Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Comparing Fractions

Some situations require us to compare fractions, that is, to rank them in order of size.

For instance, suppose that of one airline’s flights arrive on time, in contrast to of another airline’s flights. To decide which airline has a better record for on-time arrivals, we need to compare the fractions.

Or to take another example, suppose that the drinking water in your home, according to a lab report, has 2 parts per million (ppm) of lead. Is the water safe to drink? If the federal limit on lead in drinking water is 15 parts per billion (ppb), again you need to compare fractions.

One way to handle such problems is to draw diagrams corresponding to the fractions in question. The larger fraction corresponds to the larger shaded region.

For instance, the diagrams show that is greater than . The symbol > stands for “greater than.”

Both and have the same denominator, so we can rank them simply by comparing their numerators.

Note that the symbols < and > always point to the smaller number.

For like fractions, the fraction with the larger numerator is the larger fraction.


Like fractions are fractions with the same denominator.

Unlike fractions are fractions with different denominators.

To Compare Fractions

  • compare the numerators of like fractions, and
  • express unlike fractions as equivalent fractions having the same denominator and then compare their numerators.


Compare and .


These fractions are unlike because they have different denominators. Therefore we need to express them as equivalent fractions having the same denominator. But what should that denominator be?

One common denominator that we can use is the product of the denominators: 15 · 9 = 135

, 135 = 15 · 9 so the new numerator is 7 · 9, or 63.

, 135 = 9 · 15, so the new numerator is 4 · 15, or 60.

Next, we compare the numerators of the like fractions that we just found.

Because . Therefore .

Another common denominator that we can use is the least common multiple of the denominators.

The LCM is . We then compute the equivalent fractions.

Because , we know that .

Note that in Example 1 we computed the LCM of the two denominators. This type of computation is used frequently in working with fractions.


For any set of fractions, their least common denominator (LCD) is the least common multiple of their denominators.

In Example 2, pay particular attention to how we use the LCD.


Order from smallest to largest: .


Because these fractions are unlike, we need to find equivalent fractions with a common denominator. Let’s use their LCD as that denominator.

The LCD . Check: 4 and 10 are both factors of 40.

We write each fraction with a denominator of 40.

Then we order the fractions from smallest to largest. (The symbol < stands for “less than.”)


About of Earth’s surface is covered by water and is covered by desert. Does water or desert cover more of Earth?


We need to compare with .

Since . Therefore water covers more of Earth than desert does.

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