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Download Lesson 3.1 Properties of Multiplication and Division
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Lesson 3.1Properties of Multiplication and Objectives Division Solve multiplication and division equations. Rearrange formulas to solve for a given variable. Stephen is an electronics technician. He knows the current and resistance specifications for a particular computer circuit, but not the voltage. To find the voltage, he uses Ohm’s Law, which is: V I • R. In this equation, V represents the voltage, I represents the current, and R represents the electrical resistance. To find the voltage, Stephen can multiply the measured values of the current by the resistance. Suppose Stephen is able to measure the voltage V and the current I, but needs to calculate the resistance R of the circuit. How can he solve Ohm's Law for the resistance R? Multiplication and Division Properties The Commutative Property of Multiplication states that the order in which you multiply two numbers has no affect on the product. Commutative Property of Multiplication For all numbers a and b, ab ba. For example, 140 Chapter 3 Solving Equations 5•44•5 20 20 Associative Property of Multiplication For all numbers a, b, and c, The Associative Property of Multiplication states that the grouping of numbers a(bc)The (ab)c Associative Property of Multiplication states that you multiply has no affect on the product. of numbers you multiply has no affect on the product. The Identity Property of Multiplication states that any number Associative Property of Multiplication multiplied by 1 is the number itself. For numbers Property a, b, Associative andofc,Multiplication TheallAssociative states that the grouping The Property of Multiplication states that the grouping Associative Property of Multiplication of numbers you multiply has no affect on the product. of numbers you multiply no affect ona,the For has all numbers b, product. and c, a(bc) (ab)c. Identity Property of Multiplication a(bc) a,(ab)c For any number Associative Property of Multiplication Associative Property of Multiplication For example, 2 • (3 • 6) (2 • 3) • 6 a 1 a 1aa For all numbers c, Identity all numbers a, b, and and c, of Multiplication 2 For •a,18b, and 6 • 6The Property states that any 36 36 multiplied by 1 is the number itself. a(bc) (ab)c a(bc) (ab)c If the product of two numbers is 1, the numbered are called The Identity Property of Multiplication that any number multiplied by 1 Property of reciprocalsstates or multiplicative inverses. The Inverse The Identity Property of Multiplication states that any number Property of Multiplication that any is the number itself. The Identity Identity of Multiplication Multiplication states thatProperty a numberstates multiplied by number its inverse is 1. multiplied by 1 is multiplied the numberbyitself. 1 is the For number itself. any number a, Identity Property of Multiplication a 1 of a Multiplication and 1aa Inverse Property For Identity any number a, Property of Multiplication Identity Property of Multiplication For any non-zero real number a, there exists a real For any number a, a, a•1a andFor any 1 •number aIfthe a. product of two numbers is 1, the numbered are c 1 1 number such that a 1. a1a 1 a aand aor multiplicative aand 1 a reciprocals 1a a a inverses. The Inverse P states that a number multiplied by its in For example, 8 • 1 8 and 1 •Multiplication 88 The Closure Property of Multiplication and Division If the product of two numbers isof 1,two the numbered called If the product numbers isare 1, the numbered are calledstates that If the product of two numbers is 1,you the multiply numbers and are called reciprocals or the result will be a r when divide real numbers, reciprocals or multiplicative inverses. The Inverse Property of reciprocals or multiplicative inverses. The Inverse Property of Inverse Property of Multiplication multiplicative inverses. Thenumber. Inverse Property of Multiplication states that a Multiplication states that a number multiplied by its inverse is 1.number Multiplication states that aany number multiplied by its inverse is 1. exis For non-zero real a, there number multiplied by its inverse is 1. 1 1 number such that a 1. Property ofaMultiplicationa and Division Inverse Property ofClosure Multiplication Inverse Property of Multiplication Inverse Property of Multiplication For all numbers and b, the product a ba and For any non-zero real number a, there exists aa,real For any non-zero real For number a, there exists a real any non-zero realanumber there exists real the The Closure Property of Multiplication and Division a 1 1 1 are also real numbers. you ,that if multiply b a 0, number suchnumber that quotient a • 1when 1.1. such 1. and divide real numbers, the resul b a aa a number. 