Ch.4 Fractions and Mixed Numbers

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1 Ch. Fractions and Mixed Numbers. An Introduction to Fractions. Multiplying Fractions. Dividing Fractions. Adding and Subtracting Fractions. Multiplying and Dividing Mixed Numbers.6 Adding and Subtracting Mixed Numbers.7 Order of Operations and Complex Fractions.8 Solving Equations that Involving Fractions

2 . An Introduction to Fractions Assume the knowledge of positive fractions. Identify the numerator and denominator of a fraction. Simplify special fraction forms. Define equivalent fractions. Build equivalent fractions. Simplify fractions 6. Build and simplify algebraic fractions

3 . An Introduction to Fractions. Identify the numerator and denominator of a fraction Numerator and denominator Proper fraction and improper fraction. Simplify special fraction forms e.g. Simplify: a. b. c. d. 9 = 0 = = 0 undefined 7 0

4 . An Introduction to Fractions. Define equivalent fractions Equivalent Fractions Two fractions are equivalent if they represent the same number.. Build equivalent fractions Multiplying Fractions To multiply two fractions, multiply the numerators and multiply the denominators

5 . An Introduction to Fractions. Build equivalent fractions e.g. Write as an equivalent fraction with a denominator 6 of 8. Ans. 0 8 e.g. Write 7 as an equivalent fraction with a denominator of. Ans. 7

6 . An Introduction to Fractions. Simplify fractions Simplest Form of a Fraction A fraction is in simplest form, or lowest form, when the numerator and denominator have no common factors other than. e.g.6 Are the following fractions in simplest form? 8 a. b. Ans. no 9 Ans. yes 6

7 . An Introduction to Fractions. Simplify fractions e.g.7 Simplify each fraction: a. b. 9 Ans. 9 8 e.g.8 Simplify each fraction: a. b. (using prime factorization) 7 Ans. 8 Ans. simplest form 7

8 . An Introduction to Fractions 6. Build and simplify algebraic fractions Fractions that contains variable(s) in numerator and/or denominators are called algebraic fractions. e.g.0 Write as an equivalent fraction with a 8 denominator of x. x Ans. x 8

9 . An Introduction to Fractions 6. Build and simplify algebraic fractions e.g. Simplify each fractions 7 m r r a. b. c. 0c d c d Ans. Ans. m c 9

10 . Multiplying Fractions Assume the knowledge of multiplying positive fractions. Multiplying fractions. Simplify answers when multiplying fractions. Multiplying algebraic fractions. Evaluate exponential expressions that have fractional bases. Solve application problem by multiplying fractions 6. Find the area of a triangle 0

11 . Multiplying Fractions. Multiplying fractions e.g. Multiply: 8 9 Ans. 7 Comments: the sign rules for multiplying integers also holds for multiplying fractions! e.g. Multiply: (integer with fraction) 9 9 Ans.

12 . Multiplying Fractions. Simplify answers when multiplying fractions e.g. Multiply: 6 Ans. 6 Comments: don t multiply the numerator and denominator and then simplify.

13 . Multiplying Fractions. Multiply algebraic fractions e.g.6 Multiply and simplify: 9 b 6b 9 c 8cd Ans. 0 d 9c a. b. e.g.7 Multiply and simplify: (integer with fraction) m 6 m a. m b. c Ans. Ans. m Ans.

14 . Multiplying Fractions. Evaluate exponential expressions that have fractional bases e.g.8 Evaluate each expression: a. b. Ans. c. Ans. 9 9 e.g.9 Find the power: x Ans. x 9

15 . Multiplying Fractions. Solve application problem by multiplying fractions e.g.0 COLLEGE LIFE It takes of active members of Alpha Gamma Sigma to vote to accept a pledge into their fraternity. How many votes are needed if the fraternity has 8 active members? Ans. 6 votes

16 . Multiplying Fractions 6. Find the area of a triangle e.g. Find the area of the triangle shown below. 6 ft ft Ans. square feet 6

17 . Dividing Fractions. Find the reciprocal of a fraction. Dividing fractions. Dividing algebraic fractions. Solve application problems by dividing fraction 7

18 . Dividing Fractions. Find the reciprocal of a fraction Reciprocals Two numbers are reciprocals if their product is. e.g. For each number, find its reciprocal, and show that their product is. a. b. Ans. c. Ans. 8

19 . Divide fractions. Dividing Fractions Dividing Fractions (just like before) To divide two fractions, multiply the st fraction by the reciprocal of the nd fraction. Simplify if possible. e.g. Divide and simplify: 9 Ans. 7 8 e.g. Divide and simplify: 0 90 Ans. 7 9

20 . Divide fractions. Dividing Fractions e.g. Divide and simplify: 9 8 Ans. e.g.6 Divide and simplify: 6 ( 8) Ans. 8 0

21 . Dividing Fractions. Dividing algebraic fractions We can find reciprocal of an algebraic fraction in exactly the same way as we find reciprocal of an fraction. (flip) e.g.7 Divide: 9 9x Ans. x 0 e.g.8 Divide and simplify: 6 a b a a Ans. b b

