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McGraw-Hill/Irwin ©2008 The McGraw-Hill Companies, All Rights Reserved Chapter 2 Fractions

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McGraw-Hill/Irwin ©2008 The McGraw-Hill Companies, All Rights Reserved Recognize the three types of fractions Convert improper fractions to whole or mixed numbers and mixed numbers to improper fractions Convert fractions to lowest and highest terms Fractions #2 Learning Unit Objectives Types of Fractions and Conversion Procedures LU2.1

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McGraw-Hill/Irwin ©2008 The McGraw-Hill Companies, All Rights Reserved Add like and unlike fractions Find the least common denominator (LCD) by inspection and prime numbers Subtract like and unlike fractions Add and subtract mixed numbers with the same or different denominators Fractions #2 Learning Unit Objectives Adding and Subtraction of Fractions LU2.2

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McGraw-Hill/Irwin ©2008 The McGraw-Hill Companies, All Rights Reserved Multiply and divide proper fractions and mixed numbers Use the cancellation method in the multiplication and division of fractions Fractions #2 Learning Unit Objectives Multiplying and Dividing Fractions LU2.3

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2-5 Types of Fractions 3, 4, 12, 11 15 8 26 35 Proper Numerator Denominator Proper fractions have a value less than 1; its numerator is smaller than its denominator.

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2-6 Types of Fractions 19, 9, 13, 42 19 4 10 29 Improper Denominator Improper Fractions have a value equal to or greater than 1; its numerator is equal or greater than its denominator. Numerator

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2-7 Types of Fractions 5, 3, 2, 1 8 17 9 8 2 9 15 4 Mixed Numbers Mixed numbers are the sum of a whole number greater than zero and a proper fraction

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2-8 Converting Improper Fractions to Whole or Mixed Numbers 2 Steps 1. Divide the numerator by the denominator 2. a. If you have no remainder, the quotient is a whole number 2 b. If you have a remainder, the quotient is a mixed number 15 16 1 5 5 3 R 1 5 16 15 1 = 3 = 1

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2-9 Converting Mixed Numbers to Improper Fractions Mixed Numbers 3 Steps 1. Multiply the denominator of the fraction by the whole number. 2. Add the product from Step 1 to the numerator of the old fraction. 3 Place the total from Step 2 over the denominator of the old fraction to get the improper fraction. (8 x 6) = 48 48 + 1 = 49 49 8 1 8 6

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2-10 Reducing Fractions to Lowest Terms by Inspection Find the lowest whole number that will divide evenly into the numerator and denominator 24 24 / 6 4 30 30 / 6 5 ==

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2-11 Finding the Greatest Common Divisor Step 1. Divide the numerator into the denominator 1 24 30 24 6 4 6 24 24 0 Step 2. Divide the remainder in Step 1 into the divisor of Step 1 24 / 6 4 30 / 6 5 = Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0 24 30

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2-12 Divisibility Tests Last digit is 0,2,4,6,8 Sum of the digits is divisible by 3 Last two digits can be divided by 4 Last digit is 0 or 5 The number is even and 3 will divide into the sum of the digits The last digit is 0 12 6 14 7 36 12 69 23 140 1(40) 160 1(60) 15 3 20 4 12 2 18 3 90 9 100 10 3 + 6 = 9 / 3 = 3 6 + 9 = 15 / 3 = 5 35 7 40 8 = ===== =

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2-13 Raising Fractions to Higher Terms When Denominator is Known 2 Steps 1. Divide the new denominator by the old denominator to get the common number that raises the fraction to higher terms. 2. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator. 4 7 28 28 0 4 x 4 = 16 4 = ? 7 28 16 28

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2-14 Adding Like Fractions Add the numerators and place the total over the denominator If the total of your numerators is the same as your original denominator, convert your answer to a whole number; if the total is larger than your original denominator, convert your answer to a mixed number 2 3 5 9 9 9 += -b-b 5 6 11 2 9 9 +==1

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2-15 Least Common Denominator (LCD) The smallest nonzero whole number into which ALL denominators will divide evenly. 3 5 7 21 7 42 21 + What is the least common denominator?