1 • Property 1 For example, 77 •Multiplication The Closure Property and Division states that The of Closure of Multiplication and Division states that 7 when you multiplywhen and divide real numbers, the result will be athe real you multiply and divide real numbers, result will beand a realDi Closure Multiplication The Closure Property of Multiplication and DivisionProperty states thatof when you number. number. For all numbers a and b, the product a b a multiply and divide real numbers, 10 the result will be a real number. 2 a 5 quotient , if b 0, are also real numbers b Closure Property of Multiplication Division Closure Property of Multiplication and Division and Division Closure Propertyand of Multiplication For allFor realallnumbers a •ab, b b and the numbersaFor aand and b,the theproduct and the a b and the allb,numbers a and the product 1product 7 • 1 a aalso7real 0, 0, are are also numbers. quotient and Division of Equality quotient b ,, ifif bbquotient , if real b 3.1 numbers. 0,The areMultiplication also real numbers. b 1 For example, 7 • 1 7 10 2 5 10 1 10 •75• 50 1 and 5 2 7 10 2 5 3.1 Properties of Multiplication and Division 141 3.1 The Multiplication and Division o Example 1 Identifying Properties Which property of multiplication is illustrated below? 5 • (2 • 8) (5 • 2) • 8 Solution Solve both sides of the equation. 5 • (2 • 8) (5 • 2) • 8 5 • 16 10 • 8 80 80 The grouping of numbers multiplied had no affect on the product. This illustrates the Associative Property of Multiplication. Ongoing Assessment Which property of multiplication does the equation 3 4 4 3 illustrate? Commutative Property of Multiplication Division Property of Equality The Division Property of Equality solves problems modeled by multiplication equations. Activity Solving Multiplication Equations 1 Start with the equation 5 • 3 15. 2 Divide each side by 5. 3 What is the result? 3 3 4 Start with the equation 5x 15. 5 Divide each side by 5. 6 What is the result? x 3 You have just solved the multiplication equation 5x 15 using one of the basic properties of equality. Division Property of Equality If each side of an equation is divided by the same non-zero number, the results are equal; that is, the two sides of the equation stay equal or balanced. 142 Chapter 3 Solving Equations two sides of the equation stay equal or balanced. E 2 Solving a Multiplication Problem EXAMPLE XAMPLE 2 Solving a Multiplication Problem EXAMPLE 2 Solving a Multiplication Problem A shipping clerk must ship several packages of che A shipping clerk must ship several packages of che Example 2 Solving Problem EXAMPLE 2 Solvinga aMultiplication Multiplication A Problem shipping clerkcontainer must shipholds several packagesIfof che laboratory. The 18 pounds. a pac laboratory. The container holds 18 pounds. If a pac laboratory. The container 18 pounds. a pac 3 pounds, many chemical will If each co A shipping clerkship mustseveral ship several packages ofhow chemicals to aholdspackets A shipping clerk must packages of chemicals to a laboratory. 3 pounds, how many chemical packets will each co 3 weighs pounds, many chemical laboratory. The18container 18 pounds. If3ahow packet weighs The container holds pounds. holds If a packet pounds, how many packets will each co S OLUTION 3 pounds, chemical packets will each container hold? chemical packetshow willmany each container hold?SOLUTION SOLUTION Solve the equation 3x 18, which models this pro Solve the equation 3x 18, which models this pro SOLUTION Solution Solve the equation 3x 18, which models this pro 3x 18 Given 3x problem. 18 Given the equation 3x 18, which this Solve Solve the equation 3x 18, which modelsmodels this problem. 3x 18 Given 3x 18 x 18 3 Division Property of E 18 Given 33 Division Property of E x 13 8 3x 18 Given 3 3 Division Property of E 3x 63 Simplify. 3 x 1 8 Division Property Equality xofof 6 Equality Simplify. Division Property x6 Simplify. 3 3 Each container can hold 6 chemical packets. Simplify. Each container can hold 6 chemical packets. x 66 Simplify. Each container can hold 6 chemical packets. Check the solution by substituting 6 for x in the ori Each container can hold chemical packets. Check the solution by substituting 6 for x in the ori Each container can6hold 6 chemical packets. Check theeach solution bythe substituting x insure the ori Simplify side of equation 6tofor make at Simplify each side of the equation to make sure a t each side ofequation. theSimplify equationeach to make sure a t CheckCheck the solution by substituting 6 for x6Simplify infor thex original equation. results. the solution by substituting in the original results. results. side ofSimplify the equation make sureequation a true statement results. eachto side of the to make sure a true statement 3x 18 3x 18 results. 