Kirkwood Community College Fractions Kirkwood Community College January 26, 2009 Presented by Sanh Tran, MBA, CPIM, CTL

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

Learning Unit Objectives #2 Fractions Learning Unit Objectives Types of Fractions and Conversion Procedures LU2.1 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 2-3

Learning Unit Objectives #2 Fractions Learning Unit Objectives LU2.2 Adding and Subtraction of Fractions 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 2-4

Learning Unit Objectives #2 Fractions Learning Unit Objectives LU2.3 Multiplying and Dividing Fractions Multiply and divide proper fractions and mixed numbers Use the cancellation method in the multiplication and division of fractions 2-5

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

Numerator vs Denominator Numerator: The top portion of the fraction. Denominator: The bottom portion of the fraction. Proper fraction: Numerator < Denominator Example: 3 7

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

Improper Fraction Numerator > Denominator Numerator = Denominator Examples: 5/5 or 7/8

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

Converting Improper Fractions to Whole or Mixed Numbers 15 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 = 1 16 1 5 5 3 R 1 5 16 15 1 = 3

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. 1 8 6 (8 x 6) = 48 (8 x 6) = 48 48 + 1 = 49 49 8

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

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

Divisibility Tests = = = = = = = Sum of the digits is divisible by 3 Last two digits can be divided by 4 The number is even and 3 will divide into the sum of the digits Last digit is 0,2,4,6,8 Last digit is 0 or 5 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 =

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 4 7 28 28 4 x 4 = 16 16 28

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 + = 5 6 11 2 9 9 9 9 -b + = = 1

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?

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 1 1 3 8 9 12 + + + 24 + 9 + 8 + 6 = 47 72 72 72 72 72

Problem 2-15 Add unlike fractions: ３/7 + 4/21 = Change into common denominator: ３/7 + 4/21 = (3 x 3)/(7 x 3) + 4/21 = 9/21 + 4/21 = 13/21

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

Adding Mixed Numbers 7 7 20 20 3 12 5 20 1 5 4 20 24 4 17 . 4 1 20 5 4 4 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. 6 6 + 7 + 7 = 1 Step 1 = Step 2 18 = 18 Step 3

Problem 2-34 W1 W2 W3 2 1/4 + 1 3/4 + 5/8 = Change into fractions with common denominator: 2 (1 x 2)/(4x2) + 1 (3x2)/(4 x 2) + 5/8 = 2 2/8 + 1 6/8 + 5/8 = 3 13/8 = 4 5/8

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 - = =

Subtracting Unlike Fractions 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. 5 2 8 64 - 5 40 8 64 2 - 2 64 64 38 = 19 64 32 -

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

Subtracting Mixed Numbers When Borrowing is Necessary 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. 1 2 6 2 4 4 3 3 3 4 4 4 3 4 3 3 2 -1 -1 -1 1

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 x x = =

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

Reciprocal Reciprocal of a fraction is another fraction by switch the location of the numerator and the denominator. 3/5 Reciprocal of 3/5 is 5/3.

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

Problem 2-27 Divide by a fraction, and reduce to the lowest terms: 12/9 ÷ 4 = Note: 4 = 4/1 Reciprocal of 4/1 is ¼ 12/9 ÷ 4/1 = Instead of dividing by a fraction, multiply with the reciprocal of the fraction: 12/9 x ¼ = 1/3

Problem 2-44 90 pizzas were served Each guest ate 1/6 of a pizza. All pizzas were eaten. Reciprocal of 1/6 is 6/1 = 6 Calculate the total of attending guests: 90 ÷ 1/6 = 90 x 6/1 = 90 x 6 = 540 guests

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

Problem 2-38: Solution: 115 + 66 + 106 + 110 = 397 = 398 feet 4 8 2 1

Problem 2-46: Solution: 1 X $8 = x $8 = $12 1 2 3 $12 x 6 = $72

Problem 2-56: Solution: 9 8 3 + 5 + 6 + 4 = 18 = 19 1 4 2 - 19 4 2 8 1 3 + 5 + 6 + 4 = 18 = 19 1 4 2 23 - 19 4 2 8 1 Days left

Reference Slater, J. (2008). Practical business math procedures (9th ed.). New York: McGraw-Hill/Irwin

Homework (Total: 5 points)