Quản trị kinh doanh - Chapter two: Fractions

Chapter TwoFractions2-1Learning unit objectives2-2LU 2-1: Types of Fractions and Conversion ProceduresRecognize 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.LU 2-3: Multiplying and Dividing FractionsMultiply and divide proper fractions and mixed numbers.Use the cancellation method in the multiplication and division of fractions.LU 2-2: Adding and Subtraction of FractionsAdd 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.Types of Fractions2-31, 1, 1, 4, 184 2 12 7 55Proper fractions have a value less than 1; its numerator is smaller than its denominator.ProperNumeratorDenominatorTypes of Fractions2-4Improper FractionsImproper fractions have a value equal to or greater than 1; its numerator is equal to or greater than its denominator.14, 7, 15, 2214 6 14 19DenominatorNumeratorTypes of Fractions2-51, 9, 7, 5, 96 10 8 6 11558 33Mixed NumbersMixed numbers are the sum of a whole number greater than zero and a proper fraction 139Converting Improper Fractions to Whole or Mixed Numbers2-62 Steps1. Divide the numerator of the improper fraction by the denominator1515 = 12. a. If you have no remainder, the quotient is a whole number 16 1 5 5 3 R 15 16 15 1 = 32 b. If you have a remainder, the quotient is a mixed number.The remainder is placed over the old denominator as the proper fraction of the mixed number.Converting Mixed Numbers to Improper Fractions2-73 Steps1. Multiply the denominator of the fraction by the whole number.(8 x 6) = 48 1 86(8 x 6) = 4848 + 1 = 492. Add the product from Step 1 to the numerator of the old fraction.49 83. Place the total from Step 2 over the denominator of the old fraction to get the improper fraction.Reducing Fractions to Lowest Terms by InspectionFind the lowest whole number that will divide evenly into the numerator and denominator.24 24 / 6 430 30 / 6 5==2-8Finding the Greatest Common DivisorStep 1. Divide the numerator into the denominator. 124 30 24 6 46 24 24 0Step 2. Divide the remainder in Step 1 into the divisor of Step 1.24 / 6 430 / 6 5= Step 3. Divide the remainder of Step 2 into the divisor of Step 2. Continue until the remainder is 0.2-92430Divisibility TestsLast digit is 2,4,6,812 614 7=Sum of the digits is divisible by 336 1269 233 + 6 = 9 / 3 = 36 + 9 = 15 / 3 = 5=2-10 35 7 40 8Last two digits can be divided by 4140 1(40)160 1(60)==Divisibility Tests2-11Last digit is 0 or 515 320 4 =The number is even and 3 will divide into the sum of the digits12 218 3=The last digit is 0 90 9100 10= 47 28 28 0Raising Fractions to Higher Terms When Denominator is Known2 StepsDivide the new denominator by the old denominator to get the common number that raises the fraction to higher terms. 4 = ? 7 282-124 x 4 = 1616282. Multiply the common number from Step 1 by the old numerator and place it as the new numerator over the new denominator.Adding Like FractionsAdd 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.1 4 57 7 7+=5 6 11 29 9 9 9 +==12-137 4221Least Common Denominator (LCD)2-14What is the least common denominator?The smallest nonzero whole number into which ALL denominators will divide evenly. 3 5 7 21+Adding Unlike Fractions4 Steps1. Find the LCD. 1 1 1 13 8 9 12+++2-1524 9 8 6 4772 72 72 72 72+++=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.Prime NumbersA prime number is a whole number greater than 1 that is only divisible by itself and 1. The number 1 is not a prime number.Examples2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 432-16Adding Mixed Numbers3 Steps1. 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. 7 720 203 125 201 54 20 24 4 20 20 4 20 4466 18= 1+ 7+7Step 1Step 3Step 2 4 + 6 + 7 = 1718 1 52-17Subtracting Like Fractions=2-18 8 / 2 410 / 2 5 9 1 8 10 10 10 -=2 StepsSubtract the numerators and place the total over the denominatorIf necessary, reduce the answer to lowest terms. Subtracting Unlike Fractions 5 2 8 64-Step 1. Find the LCD.2-19 5 40 8 64 =Step 2. Raise the fraction to its equivalent value. 40 2 38 64 64 64 -=Step 3. Subtract the numerators and place the answer over the LCD. 38 = 19 64 32Step 4. If necessary, reduce the answer to lowest terms.4 Steps 1 4 2 8 _ 3 3 8 8 1 8Subtracting Mixed NumbersStep 1. Subtract fractions, making sure to find the LCD. When Borrowing Is Not Necessary:2-20666Step 2. Subtract whole numbers.Step 3. Reduce the fractions to lowest terms.3 Steps1 2 34 2 6 4 4 3 3 4 4 3 4Subtracting Mixed NumbersStep 1. Make sure the fractions have the LCD.When Borrowing Is Necessary:3 1 -1 -122-213 -1Step 2. Borrow from the whole number. Step 3. Subtract whole numbers and fractions.Step 4. Reduce the fractions to lowest terms.4 StepsMultiplying Proper Fractions 20 10 42 21=Step 1. Multiply the numerator and the denominator.5 1 4 1 6 7 =xx2-22Step 2. Reduce the answer to lowest terms.2 StepsMultiplying Mixed NumbersConvert the mixed numbers to improper fractions. Multiply the numerator and denominators.1 1 7 3 7 13 2 3 2 2 223=X1X==112-23Reduce the answer to lowest terms.Dividing Proper FractionsInvert (turn upside down) the divisor (the second fraction).Multiply the fractions.1 2 1 3 38 3 8 2 16 ==X..2-24Reduce the answer to lowest terms.Multiply Mixed Numbers1. Convert all mixed numbers to improper fractions. 2. Invert the divisor and multiply.3 5 35 6 105 34 6 4 17 34 343. Reduce the answer to lowest terms.8=X2X==32-25Solution:LCD 601 + 1 + 1 = 15 + 12 + 10 = 374 5 6 60 60 60 60 Problem 2-382-26Seventy-seven million people were born between 1946 and 1964. The U.S. Census classifies this group of individuals as baby boomers. It is said that today, and every day for the next 18 years, 10,000 baby boomers will reach 65. If 1/4 of the 65 and older age group uses e-mail, 1/5 obtains the news from the Internet, and 1/6 searches the Internet, find the LCD and determine total technology usage for this age group as a fraction. LU 2-2(1, 2)Problem 2-46101111 4Solution:x5112=52=12 2hours2-27A trip to the White Mountains of New Hampshire from Boston will take you 2 and ¾ hours. Assume you have traveled 1/11 of the way. How much longer will the trip take? LU 2-3(1, 2)Problem 2-56750 / = 750 x 9 = 6,750 bags19Solution:2-28Albertsons grocery planned a big sale on apples and received 750 crates from the wholesale market. Albertsons will bag these apples in plastic. Each plastic bag holds 1/9 of a crate. If Albertsons has no loss to perishables, how many bags of apples can be prepared? LU 2-3(1)