator smaller than its corresponding denominator. Thus, (\frac{2}{5}) and (\frac{3}{8}) are proper fractions since their respective numerators ((2) and (3)) are less than their denominators ((5) and (8)). On the other hand, an improper fraction has an absolute value greater than or equal to 1, which means its numerator is either equal to or larger than its denominator. Hence, (\frac{7}{9}), from the earlier subtraction example, represents an improper fraction.

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Description

Explore the world of fractions by learning how to add and subtract them, understand like and unlike fractions, and differentiate between proper and improper fractions. Enhance your math skills and confidence in dealing with these essential components of mathematics.