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Finding the Greatest Common Factor
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Adding Fractions
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The FOIL Method
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Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
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Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
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Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
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Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
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Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Adding Fractions

How much is ? (one half + one quarter).

Either halves or quarters – which is more convenient? Should we state how many quarters equal one half or vice versa? The former, of course: 1 half is 2 quarters . So, the becomes , and the above task becomes , meaning 2 quarters + 1 quarter = 3 quarters, i.e. .

In the above sum, 1 half + 1 quarter, the different types were associated with the different denominators (dividers), so the problem was solved by a transformation that led to identical denominators. Let us now do this for the more general case:

(where we have more than one of each type, namely: 2 thirds and 5 twelfths).

Again, we want the same denominators under both, either 3 or 12. Which shall we choose? We must remember that the values of the fractions must remain unchanged, so, if we change the denominator, we must also change the numerator (the divided). If we choose the 3 as the common denominator, we must change the 12 in the second term into a 3, that is, make the denominator 4 times smaller. To keep the value of the fraction unchanged, we must then also make the 5 on top (the numerator) 4 times smaller, resulting in 5/4 as the numerator. This is inconvenient because it creates another fraction above the dividing line.

Instead, we can choose the second denominator, the 12, as the common one. We leave the as it is and change the first denominator, the 3, to 12. This we do by increasing the 3, by multiplying it by 4. Then, to protect the value of this fraction, we must also make the numerator (the 2) 4 times bigger, this time giving a manageable whole number, 8. Note that this resulted from choosing the larger of the two denominators as the common one. And so, the above now becomes , 8 twelfths + 5 twelfths = 13 twelfths, i.e. .

But what about ?

To get the first (smaller) denominator (the 3) to be the same as the second denominator 4, the 3 has to be multiplied by 1.33…, and the same must then be done to the 2 above it, which results in a messy 2.66…! So we now need a common denominator which is neither 3 nor 4. In principle, we could choose anything to put equally at the bottom of both fractions. The only problem is that we need to adjust the numerators (to keep the values of the given fractions unchanged) and as we found, we may only do this by multiplying the numerators by whole numbers. Form this follows that the denominators, too, can be changed only by multiplying by whole numbers.

The task becomes: By what whole number do we multiply the 3 (the denominator), and by what (different) whole number do we multiply the 4 (the other denominator) so that in both cases we get the same result (namely, the same common, denominator)? The nice trick for this is to multiply the first denominator (3) by the second denominator (4) and the second denominator (4) by the first (3) ! (always, of course, multiplying the numerator by the same as the denominator).

So this is what was done:


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