22 . Dividing Fractions. Solve application problems by dividing fraction e.g.9 GOLD COINS How many 6 -ounce coins can be 7 cast from a -ounce bar of gold? 8 Ans. coins

23 . Adding and Subtracting Fractions. Add and subtract fractions that have the same denominator. Add and subtract fractions that have different denominators. Find LCD to add and subtract fractions. Add and subtract algebraic fractions. Identify the greater of two fractions 6. Solving application problems by adding and subtracting fractions

24 . Adding and Subtracting Fractions. Add and subtract fractions that have the same denominator e.g. Subtract: 9 Ans. 7 e.g. Perform the operations and simplify: (from left to right) Ans

25 . Adding and Subtracting Fractions. Add and subtract fractions that have different denominators (Just like in Math) Least Common Denominator The Least Common Denominator (LCD) for a set of fractions is the smallest number each denominator will divide exactly (divide with no remainder).

26 . Adding and Subtracting Fractions. Add and subtract fractions that have different denominators (Read the book. Just like in Math) ) Find LCD ) Build all fractions so that all have LCD as denominator ) Add or subtract ) Simplify e.g.6 Subtract: 7 Ans. 6 e.g.7 Add: + Ans. 6

27 . Adding and Subtracting Fractions. Find LCD to add and subtract fractions The Least Common Multiple (LCM) The Least Common Denominator (LCD) Listing method Prime factorization method (apply the algebraic fractions) *Group factor method (math) e.g.8 Add and simplify: + Ans. 8 6 e.g.9 Subtract and simplify: 7 Ans. 0 7

28 . Adding and Subtracting Fractions. Add and subtract algebraic fractions e.g.0 Add or subtract, simplify if possible: x x a. Ans. b. x Ans. 9d 9d 9d e.g. Add or subtract, simplify if possible: x + x + a. Ans. b. 0 6 n 7 Ans. 8 7n n 8

29 . Adding and Subtracting Fractions. Identify the greater of two fractions (did this in math) e.g. Which fractions is larger: or? 7 Ans. 9

30 . Adding and Subtracting Fractions 6. Solving application problems by adding and subtracting fractions or more hrs 0 hr hr 7 hr No TV 6 e.g. The pie chart shows how much time students are watching TV daily in a college. Find the fraction of student body that watches from to hrs of TV daily. Ans. 0

31 . Multiplying and Dividing Mixed Numbers. Identify the whole-number and fractional parts of a mixed number. Write mixed numbers as improper fractions. Write improper fractions as mixed numbers. Graph fractions and mixed numbers on a number line. Multiply and divide mixed numbers 6. Solving application problems by multiplying and dividing mixed numbers

32 . Multiplying and Dividing Mixed Numbers. Identify the whole-number and fractional parts of a mixed number We did it in math. e.g. In the illustration on the right, each square represents one whole. Write an improper fraction and a mixed number to represent the shaded portion. Ans.,

33 . Multiplying and Dividing Mixed Numbers. Write mixed numbers as improper fractions e.g. write as an improper fraction. 9 Ans Write improper fractions as mixed numbers e.g. write each improper fraction as a mixed number or a whole number. 9 8 a. Ans. c. Ans. 7 d. 9 9 Ans.

34 . Multiplying and Dividing Mixed Numbers. Graph fractions and mixed numbers on a number line e.g. Graph 9,, 8 0, 8 on a number line = 0

35 . Multiplying and Dividing Mixed Numbers. Multiply and divide mixed numbers (review) Strategy: convert to improper fractions, then multiply or divide. e.g. Multiply and simplify, if possible: a. Ans. b. c. () 7 6 Ans.

36 . Multiplying and Dividing Mixed Numbers. Multiply and divide mixed numbers (review) e.g.6 Multiply and simplify, if possible: 9 a. 8 7 Ans. b. 8 6 Ans. 6

37 . Multiplying and Dividing Mixed Numbers 6. Solving application problems by multiplying and dividing mixed numbers e.g.7 KLEENEX When unfolded and laid flat, a standard sheet of Kleenex brand tissue is 8 inches wide by 8 in long. Find the area of a sheet of Kleenex. Ans. 68 in 7

38 . Multiplying and Dividing Mixed Numbers 6. Solving application problems by multiplying and dividing mixed numbers e.g.8 HIGHWAYS It took workers days to paint the yellow centerline on a new 7 mile stretch of highway. If they covered the same distance each day, how much did they do in one day? Ans. 6 mi 8

39 .6 Adding and Subtracting Mixed Numbers. Add mixed number. Add mixed number in vertical form. Subtract mixed number. Solving application problems by adding and subtracting mixed numbers 9

40 .6 Adding and Subtracting Mixed Numbers. Add mixed number Strategy : convert to improper fractions, then add or subtract. e.g Ans. 0 Strategy : when add two positive mixed number, add the whole part and fraction part separately. Then combine (good strategy for large numbers) e.g Ans. 6 0

41 .6 Adding and Subtracting Mixed Numbers. Add mixed number in vertical form Comments: for adding positive mixed numbers. e.g. Add: Ans. 6

42 .6 Adding and Subtracting Mixed Numbers. Subtract mixed number You can always use the converting to improper fraction strategy. However, for large numbers, the treat whole number part and fraction part separately method has some advantages. Here we use the nd method (need minuend > subtrahend). e.g.8 Subtract and simplify: Ans.

