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# Fractions

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### Fractions

1. 1. Fractions 1/ 855/ 60 11/ 12 1 2/10 1½ 1/ 12
2. 2. What is a fraction? Loosely speaking, a fraction is a quantity that cannot be represented by a whole number. Why do we need fractions?Consider the following scenario.Can you finish the whole cake?If not, how many cakes did youeat?1 is not the answer,neither is 0.This suggest that we need anew kind of number.
3. 3. Definition:A fraction is an ordered pair of whole numbers, the 1st oneis usually written on top of the other, such as ½ or ¾ . a numerator b denominatorThe denominator tells us how many congruent piecesthe whole is divided into, thus this number cannot be 0.The numerator tells us how many such pieces arebeing considered.
4. 4. Examples:How much of a pizza do we have below?• we first need to know the size of the original pizza. The blue circle is our whole. - if we divide the whole into 8 congruent pieces, - the denominator would be 8. We can see that we have 7 of these pieces. Therefore the numerator is 7, and we have 7 of a pizza. 8
5. 5. Equivalent fractions a fraction can have many different appearances, theseare called equivalent fractions In the following picture we have ½ of a cakebecause the whole cake is divided into two congruentparts and we have only one of those parts. But if we cut the cake into smaller congruent pieces, we can see that 1 2 = 2 4 Or we can cut the original cake into 6 congruent pieces,
6. 6. Equivalent fractions a fraction can have many different appearances, theseare called equivalent fractions Now we have 3 pieces out of 6 equal pieces, butthe total amount we have is still the same. Therefore, 1 2 3 = = 2 4 6 If you don’t like this, we can cut the original cake into 8 congruent pieces,
7. 7. Equivalent fractions a fraction can have many different appearances, theyare called equivalent fractions then we have 4 pieces out of 8 equal pieces, but thetotal amount we have is still the same. Therefore, 1 2 3 4 = = = 2 4 6 8 We can generalize this to 1 1 n = whenever n is not 0 2 2 n
8. 8. How do we know that two fractions are the same? we cannot tell whether two fractions are the same untilwe reduce them to their lowest terms.A fraction is in its lowest terms (or is reduced) if wecannot find a whole number (other than 1) that can divideinto both its numerator and denominator.Examples: 6 is not reduced because 2 can divide into 10 both 6 and 10. 35 is not reduced because 5 divides into 40 both 35 and 40.
9. 9. How do we know that two fractions are the same?More examples: 110 is not reduced because 10 can divide into 260 both 110 and 260. 8 is reduced. 15 11 is reduced 23 To find out whether two fraction are equal, we need to reduce them to their lowest terms.
10. 10. How do we know that two fractions are the same?Examples:Are 14 and 30 equal? 21 4514 reduce 14 7 221 21 7 330 reduce 30 5 6 reduce 6 3 245 45 5 9 9 3 3Now we know that these two fractions are actuallythe same!
11. 11. How do we know that two fractions are the same?Another example: 24 30Are and equal? 40 42 24 reduce 24 2 12 reduce 12 4 3 40 40 2 20 20 4 5 30 reduce 30 6 5 42 42 6 7This shows that these two fractions are not the same!
12. 12. Improper Fractions and Mixed NumbersAn improper fraction is a fraction 5with the numerator larger than orequal to the denominator. 3Any whole number can be 4 7transformed into an improper 4 , 1fraction. 1 7A mixed number is a whole 3number and a fraction together 2 7An improper fraction can be converted to a mixednumber and vice versa.
13. 13. Improper Fractions and Mixed NumbersConverting improper fractions intomixed numbers: 5 2- divide the numerator by the denominator 1- the quotient is the leading number, 3 3- the remainder as the new numerator. 7 3 11 1More examples: 1 , 2 4 4 5 5Converting mixed numbers 3 2 7 3 17into improper fractions. 2 7 7 7
14. 14. How does the denominator control afraction? If you share a pizza evenly among two people, you will get 1 2 If you share a pizza evenly among three people, you will get 1 3 If you share a pizza evenly among four people, you will get 1 4
15. 15. How does the denominator control afraction?If you share a pizza evenly among eightpeople, you will get only 1 8It’s not hard to see that the slice you getbecomes smaller and smaller.Conclusion:The larger the denominator the smaller thepieces, and if the numerator is kept fixed, the largerthe denominator the smaller the fraction, a a i.e. whenever b c. b c
16. 16. Examples: 2 2 2Which one is larger, or ? Ans : 7 5 5 8 8 8Which one is larger, or ? Ans : 23 25 23 41 41 41Which one is larger, or ? Ans : 135 267 135
17. 17. How does the numerator affect a fraction?Here is 1/16 , here is 3/16 , here is 5/16 , Do you see a trend? Yes, when the numerator gets larger we have more pieces. And if the denominator is kept fixed, the larger numerator makes a bigger fraction.
