About Fractions
TABLE OF CONTENTS About Fractions... 1 What is a FRACTION?... 1 Introduction... 1 Introduction... 1 Forms of Fractions... 1 Different Forms of Fractions... 1 Proper Fractions... 2 Improper Fractions... 2 Mixed Fractions... 2 Converting Fractions... 2 Converting Fractions... 2 Converting an improper fraction to a mixed fraction... 2 Converting a mixed fraction to an improper fraction... 3 Converting a fraction to a percent... 3 Converting a percent to a fraction... 3 Reciprocals... 3 Simplifying Fractions... 3 Simplifying Fractions... 3 Equivalent Fractions... 4 Comparing Fractions... 4 Comparing Fractions... 4 Comparing fractions with the same denominator... 4 Comparing fractions with different denominators... 5 Comparing fractions with decimals... 5 Operations Involving Fractions... 5 Operations Involving Fractions... 5 Adding or Subtracting Fractions... 5 Multiply Fractions... 6 Dividing Fractions... 6 Glossary... 7 References... 8
About Fractions What is a FRACTION? A rational number. A form to express a portion of a whole number. Introduction Introduction In mathematics, a fraction formally called a rational number expresses a portion of the whole number. It is also used to illustrate a ratio between two numbers, as well as an expression that represents the quotient of two numbers. A fraction is expressed as, where x can be any integer, and y can be any integer except for zero. In cases where y equals 0, the fraction is undefined. Consider the fraction below: This fraction above can be read as one sixth, or one over six. Different Forms of Fractions Forms of Fractions A fraction is a generic terminology, and it can be expressed in different forms. 1
Proper Fractions A proper fraction is where the numerator is less than the denominator (i.e., p/q where p < q). Improper Fractions An improper fraction is where the numerator is greater than or equal to the denominator (i.e., p/q where p >= q). Mixed Fractions A mixed fraction consists of a whole number associated with a proper fraction. Converting Fractions Converting Fractions Some forms of fractions can be converted into different forms. Knowing the proper techniques will be critical for performing calculations that involve fractions. Converting an improper fraction to a mixed fraction Divide the numerator by the denominator: a. The resulting quotient becomes the whole number of the new fraction. b. The remainder of the quotient becomes the numerator of the new fraction. c. The denominator remains the same as before. 2
Converting a mixed fraction to an improper fraction 1) Multiply the whole number by the denominator, and add the numerator of the fraction. 2) The answer obtained in step 1 becomes the numerator of the new fraction; the denominator of the new fraction remains the same as before. Note: the conversion to and from mixed numbers is only applicable to improper fractions (and not proper fractions). Can you figure out why? Converting a fraction to a percent A percent is a special type of fraction where the denominator is 100: 1) Convert the fraction such that the denominator is equal to 100. 2) The numerator of the new fraction becomes the percent. Converting a percent to a fraction 1) The percent value is the numerator of the fraction. 2) The denominator of the fraction is 100. Reciprocals Two numbers are reciprocals of one another when their product is equal to 1. Understanding the concept of reciprocals is essential for performing calculations involving the division of two fractions. Simplifying Fractions Simplifying Fractions A fraction can be simplified when the numerator and the denominator are composite numbers. It is important to recognize fractions that can be simplified to their lowest terms. 3
Note: you should always simplify fractions to its lowest terms! To simplify a fraction: 1) Determine a common factor between the numerator and the denominator. A common factor is a number that is divisible by both numbers. (i.e., 4 is a common factor of 8 and 12). 2) Divide the numerator and the denominator by the common factor. 3) Repeat step 1 and 2 until there are no more common factors. 4) A fraction is fully simplified when no more common factor exist between the numerator and the denominator. Equivalent Fractions Equivalent fractions are two or more fractions are equivalent if and only if the fractions can be simplified to the same fraction. They express the same amount with one another. Note: equivalent fractions are also multiples of one another. Comparing Fractions Comparing Fractions Knowing how to compare fractions properly is a very important skill required for many disciplines. You will often be given two fractions, and will be asked to determine which of the two fractions is larger or smaller. A rule of thumb for comparing any form of fraction is to convert the given fractions so that the denominators are the same. Comparing fractions with the same denominator Given two fractions with the same denominator, the fraction with the larger numerator is greater than the fraction with the smaller numerator. 4
Comparing fractions with different denominators Fractions with different denominators cannot be compared directly. In these scenarios, the given fractions must be converted such that the denominators are the same. With the same denominator, you can then compare the fractions using the same method described previously. Comparing fractions with decimals A fraction can be regarded as the quotient of two numbers (the numerator and the denominator). By performing a straight-forward division between the numerator and the denominator, the fraction is converted into a decimal. Decimals can then be easily compared. Operations Involving Fractions Operations Involving Fractions Adding or Subtracting Fractions To add or subtract fractions, the denominators of the fractions must be the same: 1) Find the least common denominator. 2) Using the least common denominator, write its equivalent fractions. 3) Add or subtract the numerators. 4) The least common denominator is the denominator of the resulting fraction. 5) If the operands are mixed fractions, add or subtract the whole number accordingly. 5
Multiply Fractions To multiply fractions, the rules are more straightforward: 1) If possible, simplify the fractions to its lowest terms. 2) Multiply the two numerators, yielding the numerator of the new fraction. 3) Multiply the two denominators, yielding the denominator of the new fraction. Dividing Fractions The most important step in dividing fractions is using the reciprocal of the divisor: 1) Convert the divisor into its reciprocal form. 2) Change the division sign to a multiplication sign. 3) Apply the rules of multiplying fractions as described previously. 6
Glossary Composite Number: Denominator: Equivalent Fractions: Fraction: Improper Fraction: Integer: Mixed Fraction: Numerator: Percent: Proper Fraction: Reciprocals: a number that is divisible by 1, itself, and another integer. in a fraction A/B, the variable B is the denominator. two or more fractions are equivalent if and only if the fractions can be simplified to the same fraction. expresses a portion of the whole number. a fraction where the numerator is greater than or equal to the denominator (i.e., p/q where p >= q). a set of positive or negative whole numbers, such as -2, -1, 0, 1, 2. a fraction that consists of a whole number associated with a proper fraction. in a fraction A/B, the variable A is the numerator. a special type of fraction where the denominator is 100. a fraction where the numerator is less than the denominator (i.e., p/q where p < q). two numbers are reciprocals of one another when their product is equal to one. 7
References http://www.mathleague.com/help/fractions/fractions.htm#whatisafraction http://www.aaamath.com/fra.html#topic1 8