43 .6 Adding and Subtracting Mixed Numbers. Subtract mixed number e.g.9 Subtract and simplify: Ans. 6 8

44 .6 Adding and Subtracting Mixed Numbers. Solving application problems by adding and subtracting mixed numbers e.g.0 BAKING A baker wants to make a cookie recipe that calls for cups of flour, a bread recipe that 8 calls for cups of flour, and a cake recipe that calls for cups. How many cups of flour does she need to make all recipes? Ans. 7 8 cups

45 .6 Adding and Subtracting Mixed Numbers. Solving application problems by adding and subtracting mixed numbers e.g. REFRESHMENT How much punch is left in a 0-gallon container if 9 gallons have been used? Ans. 0 gallons

46 .7 Order of Operations and Complex Fractions. Use the order of operations rule. Solving application problems by using the order of operations rule. Evaluate formulas. Simplify complex fractions 6

47 .7 Order of Operations and Complex Fractions. Use the order of operations rule () + A comment. When evaluating fraction like, evaluate the expression above the fraction bar and the expression below the fraction bar separately. Then perform the division if possible. e.g. 6 + Ans. 6 7

48 .7 Order of Operations and Complex Fractions. Use the order of operations rule e.g Ans. 6 e.g. Add to the difference of and. 7 9 Ans

49 .7 Order of Operations and Complex Fractions. Solving application problems by using the order of operations rule e.g. MASONRY Find the height of a planter if 8 layers (called courses) of -inch-high bricks are held together by -inch-thick layers of mortar. 8 Ans. inches 9

50 .7 Order of Operations and Complex Fractions. Evaluate formulas Trapezoid: a b h e.g. The formula for the area of a trapezoid is A = h( a + b). 8 Find A if h = ft, a = ft, and b = ft. Ans. A = 6 8 ft 0

51 .7 Order of Operations and Complex Fractions. Simplify complex fractions Complex Fraction A complex fraction is a fraction whose numerator and denominator, or both, contain one or more fractions or mixed numbers. e.g. Main fraction bar Numerator Denominator +

52 .7 Order of Operations and Complex Fractions. Simplify complex fractions Main fraction bar means division, e.g. + = ( + ) ( ) e.g.6 Simplify: 8 7 Ans. 6 7 e.g.7 Simplify: Ans. 6

53 .7 Order of Operations and Complex Fractions. Simplify complex fractions e.g.8 Simplify: Ans. 6

54 .8 Solving Equations That Involve Fractions. Use the addition and subtraction properties of equality to solve equations that involve fractions. Use reciprocals to solve equations. Clear equations of fractions. Use equations to solve application problems that involve fractions

55 .8 Solving Equations That Involve Fractions. Use the addition and subtraction properties of equality to solve equations that involve fractions This is review. e.g. Solve: = x Ans. x = 9 e.g. Solve: 7 8 = m + Ans. m =

56 .8 Solving Equations That Involve Fractions. Use reciprocals to solve equations e.g. Solve and check the result: y = y = Ans. y = 8 a. b. e.g. Solve y = using a two-step process. Ans. y = 6

57 .8 Solving Equations That Involve Fractions. Use reciprocals to solve equations e.g. Solve 6 a = and check the result. Ans. = 6 6 a e.g.6 Solve 6d = and check the result. Ans. d = 08 Lastly, we use both properties of equality. e.g.7 Solve 8 9 n 7 = 6 and check the result. Ans. n = 6 7

58 .8 Solving Equations That Involve Fractions. Clear equations of fractions 7 8 e.g.8 Solve fractions. = m + by first clearing the equation of Ans. m = e.g.9 Solve x x = and check the result. 7 Ans. x = 70 e.g.0 Solve y 8 = 7 0 y and check the result. Ans. y = 8

59 .8 Solving Equations That Involve Fractions. Clear equations of fractions ) To ) are from section. Strategy for Solving Equations (for detail read the book) ) clear the equation of fractions (new) ) simplify each side of equation ) isolate the variable term on one side ) isolate the variable ) Check the result 9

60 .8 Solving Equations That Involve Fractions. Use equations to solve application problems that involve fractions e.g. AMERICAN LITERATURE A student has read the first 768 pages of the paperback version of the novel Gone With The Wind. If that is three-fourths of the book, how many pages are there in the paperback version? Ans. 0 pages 60

61 .8 Solving Equations That Involve Fractions. Use equations to solve application problems that involve fractions e.g. TRAFFIC TICKET DISMISSAL A traffic school class has three parts. In the first part, a film is shown that takes one-sixth of the class time. In the second part, the instructor lectures for 0 minutes. In the final part, a test is given that takes one-fourth of the class time. How many minutes long is the traffic school class? Ans. 80 minutes 6

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