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Addition of Algebraic Fractions
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Properties of Numbers and Definitions
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Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
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Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
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Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
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Adding and Subtracting Fractions
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Factoring Polynomials by Finding the Greatest Common Factor
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Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
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Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
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Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
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Multiplication by 429
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Introduction to Fractions
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Reducing Numerical Fractions to Simplest Form

To reduce a numerical fraction to simplest form means to rewrite it as an equivalent fraction which has the smallest possible denominator. People refer to this operation as reducing a fraction to lowest terms.

Recall that starting with a fraction, we can get an equivalent fraction by either:

(i) multiplying the numerator and denominator by the same nonzero value

or

(ii) dividing the numerator and denominator by the same nonzero value.

Obviously if the goal of simplification is to find an equivalent fraction with a smaller denominator, we will have to use the second principle: dividing the numerator and denominator by the same nonzero value.

The strategy for finding the values to divide into the numerator and denominator is quite systematic:

Step 1: Factor both the numerator and denominator into a product of prime factors using the method described in the previous note in this series.

Step 2: If the numerator and denominator both have a prime factor which is the same, then divide the current numerator and denominator by that value. The result will be the numerator and denominator of an equivalent fraction, but with a smaller denominator.

Repeat step 2 as often as possible. When the numerator and denominator have no further prime factors in common, they form the desired equivalent fraction which has the smallest denominator. We have then obtained the simplest form of the original fraction.

Example:

Reduce the fraction to simplest form.

solution:

For this example, we’ll go through the process step-by-step in some detail. Then we’ll illustrate the familiar “shortcuts” in a couple of examples.

The prime factorizations of the numerator and denominator here are easily obtained:

42 = 2 × 21 = 2 × 3 × 7

70 = 2 × 35 = 2 × 5 × 7

So, the numerator and denominator of the fraction we are given both contain the prime factor 2.

Thus

The new fraction, 21 / 35 , is simpler than the old fraction, 42 / 70 , because its denominator, 35, is smaller than the original denominator of 70. Still, 21 / 35 is equivalent to 42 / 70 because it was obtained by dividing both the numerator and denominator of 42 / 70 by the same value, 2.

But 21 / 35 is still not in simplest form because the factorization of its numerator and denominator (shown in brackets above) indicates that they both still contain a common prime factor of 7. So

Thus, we have

These two fractions, the original 42 / 70and the final 3 / 5 , are equivalent because we got 3 / 5 by dividing the numerator and denominator of 42 / 70 by the same values (2 and 7 in turn, or effectively, 14, if you think of doing it in one step). Furthermore, the numerator and denominator of 3 / 5clearly do not share any further common prime factors, and so this simplification process cannot be carried further. Therefore,

3 / 5 is the simplest form of 42 / 70 .

 

Example:

Reduce the fraction to simplest form.

solution:

Yes! This is the same problem as the first one. What we want to do here is show the “shortcut” form of the strategy for simplifying fractions. We begin as before by rewriting both the numerator and the denominator as a product of prime factors:

Now, if you study the steps of the previous example, you will see that dividing the numerator and denominator by 2 results in those two factors disappearing from each. We indicate this by drawing “slashes” through them:

Dividing the numerator and denominator in the resulting factored equation by 7 again just results in those two factors of 7 disappearing from each. So again,

Now there are no common factors left in the numerator and denominator, so the process ends, and we conclude that 3 / 5 is the simplest form of 42 / 70 .

In practice, this whole process is typically done in a single step, crossing out pairs of factors without rewriting intermediate forms:

 

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Tuesday 17th of October