3.3 Division of Fractions and of Mixed Numbers
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1 CCBC Math 0 Division of Fractions and of Mixed Numbers Section.. Division of Fractions and of Mixed Numbers Introduction: What does it mean to divide? The basic division questions asks, How many of b are in a? Where a b gives the answer. Let s give an example that would require dividing fractions. A recipe calls for butter will we need for our recipe? cups of butter. A stick of butter is equal to of a cup. How many sticks of This problem asks us How many cups (sticks of butter) are in cups of butter? This follows the How many of b are in a? question. Therefore, we divide a by b. ab We will learn how to calculate this problem later in the chapter but let s get an answer to this problem visually. How many halves are in? The recipe requires sticks of butter. Therefore, There are several ways to divide two fractions. We will concentrate on the most popular way to divide fractions. Reciprocal of a Fraction RECIPROCAL The reciprocal of a is a. The reciprocal of a b is b a. In the examples to follow, let s find the reciprocal of a whole number, the reciprocal of a fraction, and the reciprocal of a mixed number. 77
2 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. Example : What is the reciprocal of? = Begin by writing as the fraction. Reciprocal: Switch the numerator and denominator to write the reciprocal. Practice : What is the reciprocal of? Answer: Example : What is the reciprocal of? Switch the numerator and denominator. The reciprocal of is. Practice : What is the reciprocal of? Answer: Example : What is the reciprocal of? ( ) 0 Begin by converting fraction. to an improper Reciprocal: Switch the numerator and denominator to write the reciprocal. Practice : What is the reciprocal of Answer: 7 7
3 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. Dividing Fractions Division is the inverse operation of multiplication. As we learned in Chapter, addition and subtraction are inverse operations (of each other.) The inverse relationship between addition and subtraction manifests itself in the opposite sign of a number. The inverse relationship between multiplication and division manifests itself in the reciprocal of a (nonzero) number. To illustrate the concept of reciprocal, let s consider a division problem, say We know from our study of division that divided by is the quotient. We know from our study of fractions that is another way of writing that division problem. If we look closely at this problem, we see that we can rewrite as the multiplication problem because when we multiply those fractions together, we are multiplying which equals. So, divided by is the same as multiplied by the fraction. This application of the inverse property is saying that dividing by (dividing by ) is equivalent to multiplying by its reciprocal,. Thus, can also be written as the multiplication problem or. We apply the principle of reciprocals to fractions when we divide. To divide fractions multiply the first fraction by the reciprocal of the second fraction. As with multiplication, all mixed numbers need to be rewritten as improper fractions. 7
4 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. DIVIDING FRACTIONS a c a d ad b d b c bc When b, c, and d do not equal 0. Example : Divide: Multiply the first fraction by the reciprocal of the second fraction. 7 7 Multiply the fractions. Simplify the answer. Of course, you could have simplified before multiplying: you would divide both the and the by and then multiply to get your final answer. Note that the problem must be in multiplied form before you can simplify. Multiply the first fraction by the reciprocal of the second fraction. Multiply the fractions. Here we reduce before multiplying. Make sure your answer is completely simplified. 0
5 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. Practice : Divide: 0 Answer: 7 Example : Divide: 6 Multiply the first fraction by the reciprocal of the 6 6 second fraction. In multiplied form, simplify before multiplying Simplify again. Multiply the numerators together and the denominators together. Practice : Divide: Answer: 6 0
6 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. Example 6: Divide: 6 6 (6) (6) 6 Convert each mixed number to an improper fraction. 6 6 Don t try to do too many steps all at once. Rewrite the original division problem with the improper fractions. 0 7 NEVER attempt to reduce fractions until the 6 problem has been turned into multiplication. 0 6 Multiply the first fraction by the reciprocal of the 7 second fraction Simplify before multiplying. 0 Check for other places to reduce. There are none 7 here Multiply the usual way and make sure your answer is completely simplified. Practice 6: Divide: Answer: Division with Zero Earlier you learned that any number multiplied by 0 produces the result 0. You also learned that division by 0 is undefined. RULES FOR DIVIDING WHEN 0 IS IN THE PROBLEM If 0 is divided by any number (except 0), the answer is 0. 0n 0 If any number is divided by 0, the answer is undefined. In other words, there is no answer. n0 undefined Note: In this section n will be a fraction.
7 CCBC Math 0 Division of Fractions and of Mixed Numbers Section. Example 7: Divide: 0 0 Multiply the first fraction by the reciprocal of the second fraction. = 0 Whenever you multiply by 0, the result is Thus, whenever you are dividing 0 by a number (but not 0), the result is 0. Dividing by 0 (where 0 is the divisor) is undefined. It cannot be done. Therefore, if the divisor is 0, the answer is automatically undefined because you cannot divide by 0. Practice 7: Divide: 0 Answer: Example : Divide: 0 0 Express the 0 in fraction form. Multiply the first fraction by the reciprocal of the second 0 fraction. 0 0 Division by zero is not possible. The answer is undefined. Practice : Divide: 0 Answer: Undefined Watch All:
8 CCBC Math 0 Division of Fractions and of Mixed Numbers Section.. Division of Fractions and of Mixed Numbers Exercises. Without determining the answer, set up the following division problem as an equivalent multiplication problem. Then write this multiplication problem in reduced form, but do not multiply. a. b Divide: 0. Divide: 6 7. Divide:. Divide: 0. Divide:. Divide: Divide: 7 7. Divide: 0 6. Divide:. Divide:. Divide: 6. Divide:. Divide: 6. Divide: 7 6. Divide: 7. Divide:. In the problem 0, which fraction is the quotient?. In the problem 0, which fraction is the divisor? 0. In the problem 0, which fraction is the dividend? 6 0 0
9 CCBC Math 0 Division of Fractions and of Mixed Numbers Section.. Division of Fractions and of Mixed Numbers Exercises Answers. a. 7 7 b = undefined
10 CCBC Math 0 Section. to.. The numerator of 0 denominator is. Mid-Chapter Review is and the. Write the problem as a fraction.. Simplify Simplify.. Convert 7 to an improper fraction Write form. as a mixed number in simplest 7. Write the problem in reduced 0 form, but do not multiply.. Multiply.. Multiply. 0. Multiply 6.. Multiply. 6. Multiply.. Multiply 0.. In the problem 6, a. What fraction is the quotient? b. What fraction is the divisor?. Set up the division problem as 0 an equivalent multiplication problem. Then write the multiplication problem in reduced form, but do not multiply. CHAPTER 6. Divide Divide Divide. 0. Divide Divide ( ).. Divide 6.. Divide 0.. Evaluate.. Evaluate.. Evaluate. 6. Find the mean of, 7,, and. 7. Find the median of,,,,.. Find the mode of,,,, Determine the area of the rectangle. 0. Determine the area of the triangle. m ft m m ft 6
11 CCBC Math 0 Section. to.. 0, a). 6 6 M i d - C h a p te r R e v i e w Answers 6 6 b) Undefined ft ft m m 7
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