Adding and subtracting fractions is a key skill in algebra. You'll learn to work with common and different denominators, simplify , and solve equations involving fractions.
This knowledge builds on basic fraction concepts and prepares you for more advanced algebraic operations. Mastering these techniques will help you tackle more complex math problems with confidence.
Adding and Subtracting Fractions
Adding fractions with common denominators
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Add the numerators of the fractions together while keeping the the same
Simplify the resulting fraction by dividing the and by their (GCF) if possible
Examples:
83+81=84=21
125+127=1212=1
Adding fractions with different denominators
Find the (LCM) of the denominators to determine the common denominator
Multiply the numerator and denominator of each fraction by the factor needed to obtain the common denominator
Add the resulting numerators together while keeping the common denominator
Simplify the resulting fraction by dividing the numerator and denominator by their GCF if possible
Examples:
41+61=123+122=125
32+53=1510+159=1519 (This is an example of an improper fraction)
Simplifying complex fractions
Simplify the numerator and denominator of the complex fraction separately
Apply the order of operations (PEMDAS) to simplify each part
Divide the simplified numerator by the simplified denominator
Simplify the resulting fraction by dividing the numerator and denominator by their GCF if possible
Examples:
52−6121+31=30765=65÷307=65⋅730=725
32+2143−61=67127=127÷67=127⋅76=21
Solving expressions with fractions
Simplify the expression by combining like terms and performing any necessary operations
Multiply both sides of the equation by the common denominator to eliminate fractions
Solve the resulting equation using algebra techniques
Isolate the variable on one side of the equation
Perform the same operation on both sides of the equation to maintain equality
Examples:
32x+41=65
Multiply both sides by 12: 8x+3=10
Subtract 3 from both sides: 8x=7
Divide both sides by 8: x=87
43x−21=31
Multiply both sides by 12: 9x−6=4
Add 6 to both sides: 9x=10
Divide both sides by 9: x=910
Working with Mixed Numbers and Equivalent Fractions
are a combination of a whole number and a proper fraction (e.g., 3½)
To add or subtract mixed numbers, convert them to first
are fractions that represent the same value (e.g., ½ and 2/4)
Use to determine if two fractions are equivalent
are fractions where the numerator and denominator are swapped (e.g., 2/3 and 3/2 are reciprocals)
Key Terms to Review (18)
Addition of Fractions: Addition of fractions is the process of combining two or more fractions to find a single, equivalent fraction. It involves finding a common denominator and then adding the numerators of the fractions with the same denominator.
Common Denominator: A common denominator is the lowest number that can be used as the denominator for multiple fractions, allowing them to be added, subtracted, or compared. It is a crucial concept in working with fractions, solving equations with fractions or decimals, and performing operations on rational expressions.
Commutative Property: The commutative property is a fundamental mathematical principle that states the order of the operands in an addition or multiplication operation does not affect the result. It allows the terms in an expression to be rearranged without changing the final value.
Complex Fractions: A complex fraction is a fraction that has a fraction in either the numerator or denominator, or both. These fractions can be simplified by converting them into a single fraction with a numerator and denominator that do not contain any other fractions.
Cross Multiplication: Cross multiplication is a technique used to compare and manipulate fractions by establishing a relationship between the numerators and denominators of the fractions. This method is widely applied in various algebraic contexts, such as adding and subtracting fractions, solving equations with fractions or decimals, simplifying rational expressions, and solving proportion and similar figure applications.
Cross-Multiplication: Cross-multiplication is a technique used to compare the relative size of fractions or to solve for an unknown value in a proportion. It involves multiplying the numerator of one fraction by the denominator of the other fraction, and vice versa, to determine the relationship between the fractions.
Denominator: The denominator is the bottom number in a fraction that represents the total number of equal parts the whole has been divided into. It is a crucial concept in understanding and working with fractions, rational expressions, and other mathematical operations involving division.
Equivalent Fractions: Equivalent fractions are fractions that represent the same numerical value, even though the numerator and denominator may be different. They are fractions that depict the same portion or amount of a whole.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is a fundamental concept in elementary algebra that is applicable in various contexts, including whole numbers, fractions, and factoring.
Improper Fractions: An improper fraction is a fraction where the numerator is greater than the denominator. This means the value of the fraction is greater than 1, unlike a proper fraction where the numerator is less than the denominator and the value is less than 1.
Least Common Denominator: The least common denominator (LCD) is the smallest positive integer that is divisible by all the denominators of the given fractions. It is a crucial concept in elementary algebra that allows for the addition, subtraction, and simplification of fractions with unlike denominators.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers without a remainder. It is a fundamental concept in mathematics that is crucial for operations involving fractions and rational expressions.
Like Fractions: Like fractions are fractions that have the same denominator. They can be easily added or subtracted because the denominators are the same, allowing the numerators to be combined directly.
Mixed Numbers: A mixed number is a representation of a quantity that combines a whole number and a proper fraction. It is a way to express a number that is not a simple whole number or a simple fraction.
Numerator: The numerator is the part of a fraction that represents the number of equal parts being considered. It is the value that is positioned above the fraction bar and indicates the quantity or amount being referred to.
Reciprocals: Reciprocals refer to the relationship between two numbers where their product is equal to 1. In other words, if two numbers are reciprocals, multiplying them together will always result in 1. Reciprocals are an important concept in both adding and subtracting fractions, as well as solving equations using the division and multiplication properties of equality.
Subtraction of Fractions: Subtraction of fractions is the process of finding the difference between two fractions by aligning the fractions with a common denominator and then subtracting the numerators. This operation allows for the comparison and removal of fractional amounts from one another.
Unlike Fractions: Unlike fractions are fractions that have different denominators, meaning the bottom numbers of the fractions are not the same. This makes it necessary to convert the fractions to equivalent fractions with a common denominator before they can be added or subtracted.