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Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
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Adding and Subtracting Fractions

with Like Denominators

Objective Learn how to add and subtract fractions with like denominators.

This concept is extremely important, both from a conceptual point of view and as a basic skill. Fractions represent portions of a whole, and you will need to be able to manipulate fractions arithmetically. Here we will discuss arithmetic operations with fractions having the same denominators. It is very important that you keep in mind that the methods in this lesson apply only to fractions with like denominators.


Adding Fractions with Like Denominators

There are algorithms (procedures) for adding and subtracting fractions. In this lesson we will only be using the algorithm to add or subtract two fractions that have the same denominator. First let’s visually interpret the sum of two fractions.


Example 1

What is the sum of and ?


The fraction can be thought of as 1 of 5 blocks that are the same size, which together make up a whole.

Then the fraction can be thought of as 2 of 5 blocks just like those above.

Combined together, we have 3 blocks, which represents of the whole.

This shows that .


In order to add fractions with like denominators one may simply add the numerators of the two fractions and write the sum over the shared (common) denominator.


Key Idea

When adding fractions with the same denominator, add the numerators and retain the denominator.


Example 2

Find .



Why Does the Algorithm Work?

Suppose we want to add . Think of it as if we had a block that is cut into c equal pieces. So each piece is of the whole. We are adding a of these pieces to b of the pieces, for a total of a + b pieces, each of which is of the whole. So we have parts of the whole. In symbols,


Subtract Fractions with Like Denominators

The algorithm for subtracting fractions with like denominators is similar to the algorithm for the addition of two fractions with like denominators.


Key Idea

When subtracting fractions with the same denominator, subtract the numerators and retain the denominator.

Example 3

Find .


The algorithms given in this lesson only work when adding and subtracting fractions whose denominators are the same.

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