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# Fractions and multiples

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### Fractions and multiples

1. 1. Index Divisibility Fractions Simplest form Operations Exercises Factors and Fractions Matem´ticas 2o E.S.O. a Alberto Pardo Milan´s e -
2. 2. Index Divisibility Fractions Simplest form Operations Exercises 1 Divisibility 2 Fractions 3 Simplest form of a fraction 4 Operations 5 ExercisesAlberto Pardo Milan´s e Factors and Fractions
3. 3. Index Divisibility Fractions Simplest form Operations Exercises DivisibilityAlberto Pardo Milan´s e Factors and Fractions
4. 4. Index Divisibility Fractions Simplest form Operations Exercises Divisibility Factors and multiples A factor of a number n, is a number d which divides n. Read ⇐⇒ if and only if. d is a factor of n ⇐⇒ d is a divisor of n ⇐⇒ d divides n ⇐⇒ n is divisible by d ⇐⇒ n is a multiple of d. Examples: −7 divides 14 ⇐⇒ −7 is a factor of 14 ⇐⇒ 14 is divisible by −7 ⇐⇒ 14 is a multiple of −7.Alberto Pardo Milan´s e Factors and Fractions
5. 5. Index Divisibility Fractions Simplest form Operations Exercises Divisibility Primes A prime number is a positive number that has only two positive factors 1 and the number itself (1 is not considered a prime number as it only has one positive factor). A number with more than two positive factors it is called composite number. Examples: 3 is a prime number because has only two positive fac- tors (1 and 3). 6 is a composite number because has four positive factors (1, 2, 3 and 6). Two numbers are relatively prime if they have no common positive divisors except 1. Example: 6 and 25 are relatively prime because positive factors of 6 are 1, 2, 3, 6 and positive factors of 25 are 1, 5, 25.Alberto Pardo Milan´s e Factors and Fractions
6. 6. Index Divisibility Fractions Simplest form Operations Exercises Divisibility Prime decomposition Prime decomposition is to ﬁnd the set of prime factors of an integer: To factorize a number you have to express the number as a product of its prime factors. To factorize negative numbers use also −1. Examples: • 90 = 2 · 45 = 2 · 3 · 15 = 2 · 3 · 3 · 5 = 2 · 32 · 5. • −25 = −1 · 25 = −1 · 5 · 5 = −1 · 52 .Alberto Pardo Milan´s e Factors and Fractions
7. 7. Index Divisibility Fractions Simplest form Operations Exercises Divisibility GCD and LCM The Greatest Common Divisor (GCD) is the highest number that is a common factor of two or more numbers. It is clear that if GCD(a, b) = 1, a and b are relatively prime. Examples: GCD(42, 110) = 2, because positive factors of 42 are 1, 2, 3, 6, 7, 14, 21, 42, and positive factors of 110 are 1, 2, 5, 10, 11, 22, 55, 110. 12 and 35 are relatively prime, because GCD(12, 35) = 1. The Least Common Multiple (LCM) is the lowest positive number that is a common multiple of two or more numbers. Example: LCM(6, 9) = 18, because positive multiples of 6 are 6, 12, 18, 24, . . . and positive multiples of 9 are 9, 18, 27, . . .Alberto Pardo Milan´s e Factors and Fractions
8. 8. Index Divisibility Fractions Simplest form Operations Exercises FractionsAlberto Pardo Milan´s e Factors and Fractions
9. 9. Index Divisibility Fractions Simplest form Operations Exercises Fractions What’s a fraction? A fraction is a number that represents a part of something. a Fractions are written in the form , where a and b are integers, b a and the number b is not zero. Read a over b. b The number a is called numerator, and the number b is called denominator. 13 Example: , 13 is the numerator and 25 is the denominator. 25Alberto Pardo Milan´s e Factors and Fractions
10. 10. Index Divisibility Fractions Simplest form Operations Exercises Fractions Proper fractions and improper fractions A proper fraction is a fraction that is less than one. A fraction greater than one is called an improper fraction. 12 23 Examples: is a proper fraction. is an improper fraction. 17 15 Equivalent fractions Equivalent fractions are diﬀerent fractions that name the same amount. 3 6 Example: = 7 14 because they represent the same amount.Alberto Pardo Milan´s e Factors and Fractions
11. 11. Index Divisibility Fractions Simplest form Operations Exercises Fractions Amplify and reduce fractions To ﬁnd equivalent fractions, multiply or divide the numerator and the denominator by the same number. If you multiply by the same number, you amplify the fraction. If you divide by the same factor, you reduce the fraction. Examples: 6 • To amplify we 9 6 60 multiply the numerator and the denominator by 10: = . 9 90 6 • To reduce we 9 6 2 divide the numerator and the denominator by 3: = . 9 3Alberto Pardo Milan´s e Factors and Fractions
