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Exponential Decay
Negative Exponents
Multiplying and Dividing Fractions 4
Evaluating Expressions Involving Fractions
The Cartesian Coordinate System
Adding and Subtracting Fractions with Like Denominators
Solving Absolute Value Inequalities
Multiplying Special Polynomials
FOIL Method
Inequalities
Solving Systems of Equations by Graphing
Graphing Compound Inequalities
Solving Quadratic Equations by Completing the Square
Addition Property of Equality
Square Roots
Adding and Subtracting Fractions
The Distance Formula
Graphing Logarithmic Functions
Fractions
Dividing Mixed Numbers
Evaluating Polynomials
Power of a Product Property of Exponents
Terminology of Algebraic Expressions
Adding and Subtracting Rational Expressions with Identical Denominators
Solving Exponential Equations
Factoring The Difference of 2 Squares
Changing Fractions to Decimals
Solving Linear Equations
Using Patterns to Multiply Two Binomials
Completing the Square
Roots of Complex Numbers
Methods for Solving Quadratic Equations
Conics in Standard Form
Solving Quadratic Equations by Using the Quadratic Formula
Simplifying Fractions 2
Exponential Notation
Exponential Growth
The Cartesian Plane
Graphing Linear Functions
The Slope of a Line
Finding Cube Roots of Large Numbers
Rotating Axes
Common Mistakes With Percents
Solving an Equation That Contains a Square Root
Rational Equations
Properties of Common Logs
Composition of Functions
Using Percent Equations
Solving Inequalities
Properties of Exponents
Graphing Quadratic Functions
Factoring a Polynomial by Finding the GCF
The Rectangular Coordinate System
Adding and Subtracting Fractions
Multiplying and Dividing Rational Expressions
Improper Fractions and Mixed Numbers
Properties of Exponents
Complex Solutions of Quadratic Equations
Solving Nonlinear Equations by Factoring
Solving Quadratic Equations by Factoring
Least Common Multiples
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Solving Exponential Equations
Solving Linear Equations
Multiplication Property of Equality
Multiplying Mixed Numbers
Multiplying Fractions
Reducing a Rational Expression to Lowest Terms
Literal Numbers
Factoring Trinomials
Logarithmic Functions
Adding Fractions with Unlike Denominators
Simplifying Square Roots
Adding Fractions
Equations Quadratic in Form
Dividing Rational Expressions
Slopes of Parallel Lines
Simplifying Cube Roots That Contain Variables
Functions and Graphs
Complex Numbers
Multiplying and Dividing Fractions 1
Composition of Functions
Intercepts of a Line
Powers
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Multiplying Two Numbers with the same Tens Digit and whose Ones Digits add up to 10
Factoring Trinomials
Exponents and Polynomials
Decimals and their Equivalent Fractions
Negative Integer Exponents
Adding and Subtracting Mixed Numbers
Solving Quadratic Equations
Theorem of Pythagoras
Equations 1
Subtracting Fractions
Solving Quadratic Equations by Graphing
Evaluating Polynomials
Slope
Angles and Degree Measure
   
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Adding and Subtracting Fractions

It is not difficult to visualize what it means to add two fractions, as in

As explained before, each of the two fractions represents dividing a whole into a certain number of equal-sized pieces, and including a certain number of those pieces. So, we can represent the two fractions above pictorially as

Thus, 3 / 4 represents 3 pieces, each of which is 1 / 4 of the whole, and 2 / 3 represents 2 pieces, each 1 / 3 of the whole. Then, to form the sum

we need to come up with a fraction representing how much of a whole we get when we combine all five shaded pieces in the diagrams above. The difficulty is that we can’t just count up the number of shaded pieces to be 5 and the total number of pieces to be 7, and so declare the answer to be 5 / 7 (from ), since the pieces are not all the same size to begin with. In fact, you can see that this answer would clearly be wrong, because it represents less than a complete whole, and yet it is obvious from the pictures that the combined shaded areas are much more than a complete whole.

There are several errors in the incorrect procedure just described. The most obvious is that the five smaller shaded bits in the diagram are not all the same size, and so simply counting a total of 5 pieces is not a correct reflection of the combined size of the two fractions. However, there is a way around this problem.

First, divide each of the quarters in the diagram for 3 / 4 into three smaller, equal-sized pieces:

In fact, this diagram is now illustrating both 3 / 4 as well as 9 / 12, because now the whole can also be viewed as being partitioned into 12 equal pieces, of which 9 are shaded.

Secondly, divide each of the thirds in the diagram for 2 / 3 into four smaller equal-sized pieces:

Again, this results in the whole being divided into twelve equal pieces of which eight are shaded.

But now, both diagrams have shaded pieces which are the same size! Each is 1 / 12 of the whole. When the total number of these equal-sized shaded pieces is tallied, we get

9 + 8 = 17

shaded pieces, each of size 1 / 12 of the whole. Thus, we apparently get that

You can verify that this is a plausible result by using your calculator to turn the fractions into decimal numbers and checking the addition.

What this really shows is that we can add two fractions together only if they have the same denominator (and the same rule will apply to subtraction):

because for the sum of the numerators (or the difference of the numerators) to be meaningful, they must refer to pieces of the whole all of which are the same size (that is the same thing as saying that the denominators of the fractions are the same).

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