3x 18 3 •3x6 18 3 • 6 18 3 •186 18 ✓ 3 3x • 6 18 18 18 18 ✓ 18 18 ✓ 3 •18 6 18 18 ✓ E 3 Solving a Multiplication Problem 18 18 ✓ EXAMPLE XAMPLE 3 Solving a Multiplication Problem EXAMPLE 3 Solving a Multiplication Problem Example 3 Solving a Multiplication ReferProblem to the equation for Ohm’s Law in opening pa Refer to the equation for Ohm’s Law in opening pa EXAMPLE 3 Solving a Multiplication Problem Refer to the equation for Ohm’s Law opening pa can Rebecca solve Ohm’s Law for theinresistance R can Rebecca solve Ohm’s for the resistance R Refer to the equation for Ohm’s Law in the opening paragraph of thisLaw lesson. can Rebecca solve Ohm’s Law for the resistance R Refer to the solve equation for Ohm’s in openingR? paragraph. How How can Stephen Ohm’s Law forLaw the resistance S can Rebecca solve Ohm’s Law for the resistance R? SOLUTION OLUTION SOLUTION Use the Division Property of Equality to divide bot Solution Use the Division Property of Equality to divide bot SOLUTION Use the Division equation by I. Property of Equality to divide bot equation by I. Use the Division Property of Equality equation by I. Use the Division Property of Equality to divide both sides of the VI•R to divide both sides of the equation by I. VI•R equation by I. V II •• RR V I • R V VI I •I R VI•R I I V I I V R V I•R VI R I R I I I V R I Ongoing Assessment 3.1 Solving Multiplicatio a. Solve the equation 4x 24. 6 3.1 Solving Multiplication Equations 3.1 Solving Multiplicatio 3.1 Solving Multiplicatio 149 b. Solve the equation 6x 21. Critical Thinking Why do you choose the coefficient of the variable as the divisor when you are solving a multiplication equation? The variable is isolated on one side of the equal sign. 3.1 Properties of Multiplication and Division 143 Example 4 Finding the Degrees in a Triangle A sheet metal worker is making a sign in the shape of an equilateral triangle. How many degrees are in each angle of the triangle? Solution It usually helps to draw a picture when solving geometry problems. x x x Equilateral Triangle The sum of the angle measures of a triangle is 180°. Let x represent each angle measure. Because an equilateral triangle has three angles with equal measures, you can write the following equation: 3x 180 x 60 SW7216/Cord Algebra Figure 4.12.TA Thus, each angle is 60°. Multiplication Property of Equality The basic property of equality for solving division equations is the Multiplication Property of Equality. Multiplication Property of Equality If each side of an equation is multiplied by the same number, the two sides of the equation stay equal or balanced. 144 Chapter 3 Solving Equations Example 5 Solving a Multiplication EEXAMPLE XAMPLE 5 5Solving Solving a Multiplication a Multiplication Problem ProblemProblem An auto assembly plant hasjust juststarted started aa night The plant operates with An Anauto auto assembly assembly plant plant hashas just started a night night shift.shift. The shift.plant The plant teams of 6teams people How many people are needed work night shift if operates operates with with teams of each. 6ofpeople 6 people each. each. How How many many people people aretoneeded arethe needed there are 7 teams? to towork work the the night night shift shift if there if there are 7are teams? 7 teams? EXAMPLE 5 Solving a Multiplication Problem Solving a Multiplication Problem OLUTION SSOLUTION Solution An auto assembly mbly plant has just started a night shift. The plantplant has just started a night shift. The plant t many t people are needed with that teams of 6 people each. How h teams of 6 people each. Howoperates many people are needed where The The equation equation that models models thisthis problem is 7 is 6, 6, 6,where where t is the ttotal isthethe total The equation that models thisproblem problem is t is total number of 7 to work the night shift if there are 7 teams? night shift if there are 7 teams?number number ofofpeople people on thethe night night shift. shift. To this solve Toproblem, solve this problem, this theuse the Property people on the on night shift. To solve useproblem, theuse Multiplication Multiplication Multiplication Property Property of Equality of Equality to isolate to isolate the variable. the variable. of Equality to isolate the variable. SOLUTION t t t 6 6 Given Given Given t The equation that 7 models n that models this problem is 7 6, where t is 7the total this problem is 7 6, where