HFCC Math Lab Arithmetic - 4. Addition, Subtraction, Multiplication and Division of Mixed Numbers

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1 HFCC Math Lab Arithmetic - Addition, Subtraction, Multiplication and Division of Mixed Numbers Part I: Addition and Subtraction of Mixed Numbers There are two ways of adding and subtracting mixed numbers. One way is to express each mixed number as an improper fraction and then add or subtract the fractions. Ex : Add If possible, express your answer as a mixed number. 5 First, we write each mixed number as an improper fraction So, 5 The LCD of the fractions is Changing each fraction to an equivalent fraction with 5 as the denominator, we have Therefore, Now, express 6 5 as a mixed number Ex : Subtract 5 If possible, express your answer as a mixed number. First we write each mixed number as an improper fraction So, The LCD of the fractions is. Revised 0/0

2 Changing 9 to an equivalent fraction with as the denominator, we have (The denominator is already.) Therefore, Reduce to the lowest terms and then change to a mixed number Ex : Subtract If possible, express your answer as a mixed number. First we write the whole number and the mixed number as improper fractions. 5 So, 5 The LCD of the fractions is. Changing to an equivalent fraction with as the denominator, we have (The denominator is already.) Therefore, Note: It is a common mistake to say that 5. Be careful! The correct answer is 6. Revised 0/0

3 Ex : Perform the indicated operations. 5 (Remember the Order of Operations: Add and subtract from left to right.) 0 5 First we write the whole number and the mixed numbers as improper fractions So, The LCD of the fractions is Changing each fraction to an equivalent fraction with 0 as the denominator, we have Therefore, Reduce 5 0 to lowest terms and then change to a mixed number Note: The above method of adding and subtracting mixed numbers becomes cumbersome and time consuming when the whole number parts of the mixed numbers are large numbers. For such problems, it is strongly recommended that you use the following method. Revised 0/0

4 Another way to add mixed numbers Step : Step : Add the fractional parts of the mixed numbers separately. Reduce the fraction obtained in step to lowest terms. If it is an improper fraction, change it to a mixed number. Note: If the fraction obtained in step is an improper fraction, you may first change it to a mixed number and then reduce the fractional part of the mixed number to lowest terms. Step : Add the whole number parts of the given mixed numbers. Step : Add the sum from step to the fraction or the mixed number obtained in step. Write your answer as a mixed number. Ex 5: Add and simplify Step : Add the fractions 5 and The LCD of the fractions is Step : 8 is already in lowest terms and cannot be expressed as a mixed number. Step : Add the whole numbers. 5 5 Step : Add 5 to (Remember, whenever we add a mixed number and 8 8 a proper fraction, we omit the + and write the sum as a mixed number.) Revised 0/0

5 Ex 6: Add and simplify. 9 Step : Add the fractions and. The LCD of the fractions is. 0 Step : Reduce 0 to lowest terms and then change to a mixed number Step : Add the whole numbers Step : Add 66 to. 66 6, so Ex : Add and simplify: First, we reduce the fractional part of each mixed number to lowest terms Therefore, Steps & : Add the fractional parts and reduce. Step : Add the whole numbers Step : Add the results of Step and Step Revised 0/0 5

6 Another way to subtract mixed numbers Step : Write the fractional part of each mixed number as an equivalent fraction, using the LCD as the new denominator. Step : Look at the two numerators obtained in Step. If the first numerator is greater than the second, then (A) (B) (C) Subtract the fractional parts separately. Reduce to lowest terms. Subtract the whole number parts of the given mixed numbers. The final answer is the mixed number obtained by adding the whole number in (B) and the fraction in (A). If the first numerator in Step is smaller than the second, we omit Step and go to Step. Step : Borrow from the whole number part of the first mixed number. Decrease the whole number part by. Add this to the fractional part of the first mixed number and write it as an improper fraction. Then (A) (B) (C) Subtract the fractional parts separately. Reduce to lowest terms. Subtract the whole number parts. The final answer is the mixed number obtained by adding the whole number in (B) and the fraction in (A). Ex 8: Subtract and simplify Step : The LCD of the fractional parts is Since the first numerator, 5, is greater than the second numerator,, we complete Step. Step : (A) Subtract the fractional parts and reduce to lowest terms: Revised 0/0 6