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2-16 Adding Unlike Fractions 4 Steps 1. Find the LCD 2. Change each fraction to a like fraction with the LCD. 3. Add the numerators and place the total over the LCD. 4. If necessary, reduce the answer to lowest terms. 1 1 3 8 9 12 +++ 24 + 9 + 8 + 6 = 47 72 72 72 72 72

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2-17 Prime Numbers A whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number. Examples 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43

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2-18 Adding Mixed Numbers 3 Steps 1. Add the fractions. 2. Add the whole numbers. 3. Combine steps 1 & 2. Be sure you do not have an improper fraction in your final answer. If necessary, reduce the answer to lowest terms. 7 7 20 3 12 5 20 1 5 4 20 24 4 20 20 17. 4 1 20 5 44 66 77 1 18 = = = ++ Step 1 Step 2 Step 3

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2-19 Subtracting Like Fractions Step 1 - Subtract the numerators and place the total over the denominator Step 2 - If necessary, reduce the answer to lowest terms 9 1 8 / 2 4 10 10 10 / 2 5 = - =

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2-20 Subtracting Unlike Fractions 5 2 8 64 - 5 40 8 64 2 - 2 64 64 38 = 19 64 32 Step 1. Find the LCD Step 2. Raise the fraction to its equivalent value. Step 3. Subtract the numerators and place the answer over the LCD. Step 4. If necessary, reduce the answer to lowest terms. -

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2-21 Subtracting Mixed Numbers 1 4 2 8 - 3 8 8 1 8 Step 1. Subtract fractions, making sure to find the LCD. Step 2. Subtract whole numbers. Step 3. Reduce the fractions to lowest terms. When Borrowing is Not Necessary 66 6

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2-22 Subtracting Mixed Numbers 1 2 6 2 4 4 3 3 3 4 4 4 3 4 Step 1. Make sure the fractions have the LCD. Step 2. Borrow from the whole number. Step 3. Subtract whole numbers and fractions. Step 4. Reduce the fractions to lowest terms. When Borrowing is Necessary 33 1 2

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2-23 Multiplying Proper Fractions Step 1. Multiply the numerator and the denominators Step 2. Reduce the answer to lowest terms 5 1 4 20 10 1 6 7 42 21 ==xx

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2-24 Multiplying Mixed Numbers Convert the mixed numbers to improper fractions Multiply the numerator and denominators 1 1 7 3 7 1 3 2 3 2 2 2 2 3 = X 1 X == 1 1 Reduce the answer to lowest terms

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2-25 Dividing Proper Fractions 1 2 1 3 3 8 3 8 2 16 = = X Invert (turn upside down) the divisor (the second fraction ) Multiply the fractions.. Reduce the answer to lowest terms

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2-26 Dividing Mixed Numbers Convert all mixed numbers to improper fractions Invert the divisor and multiply 3 5 35 6 105 3 4 6 4 17 34 34 8 = X 2 X = = Reduce the answer to lowest terms 3

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2-27 Problem 2-31: Solution: 2323 16 3 8 x = 3 16 15 1 5 ounces 3434 2323 6 12 1212 X = = cup 1212 2323 2626 1313 x = = cup 2323 4343 2 x = 3 4 3 1 1 teaspoon Cream cheese: Butter: Sugar: Vanilla:

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2-28 Problem 2-38: 115 + 66 + 106 + 110 = 397 = 398 feet 4848 2828 1818 2828 9898 1818 Solution:

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2-29 Problem 2-46: 1 X $8 = x $8 = $12 1212 3232 Solution: $12 x 6 = $72

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2-30 Problem 2-56: 9898 3 + 5 + 6 + 4 = 18 = 19 1818 4848 2828 2828 1818 23 - 19 4 2828 1818 1818 Days left Solution:

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