18. 18. Examples: 7 5 7Which one is larger, or ? Ans : 12 12 12 8 13 13Which one is larger, or ? Ans : 20 20 20 45 63 63Which one is larger, or ? Ans : 100 100 100
19. 19. Comparing fractions with differentnumerators and different denominators.In this case, it would be pretty difficult to tell just fromthe numbers which fraction is bigger, for example 3 5 8 12 This one has less pieces This one has more pieces but each piece is larger but each piece is smaller than those on the right. than those on the left.
20. 20. 3 5 8 12One way to answer this question is to change the appearanceof the fractions so that the denominators are the same.In that case, the pieces are all of the same size, hence thelarger numerator makes a bigger fraction.The straight forward way to find a common denominator is tomultiply the two denominators together: 3 3 12 36 5 5 8 40 and 8 8 12 96 12 12 8 96Now it is easy to tell that 5/12 is actually a bit bigger than 3/8.
21. 21. A more efficient way to compare fractionsWhich one is larger, From the previous example, we see that we don’t really have to know what the common denominator 7 5 or ? turns out to be, all we care are the numerators. 11 8 Therefore we shall only change the numerators by cross multiplying. 7 5 11 8 7 × 8 = 56 11 × 5 = 55 7 5 Since 56 > 55, we see that 11 8 This method is called cross-multiplication, and make sure that you remember to make the arrows go upward.
22. 22. Addition of Fractions- addition means combining objects in two or more sets- the objects must be of the same type, i.e. we combine bundles with bundles and sticks with sticks.- in fractions, we can only combine pieces of the same size. In other words, the denominators must be the same.
23. 23. Addition of Fractions with equal denominatorsExample: 1 3 8 8 + = ?Click to see animation
24. 24. Addition of Fractions with equal denominatorsExample: 1 3 8 8 + =
25. 25. Addition of Fractions with equal denominatorsExample: 1 3 8 8 + = (1 3) which can be simplified to 1 The answer is 8 2(1+3) is NOT the right answer because the denominator tells(8+8)us how many pieces the whole is divided into, and in thisaddition problem, we have not changed the number of pieces inthe whole. Therefore the denominator should still be 8.
26. 26. Addition of Fractions with equal denominatorsMore examples 2 1 3 5 5 5 6 7 13 3 1 10 10 10 10 6 8 14 15 15 15
27. 27. Addition of Fractions with different denominatorsIn this case, we need to first convert them into equivalentfraction with the same denominator.Example: 1 2 3 5An easy choice for a common denominator is 3×5 = 15 1 1 5 5 2 2 3 6 3 3 5 15 5 5 3 15Therefore, 1 2 5 6 11 3 5 15 15 15
28. 28. Addition of Fractions with different denominatorsRemark: When the denominators are bigger, we need to find the least common denominator by factoring.If you do not know prime factorization yet, you have tomultiply the two denominators together.
29. 29. More Exercises: 3 1 3 2 1 6 1 6 1 7 = = = 4 8 4 2 8 8 8 8 8 3 2 3 7 2 5 21 10 21 10 31 = = = 5 7 5 7 7 5 35 35 35 35 5 4 5 9 4 6 45 24 = = 6 9 6 9 9 6 54 54 45 24 69 15 = 1 54 54 54
30. 30. Adding Mixed Numbers 1 3 1 3Example: 3 2 3 2 5 5 5 5 1 3 3 2 5 5 1 3 5 5 4 5 5 4 5 5
31. 31. Adding Mixed NumbersAnother Example: 4 3 4 3 2 1 2 1 7 8 7 8 4 3 2 1 7 8 4 8 3 7 3 56 53 53 3 3 56 56
32. 32. Subtraction of Fractions- subtraction means taking objects away.- the objects must be of the same type, i.e. we can only take away apples from a group of apples.- in fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.
33. 33. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12 11 12 (Click to see animation)
34. 34. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12 11 12
35. 35. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
36. 36. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
37. 37. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
38. 38. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
39. 39. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
40. 40. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
41. 41. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
42. 42. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
43. 43. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
44. 44. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
45. 45. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
46. 46. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
47. 47. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
48. 48. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
49. 49. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
50. 50. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
51. 51. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
52. 52. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
53. 53. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12
54. 54. Subtraction of Fractions with equal denominatorsExample: 11 3 12 12 3 from 11 This means to take away 12 12 Now you can see that there are only 8 pieces left, therefore 11 3 11 3 8 2 12 12 12 12 3
55. 55. Subtraction of FractionsMore examples:15 7 15 7 8 116 16 16 16 26 4 6 9 4 7 54 28 54 28 267 9 7 9 9 7 63 63 63 63 7 11 7 23 11 10 7 23 11 10 161 11010 23 10 23 23 10 10 23 230 51 Did you get all the answers right? 230
56. 56. Subtraction of mixed numbersThis is more difficult than before, so please take notes. 1 1Example: 3 1 4 2Since 1/4 is not enough to be subtracted by 1/2, we better convertall mixed numbers into improper fractions first. 1 1 3 4 1 1 2 13 1 4 2 4 2 13 3 4 2 13 6 4 4 7 3 1 4 4