7 (B) Subtract the whole number parts: 0 (C) Ex 9: Subtract and simplify. Add the results obtained in (B) and (A): Step : The LCD of the fractional parts is Step : Since the first numerator 9, is smaller than the second numerator,, and we cannot subtract from 9, we go to Step. Step : Borrow from and decrease by. So, (Write as ) (Add 9 ) Thus, 8 Therefore, (A) Subtract the fractional parts: 9 (B) Subtract the whole number parts: 6 6 (C) Add the results obtained in (B) and (A): Revised 0/0

8 Ex 0: Subtract and simplify In this problem, we are subtracting a whole number from a mixed number. Since the 0 whole number,, does not have a fractional part, we can say:. 9 Therefore, A) Subtract the fractional parts: (B) Subtract the whole number parts: 56 (C) Add the results obtained in (B) and (A): Concept: To subtract a whole number from a mixed number, we subtract the whole number parts. Then, add this difference to the fractional part of the first mixed number to obtain the final answer. Ex : Subtract and simplify In this problem, we are subtracting a mixed number from a whole number. Since the 0 whole number, 56, does not have a fractional part, we can say: 56 = 56. We borrow 9 from 56 and decrease 56 by. Write as an improper fraction with its denominator = 9, 5 which is the same as the denominator of the fractional part of. 9 So, Therefore, (A) Subtract the fractional parts: (B) Subtract the whole number parts: 55 Revised 0/0 8

9 (C) Add the results obtained in (B) and (A) + 9 = Concept: To subtract a mixed number from a whole number, we write the whole number as a mixed number by borrowing from the whole number, decreasing the whole number by and writing as an improper fraction with denominator the same as the denominator of the fractional part in the given mixed number. Then (A) (B) (C) Subtract the fractional parts. Subtract the whole number parts. Add the results obtained in (B) and (A). Note: It is very common for students to get mixed up between Ex 0 and Ex. Be very very careful! Notice that but, The correct answer for 5 56 is. 9 9 Exercises: Perform the indicated operations by first changing mixed numbers to improper fractions. Simplify your answers and whenever possible, express your answers as mixed numbers Revised 0/0 9

10 Add and simplify without changing mixed numbers to improper fractions. Whenever possible, express your answer as a mixed number Subtract and simplify without changing mixed numbers to improper fractions. Whenever possible, express your answer as a mixed number Solutions to odd numbered problems and answers to even numbered problems. 9 5 The LCD is Revised 0/0 0

11 . 5 Reduce The LCD is The LCD is The LCD is The LCD is Revised 0/0

12 The LCD is Since 5 96, we have The LCD is Reducing fractions: and 5 6 0, we have The LCD is Revised 0/0

13 Reducing fractions: 9 and 8, we have The LCD is Since 65 =, we have Revised 0/0

14 Part II: Multiplication and Division of Mixed Numbers To multiply or divide mixed numbers, we use the following two steps. Step : Step : Express each mixed number as an improper fraction. Multiply or divide, using the procedures for multiplying and dividing fractions. Ex : Multiply and simplify. If possible, express your answer as a mixed number. 5 First, we write each of the mixed numbers as an improper fraction Ex : Multiply and simplify. If possible, express your answer as a mixed number. 8 First, we write the whole number and the mixed number as improper fractions (Divide 8 and 6 by ) Ex : Multiply and simplify. If possible, express your answers as a mixed number. 5 5 First, we write each mixed number and the whole number as an 5 improper fraction = (Divide 5 and 5 by 5. Divide 8 and by 6) Revised 0/0

15 Ex : Divide and simplify. If possible, express your answer as a mixed number. 5 8 First, we write each mixed number as an improper fraction. 5 Changing division to multiplication by the reciprocal, we have Ex 5: Divide and simplify. If possible, express your answer as a mixed number. 5 5 First, we write the whole number and the mixed number as improper 5 5 fractions. 5 Changing division to multiplication by the reciprocal, we have = (Divide 5 and by ) Ex 6: Divide and simplify. If possible, express your answer as a mixed number. 6 9 First, we write the whole number and the mixed number as improper fractions Changing division to multiplication by the reciprocal, we have (Divide 6 and 9 by 9) Revised 0/0 5

16 Ex : Divide and simplify. If possible, express your answer as a mixed number. 5 5 Since a fraction means to divide, rewrite as a division problem Change to improper fractions Rewrite as a multiplication problem (Divide 6 and 6 by. Divide 5 and 5 by 5) 9 9 Exercises: Multiply the following. Simplify your answers and whenever possible, express your answers as a mixed numbers Revised 0/0 6

17 Divide the following. Simplify your answers and whenever possible, express your answers as a mixed numbers Solutions to odd numbered problems and answers to even numbered problems Revised 0/0

18 Revised 0/0 8

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