1. FRACTIONS CHAPTER 1 SECTION 3 MTH 11203 Introductory Algebra

2. Algebra Arithmetic – all quantities are known Algebra – one or more of the quantities are unknown Variables – letter/smybols that represent the numbers that are unknown. x, y, and z are the most often letters used

3. Multiplication Symbols If a and b represent any two mathematical quantities then “a times b” can be written as follows. Example ab5x5x xy a b5 25 xx yx y a(b)5(2)5( x ) x(y)x(y) (a)b(5)2(5) x ( x ) y (a)(b)(5)(2)(5)( x )( x )( y )

4. Factors Factors are numbers or variables that are multiplied in a multiplication problem. If a b = c then a and b are factors of c Examples:  2 5 = 10, the numbers 2 and 5 are factors of the product 10  2x, means “2 times x” both the number 2 and the variable x are factors

5. Fractions Numerator – top number of a fraction Denominator – bottom number of a fraction Fractions in general are written = a ÷ b = Example: 2/5 = 2 ÷ 5 =

6. Simplified Fractions To simplify or reduce to lowest terms means that the numerator and the denominator have no common factor other than 1. Simplify 1. Find the greatest common factor (GCF), which is the largest number that will divide both the numerator and denominator without a remainder. (Appendix B) 2. Divide both the numerator and the denominator by the GCF.

7. Simplified Fractions Prime Numbers: Every integer that can only be divided by itself and 1 is prime, but 1 is not a prime number. There are 25 prime numbers between 1 and 100. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97. GCF – Greatest Common Factor: 1. Write both the numerator and denominator as a product of primes 2. Determine all the prime factors that are common to both prime factorizations 3. Multiply the common prime factors to obtain the GCF

8. Simplified Fractions GCF 4010 2 * 20 2 * 5 2 * 10 2 * 5 GCF: 2 * 5 = 10

9. Simplified Fractions GCF 60 105 2 * 30 5 * 21 2 * 15 3 * 7 3 * 5 GCF: 3 * 5 = 15

10. Simplified Fractions Examples pg 27: or

11. Multiply Fractions Evaluate means to answer the problem using the given operation. 2 + 2 means add2÷2 mean divide 2 2 mean multiply2 – 2 means subtract Whole number are 1, 2, 3, 4, 5, 6, … … is called ellipsis meaning continues indefinitely Multiple Fractions

12. Examples of Multiplying Fractions Find the product (pg 27).

13. Examples of Multiplying Fractions Find the product (pg 27). 16 divides 16 and 48 9 divides 36 and 45 OR

14. Examples of Multiplying Fractions Find the product (pg 27). 2 divides 10 and 8

15. Divide Fractions Evaluate means to answer the problem using the given operation. 2 + 2 means add2÷2 mean divide 2 ∙ 2 mean multiply2 – 2 means subtract Divide Fractions - turn it into what you already know.

16. Examples of Dividing Fractions Find the quotient (pg 27). Change to multiplication 4 divides 8 and 4 1 divides 3 and 3 Put the whole number over 1 Change to multiplication 5 divides 5 and 30

17. Add and Subtract Fractions Fractions with a common denominator can easily be added and subtracted. Add or subtract the numerator and keep the same denominator. Add Subtract

18. Examples of Adding and Subtracting Fractions Add with common denominator (pg 28) Subtract with common denominator (pg 28)

19. Adding and Subtracting Fractions To adding or subtracting fractions with unlike denominators you must first rewrite each fraction with the same or the least common denominator (LCD). LCD – Write each denominator as a product of prime factors. Determine the maximum number of times that prime number appears in the factorization. Multiply these prime numbers. Help with finding LCD is in Appendix B. GCF – Write each number as a product of prime factors. Determine the prime factors common to all numbers. Multiply the common factors. Prime Numbers are numbers that has only 2 factors. 1 and itself.  The first 7 prime numbers are 2, 3, 5, 7, 11, 13, 17

20. Adding and Subtracting Fractions Example Find the GCF and the LCD of 108 and 156 108 156 2 54 2 78 2 27 239 3 9 3 13 3 3 LCD = 2 · 2 · 3 · 3 · 3 · 13 = 1404 GCF = 2 · 2 · 3 = 12

21. Adding and Subtracting Fractions Add with unlike terms (pg 28).

22. Adding and Subtracting Fractions Add with unlike terms (pg 28).

23. Adding and Subtracting Fractions Add with unlike terms (pg 28).

24. Adding and Subtracting Fractions Subtract with unlike terms (pg 28).

25. Common Error Remember that dividing out a common factor in a numerator of one fraction and the denominator of a different fraction can only be performed when multiplying fractions. You cannot divide out common factors when adding and subtracting.

26. Mixed Numbers Mixed Number is a whole numbered followed by a fraction.  2 ½  5 ¼  3 ⅝ Changing a mixed number to a fraction  (Whole number times the denominator plus the numerator) divide by the denominator

27. Mixed Numbers

28. Changing a fraction to a mixed number

29. Adding Mixed Numbers

30. Subtracting Mixed Numbers

31. HOMEWORK 1.3 Page 27 - 28 #25, 27, 43, 51, 53, 55, 64, 66, 67, 69, 71, 72, 81, 83, 89