TOPICS: Fractions - Addition and Subtraction

Keywords

Addition of Fractions with equal denominators: [\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}]
Addition of Fractions with different denominators: [\frac{a}{c} + \frac{b}{d} = \frac{a \times d + b \times c}{c \times d}] (After finding the LCM)
Subtraction of Fractions with equal denominators: [\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}]
Subtraction of Fractions with different denominators: [\frac{a}{c} - \frac{b}{d} = \frac{a \times d - b \times c}{c \times d}] (After finding the LCM)
Simplification of Fractions: [\frac{a}{b} \rightarrow \frac{a \div x}{b \div x}] (Where (x) is a common divisor to (a) and (b))

Fraction: Represents one or more equal parts of a whole. It consists of the numerator (upper part, indicating how many parts of the whole are being considered) and the denominator (lower part, indicating into how many parts the whole is divided).
Numerator: Indicates how many parts of the whole the fraction represents.
Denominator: Indicates into how many parts the whole has been divided.
Least Common Multiple (LCM): The smallest positive integer that is a common multiple of two or more numbers. This concept is key to adding or subtracting fractions with different denominators.

Equivalent Fractions: Are fractions that, although different in their terms (numerator and denominator), represent the same amount. For example, (\frac{1}{2}) is equivalent to (\frac{2}{4}).
LCM Applied: Determining the LCM is essential when we need to perform operations with fractions that have different denominators. It allows us to convert the fractions to a common base, facilitating addition or subtraction.

Addition and Subtraction with the Same Denominator: When fractions have the same denominator, we add or subtract only the numerators, keeping the denominator unchanged.
- Step by step: ( \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} )
Finding the LCM: To add or subtract fractions with different denominators, we first find the LCM of the denominators, transform the fractions so that both have the LCM as a denominator, and then add or subtract the numerators.
- Step by step: ( \frac{a}{c} + \frac{b}{d} \rightarrow ) find the LCM of (c) and (d) ( \rightarrow ) convert to the same denominator ( \rightarrow ) add or subtract the numerators.
Simplification of Fractions: After performing the addition or subtraction, we simplify the resulting fraction by dividing both the numerator and the denominator by the greatest common divisor to both.

Addition with Equal Denominators: ( \frac{2}{5} + \frac{1}{5} = \frac{2+1}{5} = \frac{3}{5} ).
- Explanation: Both fractions have the same denominator (5), so we add only the numerators (2 and 1), resulting in (\frac{3}{5}).
Addition with Different Denominators: ( \frac{1}{3} + \frac{1}{6} ).
- Explanation: First, we find the LCM of 3 and 6 which is 6. Converting each fraction to have the denominator 6, we have that ( \frac{1}{3} = \frac{2}{6} ) and ( \frac{1}{6} = \frac{1}{6} ). We add the numerators, resulting in ( \frac{2+1}{6} = \frac{3}{6} ) which simplified is (\frac{1}{2}).
Subtraction with Different Denominators: ( \frac{3}{4} - \frac{1}{2} ).
- Explanation: The LCM of 4 and 2 is 4. Converting each fraction to have the denominator 4, ( \frac{1}{2} = \frac{2}{4} ). We perform the subtraction ( \frac{3}{4} - \frac{2}{4} = \frac{1}{4} ).

Fractions represent parts of a whole and are composed of a numerator and a denominator.
Equivalent Fractions are different in appearance but equal in value.
Least Common Multiple (LCM) is crucial for adding and subtracting fractions with distinct denominators.
The addition of fractions with the same denominator is done by adding the numerators.
To add or subtract fractions with different denominators, both are converted to the same denominator using the LCM.
Simplifying the resulting fraction is important to reach the most reduced form.

The ability to identify and operate with equivalent fractions is fundamental for working with additions and subtractions of fractions.
The process of adding and subtracting fractions requires attention to the denominators, requiring equality between them for direct operation.
Finding the LCM is an essential step to combine fractions with different denominators into a single fraction.
Simplification is the final step to present the result clearly and precisely, facilitating interpretation and subsequent use of the fraction.
Practicing these operations develops logical reasoning and mathematical ability, aiding in the resolution and elaboration of problems.