t is the total t t on the night shift. To solve this problem, use the number of7people 7 •76• 6 Multiplication Multiplication Property Property 7• • Multiplication Property of Equality eople on the night shift. To solve this problem, 7 7use the Multiplication on Property of Equality to isolate the variable.Property of Equality to isolate the variable. t 4242 Simplify Simplify Simplify. tt t 6 Given 6 Given 7 7 The Theauto autoauto assembly assembly plant plant willwill need 42 people 42 people to work the work night the shift. night shift. The assembly plant willneed need 42 people totowork the night shift. t t Multiplication Property 7 • 7 • 6 Multiplication Property • 7 • 6 7 7 EEXAMPLE XAMPLE 6 6Finding Finding thethe Speed Speed of aSpeed of Cara Carof a Car Example 6 Finding the t 42 Simplify t 42 Simplify A Atechnician technician at at anan automobile test test tracktrack is calibrating is calibrating a new a new A technician atautomobile an 3 3 The auto assembly plant will need 42 people to work the embly plant will need 42 people to work the night shift. hours). speedometer. speedometer. She She drives drives the the car car 45 miles 45 miles in 45in minutes 45 minutes ( night (shift. automobile test track 4 hours). 4 What What isthe thespeed speed of thethe car?car? isiscalibrating aofnew EXAMPLE 6 Finding the Speed of a Car Finding the Speed of a Car speedometer. She drives OLUTION OLUTION S S A technician atmiles an automobile test track is calibrating a new the cara45 in 45 at an automobile test track is calibrating new 3 speedometer. She drives the car 45the miles in 45 minutes ( 34(t). hours). hours). minutes (distance hour). What is . She drives the car 45 miles inTo 45find To find the thedistance (d ), (d multiply ), multiply rate the (r) rate and (r) the and time the time You(t). You 4 What is the speed of the car? the speed of the car? peed of the car? can canexpress expressthis this relationship relationship as the as equation the equation SOLUTION Solution d drt rt or or rt rt d d To find thetime distance (dvalues ), multiply theand rate (r) time (t). You Now Now substitute substitute the the values for d and d t into into equation thethe equation andtime solve. and istance (d), multiply the rate (r) and the You To find the(t). distance (dfor ), multiply thet the rateand (r) and the (t). solve. You can express can express this relationship as the equation his relationship as the equation this relationship as the equation rt rt d d Distance Distance Formula Formula d rt or rt d d rt or rt d. d rt or rt d 33 r r 45 45 Given Given 4 4 values for d and t into the equation and solve. Now substitute Now substitute the values for d and t into the equation and solve. ute the values for d and t into the equation andthe solve. 443 3 4 4 • • r •45• 45 Distance r Multiplication Multiplication Property Property Distance Formula Formula 4 d3 3 rt d Distance Formula 3 3 4 rt 3 3 r 60 60 Reciprocal Reciprocal Property; Property; Simplify. Simplify. r r 45 Given Given r 45 Given 4 4 The Thespeed speedof the carcar is is 60 miles miles per hour. per hour. 3the 4 of 4 60 3 4 • r • 45 Multiplication Propertyof Equality Multiplication Property • r • 45 Multiplication Property 3 4 3 4 3 r 60 ReciprocalProperty; Property;Simplify. Simplify. Reciprocal r 60 Reciprocal Property; Simplify. of theofcar 60ismiles per hour. The speed theiscar 60 miles per hour. the car is 60 miles per hour. The speed 3.13.1TheThe Multiplication Multiplication Division and Division Equality of Equality 151 151 3.1 and Properties ofofMultiplication and Division 145 LLESSON A LLLESSON AA ASSESSMENT ESSON SSESSMENT Lesson Assessment ASSESSMENT ESSON ESSON SSESSMENT SSESSMENT Think and Discuss see Think Think and and Discuss Discuss see see seemargin margin margin see margin margin Think and Discuss Think and Think Discuss and Discuss see margin see margin 111 How is the Division Property of Equality toaasolve How is isHow the the Division Property Property of of Equality Equality used usedused to toused solve solve 1. 111How isDivision the Division Property of Equality to solve a aa 1 How is How the Division is the Division Property Property of Equality of Equality used toused solve toa solve multiplication equation? multiplication multiplication equation? equation? multiplication equation? multiplication multiplication equation? equation? 222 How is the Identity Property of Multiplication in solving How is isHow the the Identity Property Property of of Multiplication Multiplication used usedused in inused solving solving 2. 