12. 12. Index Divisibility Fractions Simplest form Operations Exercises Simplest form of a fractionAlberto Pardo Milan´s e Factors and Fractions
13. 13. Index Divisibility Fractions Simplest form Operations Exercises Simplest form of a fraction Lowest Terms Fraction A lowest terms fraction is a fraction that can not be reduced anymore. To write a fraction in the simplest form ﬁnd the lowest terms fraction. To reduce a fraction to the lowest terms fraction, we can use two methods: • Divide the numerator and the denominator by the Greatest Common Factor. • Divide the numerator and the denominator by any common factor and keep dividing until they are relatively prime.Alberto Pardo Milan´s e Factors and Fractions
14. 14. Index Divisibility Fractions Simplest form Operations Exercises Simplest form of a fraction Examples: 70 • To obtain the lowest terms fraction of we can divide 105 70 2 70 and 105 by GCD(70, 105) = 35, so = . 105 3 70 • To reduce to the lowest terms fraction we can di- 105 vide by 5 and keep dividing until there are no common factors 70 14 2 but one = = , (note 2 and 3 are relatively prime ). 105 21 3Alberto Pardo Milan´s e Factors and Fractions
15. 15. Index Divisibility Fractions Simplest form Operations Exercises OperationsAlberto Pardo Milan´s e Factors and Fractions
16. 16. Index Divisibility Fractions Simplest form Operations Exercises Operations Add and subtract To add or subtract fractions with like denominators, add the numerators and keep the same denominator. To add or subtract fractions with unlike denominators, ﬁrst amplify these fractions to obtain like denominators (using the LCM or any multiple). Simplify, if possible. Examples: 4 8 12 4 • + = = . 15 15 15 5 3 1 3 1 6 5 11 • To add + using the LCM: + = + = . 15 6 15 6 30 30 30 Because LCM(15, 6) = 30. • Using another common multiple of 15 and 6, for example 60: 3 1 12 10 22 11 + = + = = . 15 6 60 60 60 30Alberto Pardo Milan´s e Factors and Fractions
17. 17. Index Divisibility Fractions Simplest form Operations Exercises Operations Multiply and divide To multiply fractions, multiply the numerators and multiply the denominators then simplify. 12 3 12 · 3 36 18 Example: · = = = . 5 14 5 · 14 70 35 You can get the reciprocal of a fraction by switching its numerator and denominator. 14 21 Example: The reciprocal of is . 21 14 To divide by a fraction,multiply by its reciprocal and simplify,if possible. 2 4 2 17 2 · 17 34 17 Example: : = · = = = . 8 17 8 4 8·4 32 16Alberto Pardo Milan´s e Factors and Fractions
18. 18. Index Divisibility Fractions Simplest form Operations Exercises ExercisesAlberto Pardo Milan´s e Factors and Fractions
19. 19. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 1 There are 28 students in our class and we want to divide it into groups with equal number of students. How many ways can the class be divided into groups? What are they?Alberto Pardo Milan´s e Factors and Fractions
20. 20. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 2 Mary wants to serve hot dogs for 48 people. Sausages come in packages of 8 and hot dog buns come in packages of 12. She wants to have enough to serve everyone and have none left over, how many packages of sausages and hot dog buns should she purchase?Alberto Pardo Milan´s e Factors and Fractions
21. 21. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 3 Peter works in a ﬂorist. Today He is making identical ﬂoral arrangements for a bridal party. He has 84 daisies, 66 lilies, and 30 orchids. He wants each arrangement to have the same number of each ﬂower. What is the greatest number of arrangements that he can make if every ﬂower is used?Alberto Pardo Milan´s e Factors and Fractions
22. 22. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 4 Samantha loves the sea. She has kayaking lessons every ﬁfth day and diving lessons every seventh day. If she had a kayaking lesson and a diving lesson on June the sixth, when will be the next date on which she has both kayaking and diving lessons?Alberto Pardo Milan´s e Factors and Fractions
23. 23. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 5 There are two ﬂashing neon lights. One blinks every 4 seconds and the other blinks every 6 seconds. If they are turned on exactly at the same time, how many times will they blink at the same time in a minute?Alberto Pardo Milan´s e Factors and Fractions
