Section. PRE-ACTIVITY PREPARATION Reducing Fractions You must often use numbers to communicate information to others. When the message includes a fraction whose components are large, it may not be easily understood. In that case, you might use a simplified form of the fraction to most effectively convey the same information. The following example illustrates the practicality of the mathematical skill of reducing fractions to their simplest form. Consider, for an example, the five hundred twenty out of seven hundred eighty third graders in a certain school district who ride the bus to school (20/780). The same fraction in its simplest form is 2/ two thirds of the third graders ride the bus to school. This simplest form more efficiently communicates the same information to prospective school parents. LEARNING OBJECTIVES Reduce fractions to lowest terms by dividing or canceling out common factors. Use the meaning of equivalent fractions for validation. TERMINOLOGY PREVIOUSLY USED NEW TERMS TO LEARN denominator divisibility tests factor numerator prime factorization cancel common factor cross-multiply cross-product does not equal sign equivalent fraction lowest terms reduce simplify 29
20 Chapter Fractions BUILDING MATHEMATICAL LANGUAGE Fractions are equivalent (equal) when they represent the same part of a whole or group, the same division, or the same ratio. 2 For example, is equivalent to 2 4 VISUALIZE One way to check whether two given fractions are equivalent is to apply the following test. Equality Test for Fractions If two fractions are equal, their cross-products will be equal. A cross-product is found by multiplying the numerator of one fraction by the denominator of the other. 20 2 For example, to determine whether the fractions in the Introduction, and, are equivalent fractions, you could calculate the cross-products (cross-multiply): 780 20 2 Does 20 equal 2 780 780 Yes,,0,0 The two fractions are equivalent. 2 4 Now consider and. 9 These two fractions are not equivalent because they do not pass the Test for Equality. That is, 2 9 4 Read, 2 times 9 does not equal 4 times 8 20 Read, 8 does not equal 20. A common factor of two or more numbers is a factor that they share. That is, it divides evenly into each of the numbers. For example, and are the two common factors of 2, 8, and. To reduce a fraction is to rewrite it as an equivalent fraction with a smaller numerator and smaller denominator. To reduce a fraction to its lowest terms or to simplify a fraction is to write the equivalent fraction whose numerator and denominator have no common factors. 2 Example: in reduced form. 4 2
Section. Reducing Fractions 2 In addition to the divisibility tests for the numbers 2,, and which you used to determine the prime factorization of a number, there are several other divisibility tests which you might use to determine common factors: Additional Divisibility Tests If the final two digits of a number form a number divisible by 4, then the original number is divisible by 4. If a number is divisible by both 2 and, then it is divisible by. If the sum of the digits of a number is divisible by 9, then the original number is divisible by 9. If the number ends in 0, then it is divisible by 0; if it ends in 00, it is divisible by 00; in 000, it is divisible by 000, and so on. METHODOLOGY This methodology breaks down the numerator and denominator of a fraction to their prime factorizations, in order to easily see their common factors. It is particularly useful to use when the common factors of the original numerator and denominator are not readily apparent to you. Be sure to note its shortcut options! Reducing a Fraction 0 Example : Reduce to its lowest terms. 42 Example 2: Reduce to its lowest terms. 40 Try It! Steps in the Methodology Example Example 2 Step Prime factor numerator Step 2 Prime factor denominator Determine the prime factorization of the numerator. Shortcut: Quick reduction (see page 2, Model 2) Determine the prime factorization of the denominator. 2 0 2 2 8 9
22 Chapter Fractions Steps in the Methodology Example Example 2 Step Write as prime factorization. Step 4 Cancel. Step Multiply remaining factors. Step Present the answer. Step 7 Validate your answer. Re-write the fraction using the prime factorizations. Cancel each common numerator factor with its matching denominator factor. Why can you do this Multliply the remaining numerator factors to get the new numerator and the remaining denominator factors to get the new denominator. Present your answer. Validate by using the Equality Test for Fractions. Compare the cross-products of the original fraction and the reduced fraction. The cross products must be equal. Also, there should be no common factors between the numerator and denominator of the final answer. 2 2 2 2 2 2 2 0 2 no common factors 0 80 80 Why can you do Step 4 Recall the Special Property of Division that states that any number divided by itself equals. any number that same number That is, 2, 2,, and so on. For any fraction, then, if a factor in the numerator is equal to a factor in the denominator, you can apply this property and replace the two factors with the number (or /), a procedure called canceling. Example: 0 2 2 or or, by the Identity Property of Multiplication, simply You will get the same result if you divide both the numerator and denominator by the same common factor. 0 2 0 2 OR, with canceling notation, 0.