222How isIdentity the Identity Property of Multiplication in solving a 2 How is How the Identity is the Identity Property Property of Multiplication of Multiplication used inused solving in solving 333 3 444 4 a multiplication equation? aa multiplication multiplication equation? equation? multiplication equation? a multiplication a multiplication equation? equation? is the Multiplication Property a division How How is isHow the the Multiplication Property Property used usedofto toused solve solvetoaasolve division division 3. 333How isMultiplication the Multiplication Property to solve a How is How the Multiplication is the Multiplication Property Property used toEquality used solvetoaused solve division a division equation? Give an example. equation? equation? Give Give an an example. example. division equation? Give an example. equation? equation? Give anGive example. an example. it necessary toyour check your answer a problem after Why Why is is Why itit necessary necessary to to check check your answer answer to to aa problem problem after afterafter 4. 444Why itis to check your answer ato you Why is Why it is necessary isnecessary it necessary to check toyour check answer your answer to a to problem toproblem a problem after after youthe solve the equation? you you solve solve the equation? equation? solve the equation? you solve youthe solve equation? the equation? Practice and Problem Solving Practice Practice and and Problem Problem Solving Solving Practice and Problem Solving Practice Practice and Problem and Problem Solving Solving State the property is illustrated. State State the the property property that that is is that illustrated. illustrated. 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Solve forequation the indicated variable. Give a reason for each step. 34. Solve for i in the equation V ir. A 33. Solve for L in the equation A LW. L W 35. Solve for p in the equation i prt.V 34. Solve for i in the equation V ir. i r For each problem, write and solve an equation. Check your answer. i 35. Solve for p in the equation i prt. p rt 36. How many hours does it take a jetproblem, to fly 729 write miles and at a rate ofan equation. Check For each solve 485 miles per hour? 485t 729; 1.5 hours your answer. 36. How many hours does it take a jet to fly 729 miles at a 37. A brick mason is building a patio in rate of 485 miles per hour? 485m 729; 1.5 hours the shape of a parallelogram. Paul needs the area of the patio toinbethe shape of a 37. A brick mason is building a patio 180 square If the parallelogram. Paulfeet. needs the base area of the patio to be patiofeet. is 12Iffeet, whatof is the the patio height? 180 square the base is 12 feet, what is the 12x 180; 15 feet height? 12x 180; 15 feet sum of the angles a square is 360°. Each of the angles in a 38.38. TheThe sum of the angles in ainsquare is 360°. Each of the square has has the same measure. Show thatthat eacheach angle of a square angles in a square the same measure. Show 90°. 4x 360°; angle ofmeasures a square measures 90°.x 4x90° 360°; x 90° Latisha saved $230 a new compact stereo. 39.39. Latisha hashas saved $230 for for a new compact stereo. So So far,far, Latisha 2 Latishahas has 3 of the amount she needs to buy the stereo. What is the of theofstereo? What iscost the cost the stereo? 2x 230; $345 3 Twenty gallons of paint cost a painting contractor $170. 40.40. Twenty gallons of paint cost a painting contractor $170. a. Let a represent the cost of one gallon. Write a multiplication a. Let a represent the cost of one gallon. Write a equation relating the cost of 20 gallons of paint to the multiplication equation relating the cost of 20 gallons of paint total cost. 20a 170 to the total cost. 20a total costUse 170 b. the equation you wrote in part a to find the cost of one b. Use the gallon equation you wrote part a to find$8.50 the cost of of paint to theinnearest cent. one gallon of paint to the nearest cent. $8.50 Mixed ReviewMixed Review distance from Tom’s apartment to the library 41.41. TheThe distance from Tom’s apartment to the library is is 2.4 kilometers. How far How is thisfar distance in meters? 2,400 2.4 kilometers. is this distance in meters? 2,400 meters A carton of butter weighs 2.25 kilograms. What is this weight 42.42. A carton of butter weighs 2.25 kilograms. What is this weight in in grams? grams? 2,250 2,250 grams A container holds milliliters of water. How many liters does 43.43. A container holds 520520 milliliters of water. How many the glass hold? 0.52 liters does the glass hold? 0.52 liters Paul weighs 90,250 grams. What is Paul’s weight 44.44. Paul weighs 90,250 grams. What is Paul’s weight in in kilograms? 90.25 kilograms? 90.25 kilograms 3.1 Properties of Multiplication and Division 147 Chapter 3 Solving Equations