24. 24. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 6 Peter sells books. He made 240e selling children’s books, 140e from cookbooks, and 280e from paperback books. He gets exactly the same beneﬁt from each book. What is the most that Peter could get for each book? How many books could Peter have sold then?Alberto Pardo Milan´s e Factors and Fractions
25. 25. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 7 Complete: 3 = 4 A half= One and a half= Three ﬁfths= 5 1 = 1 4 = 2 = 3 5 Two thirds= Two and a quarter= Two quarters= 1 2 = 1 4 = 3 = 5 7 A quarter= Two and four tenths= An/one eighth= 1 3 1 = = 12 6Alberto Pardo Milan´s e Factors and Fractions
26. 26. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 8 Write a fraction in simplest form beside each sentence: There are forty-two girls out of seventy-two people: The baby snake was just four and three quarters inches long: Fifteen out of the twenty students has dogs as pets: I walk ﬁve and a quarter miles a day:Alberto Pardo Milan´s e Factors and Fractions
27. 27. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 9 An orchard has 72 peach trees 15 apple trees, and 23 lemon trees. In simplest form, what fraction of the trees are peach trees?Alberto Pardo Milan´s e Factors and Fractions
28. 28. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 10 George swims seven eighths miles and Mat swims ten twelfths miles. Who swims farther?Alberto Pardo Milan´s e Factors and Fractions
29. 29. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 11 Compare these fractions using the least common denominator: 2 3 9 7 −1 −2 13 , , , , , , . −6 4 15 5 −3 10 12Alberto Pardo Milan´s e Factors and Fractions
30. 30. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 12 Lance and Frank ate seven twelfths of a pizza. If Frank ate a third of the pizza, how much of the pizza did Lance eat?Alberto Pardo Milan´s e Factors and Fractions
31. 31. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 13 Tangram means seven boards of skill. The Tangram is a Chinese puzzle consisting of seven ﬂat shapes (called tans) which are put together to form diﬀerent shapes. Study the tans at the right forming a square. The side of the square is 1 cm, so the area is 1 cm2 . Find the fractional value of each piece.Alberto Pardo Milan´s e Factors and Fractions
32. 32. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 14 Of the 20 students, a quarter of the students wear jeans, a ﬁth wear shorts and a tenth wear a skirt. How many students wear something else?Alberto Pardo Milan´s e Factors and Fractions
33. 33. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 15 My car´s gas tank was empty and now is full but I paid forty euros for it. To go to Marbella I need two ﬁfths of the tank, how much money is this fraction of the tank?Alberto Pardo Milan´s e Factors and Fractions
34. 34. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 16 Peter cut six apples into quarters, how many pieces of apple did he have?Alberto Pardo Milan´s e Factors and Fractions
35. 35. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 17 1 Mike cleaned of the 24 m2 yard . How many are left? 4Alberto Pardo Milan´s e Factors and Fractions
36. 36. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 18 3 Thomas drunk cup of milk, and now he has 24 cl left. How 5 many ml did the cup have originally?Alberto Pardo Milan´s e Factors and Fractions
37. 37. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 19 2 Eric is reading a book with 195 pages. He read of the pages 5 today. How many are left?Alberto Pardo Milan´s e Factors and Fractions
38. 38. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 20 1 Of Paul’s stone collection are white stones. He has total of 75 5 stones. How many of them are not white?Alberto Pardo Milan´s e Factors and Fractions
39. 39. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 21 1 Mom baked cookies, and gave of them to Beth, and of the 2 2 remaining ones, she gave to Seth. The rest, which was 4 3 cookies, she ate herself. How many did she bake originally?Alberto Pardo Milan´s e Factors and Fractions
40. 40. Index Divisibility Fractions Simplest form Operations Exercises Exercises Exercise 22 1 This time Mom baked again cookies, and gave of them to Beth, 2 1 and of the remaining ones she gave to Seth. There were 2 3 cookies left. How many did she bake originally?Alberto Pardo Milan´s e Factors and Fractions