1. MULTIPLYING AND DIVIDING FRACTIONS The Number System 1 © 2013 Meredith S. Moody

2. Objective: You will be able to… Convert fractions to their reciprocals and back Divide a fraction by another fraction Solve problems by dividing fractions by fractions 2 © 2013 Meredith S. Moody

3. The Fraction When we divide a whole into equal parts, we represent those parts as fractions 3 © 2013 Meredith S. Moody

4. Part of a Part Why would we need to divide a fraction by a fraction? We may not have a whole number to divide into pieces How much is half of a half? What about a third of a half? What about a half of a fourth? 4 © 2013 Meredith S. Moody

5. Example 1 Let’s say you have a cake for a party The day of the party, you serve half of the cake to your guests, but have half left over The next day, you have some friends over and you want to serve the other half of the cake, but you want everyone to have the same number of pieces Here, you would have to divide a fraction ( ½ of the cake) by a whole number (the number of friends you have over) 5 © 2013 Meredith S. Moody

6. Example 1, visual Drawing pictures is helpful when dividing a fraction by a fraction You start with a whole, and then divide it in half 1 whole cake½ of the cake 6 © 2013 Meredith S. Moody

7. Example 1, visual, continued When half the cake is gone, you only have the leftover half to work with If you have 1 friend over, you have to split the half into 2 pieces (1 for you, 1 for the friend) ½ of a ½ 7 © 2013 Meredith S. Moody

8. Example 1, visual, continued You can look at how the smallest piece fits into the whole: ½ of a ½ is equal to ¼ of the whole ½ of a ½ 8 © 2013 Meredith S. Moody

9. Example 1, Algebraic translation What operation represents ‘½’ something? ½ of a 1 whole is the same thing as dividing 1 whole by 2 This is because the fraction bar represents the operation of division ( ½ = 1 ÷ 2) So if we want to ‘half’ something, we divide by 2 So, ‘half’ing a half is = ½ ÷ 2 ½ ÷ 2 = ¼ 9 © 2013 Meredith S. Moody

10. Reciprocals Reciprocal fraction: The result of interchanging the numerator and denominator in a fraction A whole number can be represented as a fraction over ‘1’ (because a number divided by ‘1’ is itself) Therefore, the whole number 5 = 5 / 1 and the reciprocal = 1 / 5 10 © 2013 Meredith S. Moody

11. Example 2, visual Working with halves is easier than other fractions What if I wanted to know what a third of a half is? 1 whole cake ½ of the cake 1/3 of the ½ 1/6 of the whole 11 © 2013 Meredith S. Moody

12. Example 2, Algebraic translation Here, I took ½ and divided by 3 to find 1/3 of the ½ ½ ÷ 3 = 1/6 ½ ÷ 3/1 = 1/6 Do you see a pattern yet? 12 © 2013 Meredith S. Moody

13. Example 3 What if I wanted to find something more complicated? What if the first day of my party, I served 1/3 of the cake, so I had 2/3 left over The next day, I had 8 friends over, and I wanted each person to have a slice, but three said ‘no thanks’ – so I only served 5/8 of the 2/3 What is the leftover 3/8 of the 2/3? So I want to know: what is 3/8 of 2/3? 13 © 2013 Meredith S. Moody

14. Example 3, visual Here is the visual representation of 3/8 of 2/3 1 whole cake the cake2/3 of is left whole of cake the 6/24 3/8 of 2/3 of the cake 14 © 2013 Meredith S. Moody

15. Example 3, Algebraic translation I divided 2/3 into 8 pieces  2/3 ÷ 8 But I wasn’t interested in just 1 of those pieces, I wanted to figure out what 3 of them were I didn’t want 2/3 ÷ 8/1 I wanted 2/3 ÷ 8/3 So 2/3 ÷ 8/3 = 6/24 Do you see a pattern yet? 15 © 2013 Meredith S. Moody

16. Seeing the Pattern Let’s review: ½ ÷ 2 = ½ ÷ 2/1 = ¼ ½ ÷ 3 = ½ ÷ 3/1 = 1/6 2/3 ÷ 8/3 = 6/24 What is the pattern? That’s right! The answer is the dividend (1 st fraction) multiplied by the reciprocal of the divisor (2 nd fraction)! 16 © 2013 Meredith S. Moody

17. Using Reciprocals We can use reciprocals and our understanding of multiplication to divide a fraction by a fraction ½ ÷ 2 = ½ ÷ 2/1 = ¼ When I divide a ½ by two, I’m finding ½ of the ½ What operation does ‘of’ represent? That’s right! Multiplication! ½ ÷ 2 = ½ ÷ 2/1 = ½ x ½ = ¼ 17 © 2013 Meredith S. Moody

18. Check the Pattern Does this pattern always work? Let’s check our second cake scenario, where we wanted to find 1/3 of ½ We took ½ the cake and cut it into 3rds and looked at one of those pieces: ½ ÷ 3 = ½ ÷ 3/1 = 1/6 1/3 of ½ = 1/3 x ½ = 1/6 Does the pattern fit? Yes! 18 © 2013 Meredith S. Moody

19. Check the Pattern Let’s check our last cake scenario, where we wanted to find 3/8 of 2/3 We took 2/3 of the cake and cut it into 8ths and looked at 3 of those pieces 2/3 ÷ 8/3 = 6/24 3/8 of 2/3 = 3/8 x 2/3 = 6/24 Does the pattern fit? Yes! 19 © 2013 Meredith S. Moody

20. Algebraic Rule Based on the pattern we found, we can write a rule for dividing a fraction by another fraction: a/b ÷ c/d = a/b x d/c = ad/bc 20 © 2013 Meredith S. Moody

21. You try Evaluate the following expressions. Remember to write your answer in lowest terms! ½ ÷ ⅛  8/2 = 4 ⅞ ÷ ¼  28/8 = 7/4 21 © 2013 Meredith S. Moody

22. Mixed Numbers What if I have mixed numbers as part of my problem? Convert the mixed number to an improper fraction, then solve 22 © 2013 Meredith S. Moody

23. You try 6 and ¼ ÷ 3 and 5/9  25/4 ÷ 32/9  25/4 x 9/32  225/256 3 and 2/7 ÷ 2 and 5/6  21/2 ÷ 17/6  21/2 x 6/17  126/34  3 and 24/34  3 and 12/17 23 © 2013 Meredith S. Moody