Multiplying Fractions MPM1D: Principles of Mathematics Recap - PDF document

n u m e r a c y n u m e r a c y Multiplying Fractions MPM1D: Principles of Mathematics Recap Evaluate 12 25 × 15 32 . Since the GCF of 12 and 32 is 4, and the GCF of 25 and 15 is 5, reduce these values before multiplying. Working with Fractions Part 2: Adding, Subtracting and Negative Fractions 25 × 15 12 32 = 3 5 × 3 8 = 9 J. Garvin 40 J. Garvin — Working with Fractions Slide 1/16 Slide 2/16 n u m e r a c y n u m e r a c y Lowest Common Multiple Lowest Common Multiple Consider the two values 6 and 8. Example Determine the LCM of 8 and 20. Multiples of 6 are 6, 12, 18, 24, 30, . . . Multiples of 8 are 8, 16, 24, 32, 40, . . . Multiples of 8 are 8, 16, 24, 32, 40, 48, . . . Examining the two lists of multiples, 24 is the first value that Multiples of 20 are 20, 40, 60, 80, . . . appears in both lists. Examining the multiples, the first (and smallest) multiple The smallest value that a multiple of two other values is that is common to both values is 40. Thus, 40 is the LCM of called the lowest common multiple (LCM). 8 and 20. This shouldn’t be confused with the greatest common factor, which is the largest value that divides into two other values. J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 3/16 Slide 4/16 n u m e r a c y n u m e r a c y Adding/Subtracting Fractions Adding/Subtracting Fractions Example When adding or subtracting fractions, we need to ensure that the denominators are the same value. 16 + 5 3 Evaluate 16 . Recall that a fraction represents some part of a whole. Since the denominators are already the same, add the The numerator represents some number of pieces of the numerators. whole, while the denominator represents the total number of pieces made from the whole. 16 + 5 3 16 = 8 16 When a common denominator is used, we can simply count = 1 up the total number of pieces via addition or subtraction. 2 By finding the LCM of two denominators being added or Don’t forget to reduce if possible. subtracted, we may use it as the common denominator. J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 5/16 Slide 6/16

n u m e r a c y n u m e r a c y Adding/Subtracting Fractions Adding/Subtracting Fractions Example Example 10 − 3 7 Evaluate 3 8 + 7 Evaluate 5 . 12 . Since 12 is not a multiple of 8, we need to find a common In this case, the denominators are different, but since 10 is a multiple of 5 (by a factor of 2), we can easily convert 3 denominator before we can add the numerators. 5 to 3 × 2 5 × 2 = 6 10 . The LCM of 8 and 12 is 24, so we can use this value as the denominator. 10 − 3 7 5 = 7 10 − 6 Since 3 × 8 = 24 and 2 × 12 = 24, we must multiply the 10 numerators by these values. = 1 10 3 8 + 7 12 = 9 24 + 14 24 Always check to see if one denominator is a multiple of the = 23 other first. 24 J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 7/16 Slide 8/16 n u m e r a c y n u m e r a c y Adding/Subtracting Fractions Adding/Subtracting Fractions If we did not identify the LCM of 8 and 12 as 24, it would Example have been possible to solve the problem by creating a 24 − 23 9 Evaluate 30 . common denominator equal to the product of the two denominators. The LCM of 24 and 30 is 120. 3 8 + 7 12 = 3 × 12 8 × 12 + 7 × 8 24 − 23 9 30 = 45 120 − 92 12 × 8 120 = 36 96 + 56 = − 47 96 120 = 92 96 Note that, in this case, our answer is negative. = 23 While the fraction may also be written as − 47 120 , it is usually 24 written with the negative sign preceding the entire fraction. Note that using this approach generally involves larger values that must be reduced later. Using the LCM instead may eliminate this need. J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 9/16 Slide 10/16 n u m e r a c y n u m e r a c y Negative Fractions Negative Fractions Example Operations with negative fractions are the same as those with negative integers. − 3 − 12 � � � � Evaluate . 8 5 • multiplying or dividing two fractions with the same sign Since we are multiplying two negatives, our final answer will produces a positive result. be positive. • multiplying or dividing two fractions with different signs produces a negative result. Reduce the 8 and 12 by a factor of 4, the GCF. • adding a negative fraction is the same as subtracting a � � � � � 3 � � 3 � − 3 − 12 positive fraction. = 8 5 2 5 • subtracting a negative fraction is the same as adding a = 9 positive fraction. 10 J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 11/16 Slide 12/16

n u m e r a c y n u m e r a c y Negative Fractions Negative Fractions Example Example Evaluate 5 − 3 Evaluate 2 5 + 18 − 27 � � � � . . 7 ÷ 6 − 10 14 Perform the division first, according to the order of Since we are subtracting a negative, add a positive instead. operations. Use 30, the LCM of 6 and 10, as a common denominator. Since we are dividing a positive by a negative, our result will � � 5 − 3 = 25 30 + 9 be negative. 6 − 10 30 2 5 + 18 � − 27 � = 2 � 18 7 × 14 � = 34 7 ÷ 5 − 14 27 30 � 2 � = 2 1 × 2 = 17 5 − 3 15 = 2 5 − 4 3 J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 13/16 Slide 14/16 n u m e r a c y n u m e r a c y Negative Fractions Questions? Use 15 as a common denominator to add. 2 5 − 4 3 = 6 15 − 20 15 = − 14 15 J. Garvin — Working with Fractions J. Garvin — Working with Fractions Slide 15/16 Slide 16/16