Section. Reducing Fractions 2 MODELS Model Reduce to lowest terms: 84 Step Step 2 Step Step 4 Steps & 2 84 2 42 2 7 7 7 9 9 Step 7 Validate: 2 2 7 7 9 2 2 7 7 9 84 2 9 84 9 2,9,9 2 9 4 2 9 9 Answer: 2 9 2 2 9 no common factors A Model 2 Shortcut: Quick Reduction Shortcut: Before you do Steps and 2, first divide out the factor(s) you readily recognize as being common to both the numerator and denominator. Simplify: 440 870 Shortcut Version (optional) Step Step 2 THINK both divisible by 0 2 440 2 870 440 0 44 2 220 9 870 0 87 2 0 87 Step Step 2 7 7 2 44 87 2 22 7 7 Step 2 2 2 2 2 Step 2 7 7 2 2 2 Step 4 Step 4 2 7 2 2 7 4 Steps & Answer: Steps & Answer: 7 4 7 Step 7 Validate: 440 870 4 440 7 4 870 4 2 2 7 7,480 7,480 7 7 no common factors
24 Chapter Fractions B 48 Reduce to lowest terms: 4 48 8 2 Use the shortcut to divide out the common factors. 4 8 8 8 2 4 Answer Validate: 48 48 4 4 4 4 92 92 4 2 2 no common factors ADDRESSING COMMON ERRORS Issue Incorrect Process Resolution Correct Process Validation Not reducing all the way to lowest terms when simplifying a fraction Simplify: 0 240 0 0 240 0 24 0 24 240 00 00 For this issue, validating by cross-multiplying alone does not catch the error since both fractions are, in fact, equivalent. Always do a prime factorization of your final answer to assure that there are no remaining common factors to cancel. Simplify: 0 240 0 0 240 0 24 24 2 2 2 8 0 240 8 0 8 240 200 200 8 2 2 2 no common factors Mismatching the factors when canceling 0 0 Reduce: Use effective notation. Reduce: 2 2 0 Factors must be 2 canceled in pairs: one numerator factor with only one denominator 0 2 2 2 2 2 factor, with each pair 2 7 2 7 equaling one (). 0 2 7 0 2 0 2 20 20 0 2 2 7 no common factors PREPARATION INVENTORY Before proceeding, you should have an understanding of each of the following: the terminology and notation associated with reducing fractions the meaning of equivalent fractions what it means to reduce (simplify) a fraction to its lowest terms why you can cancel common factors when reducing the validation of the final reduced fraction
Section. ACTIVITY Reducing Fractions PERFORMANCE CRITERIA Reducing a fraction to its lowest terms correct reducing techniques validation of the final answer CRITICAL THINKING QUESTIONS. What is a fully reduced fraction Fractions are in its lowest terms if there are no factors common to both the numerator and denominator. 2. How do you validate that fractions are equivalent Find the cross-products by multiplying the numerator of the fi rst fraction times denominator of the second fraction, then, denominator of the fi rst fraction times numerator of the other fraction, then, comparing to make sure the products are equal.. When reducing to lowest terms, what is the result when all the factors in the numerator cancel out The result is that the numerator (or denominator) becomes. 2
2 Chapter Fractions 4. When reducing to lowest terms, what is the result when all the factors in the denominator cancel out The result will be the product of the factors left in the numerator over. But we know that by the Division Property of One that any number divided by one is equal to the beginning number. For example: 2 2. How can you tell if a fraction is in lowest terms You can tell a fraction is in lowest terms, if you cannot fi nd a common factor that will divide into both the numerator and the denominator. If the numerator is larger than the denominator, you must change to a mixed number.. How can you be sure that your reduced fraction answer is correct Set the original fraction equal to the reduced answer. If the products of the cross multiplication are the same, then the reduced answer is correct. TIPS FOR SUCCESS Know and use Divisibility Tests to quickly cancel common factors. When you cannot readily recognize common factors, use the prime factorization of the numerator and denominator to reduce (by using the Methodology for Reducing a Fraction). Cross multiply to test the equivalency of your reduced fraction. Even when you do quick reduction by common factors, always prime factor your final answer to assure that there are no remaining common factors.
Section. Reducing Fractions 27 DEMONSTRATE YOUR UNDERSTANDING Reduce each of the following to lowest terms. If improper, write as a mixed number with its fraction in lowest terms. Fraction Factorization Reduced Fraction Validation ) 24 9 2) 28 42 ) 4 0 4) 780 820 ) 8 02
28 Chapter Fractions IDENTIFY AND CORRECT THE ERRORS In the second column, identify the error(s) you find in each of the following worked solutions. If the answer appears to be correct, validate it in the second column and label it Correct. If the worked solution is incorrect, solve the problem correctly in the third column and validate your answer in the last column. Worked Solution What is Wrong Here Identify Errors or Validate Correct Process Validation ) Reduce to lowest terms: 2 00 2 00 4 2 4 00 00 00 4 2 2 fully reduced Correct 2) Reduce to lowest terms: Can be reduced 0 further. 20 ) Reduce to lowest terms: Cannot cancel 20 the on the bottom 0 (denominator) twice. ADDITIONAL EXERCISES Reduce to lowest terms and validate your answers. already reduced 42 7 400 8 7 27 27 ) 2) ) 4) ) 08 8 700 2 20 2 20 27