Lesson 3.1 Objectives Solve multiplication and division equations. Rearrange formulas to solve for a given variable. Properties of Multiplication and Division 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, 5 4 4 5 20 20 140 Chapter 3 Solving Equations
The Associative Property of Multiplication states that the grouping of numbers you multiply has no affect on the product. Associative Property of Multiplication For all numbers a, b, and c, a(bc) (ab)c. For example, 2 (3 6) (2 3) 6 2 18 6 6 36 36 The Identity Property of Multiplication states that any number multiplied by 1 is the number itself. Identity Property of Multiplication For any number a, a 1 a and 1 a a. For example, 8 1 8 and 1 8 8 If the product of two numbers is 1, the numbers are called reciprocals or multiplicative inverses. The Inverse Property of Multiplication states that a number multiplied by its inverse is 1. Inverse Property of Multiplication For any non-zero real number a, there exists a real number 1 a such that a 1 a 1. For example, 7 1 7 1 The Closure Property of Multiplication and Division states that when you multiply and divide real numbers, the result will be a real number. Closure Property of Multiplication and Division For all real numbers a and b, the product a b and the quotient b a, if b 0, are also real numbers. For example, 10 5 50 and 1 0 2 5 3.1 Properties of Multiplication and Division 141
Example 1 Identifying Properties Which property of multiplication is illustrated below? 5 (2 8) (5 2) 8 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
Example 2 Solving a Multiplication Problem A shipping clerk must ship several packages of chemicals to a laboratory. The container holds 18 pounds. If a packet weighs 3 pounds, how many chemical packets will each container hold? Solve the equation 3x 18, which models this problem. 3x 18 Given 3 x 1 8 Division Property of Equality 3 3 x 6 Simplify. Each container can hold 6 chemical packets. Check the solution by substituting 6 for x in the original equation. Simplify each side of the equation to make sure a true statement results. 3x 18 3 6 18 18 18 Example 3 Solving a Multiplication Problem Refer to the equation for Ohm s Law in the opening paragraph of this lesson. How can Stephen solve Ohm s Law for the resistance R? Use the Division Property of Equality to divide both sides of the equation by I. Ongoing Assessment V I R V I I R I V I R a. Solve the equation 4x 24. 6 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? 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 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 Problem An auto assembly plant has just started a night shift. The plant operates with teams of 6 people each. How many people are needed to work the night shift if there are 7 teams? t The equation that models this problem is 7 6, where t is the total number of people on the night shift. To solve this problem, olv use the Multiplication Property of Equality to isolate the variable. 7 t 6 7 7 t 7 6 t 42 Given Multiplication Property of Equality Simplify. The auto assembly plant will need 42 people to work the night shift. Example 6 Finding the Speed of a Car A technician at an automobile test track is calibrating a new speedometer. She drives the car 45 new miles in 45 minutes ( 3 hour). What is 4 the speed of the car? To find the distance (d), multiply the rate (r) and the time (t). You can express this relationship as the equation d rt or rt d. Now substitute the values for d and t into the equation and solve. rt d r 3 4 45 4 3 3 4 r 4 3 45 r 60 Distance Formula Given Multiplication Property of Equality Reciprocal Property; Simplify. The speed of the car is 60 miles per hour. 3.1 Properties of Multiplication and Division 145
Lesson Assessment Think and Discuss 1. How is the Division Property of Equality used to solve a multiplication equation? 2. How is the Identity Property of Multiplication used in solving a multiplication equation? 3. How is the Multiplication Property of Equality used to solve a division equation? Give an example. 4. Why is it necessary to check your answer to a problem after you solve the equation? Practice and Problem Solving State te the the property that is illustrated. ustrated. 1 5. 365 3 65 1 Inverse Prop. 6. 1 2 2 3 1 3 Closure Prop. of Multiplication verse Prop. of Multi. of Multiplication Closure Prop 7. 1 8. 5 ( 7 3) (5 7) 3 Identity Prop. of Multiplication. Associative Prop. of Multiplication Solve each equation. Check the solution. 9. 8c 64 c 8 10. 8c 64 c 8 11. 8c 64 c 8 12. 8c 64 c 8 m 13. 26 m 52 14. 5 2 n 45 n 63 7 15. 15t 140 16. 8 k 40 k 25 5 3 17. 2.5y 20 y 8 18. 3 a 15 a 25 5 19. 8 12b 20. 7.5 2.5d d 3 21. 72 3.6d d 20 22. 10q 5 23. 54 9 y y 24 24. 0.3f 9 f 30 4 b 25. 7 b 84 26. 5r 27 1 2 27. 12m 80 28. 125 5y y 25 29. 9x 66 30. 9p 87 31. 80 6z 32. 36 1 2 3 d d 21 3 5 146 Chapter 3 Solving Equations
Solve each equation for the indicated variable. 33. Solve for L in the equation A LW. 34. Solve for i in the equation V ir. 35. Solve for p in the equation i prt. For each problem, write and solve an equation. Check your answer. 36. How many hours does it take a jet to fly 729 miles at a rate of 485 miles per hour? 485t 729; 1.5 hours 37. A brick mason is building a patio in the shape of a parallelogram. Paul needs the area of the patio to be 180 square feet. If the base of the patio is 12 feet, what is the height? 12x 180; 15 feet 38. The sum of the angles in a square is 360. Each of the angles in a square has the same measure. Show that each angle of a square measures 90. 4x 360 ; x 90 39. Latisha a has saved $230 for a new compact stereo. So far, Latisha has 2 3 of the amount she needs to buy the stereo. What is the cost e co of the stereo? 40. Twenty gallons of paint cost a painting contractor $170. a. Let a represent the cost of one gallon. Write a multiplication equation relating the cost of 20 gallons of paint to the total cost. 20a 170 b. Use the equation you wrote in part a to find the cost of one gallon of paint to the nearest cent. $8.50 Mixed Review 41. The distance from Tom s apartment to the library is 2.4 kilometers. How far is this distance in meters? 2,400 42. A carton of butter weighs 2.25 kilograms. What is this weight in grams? 2,250 43. A container holds 520 milliliters of water. How many liters does the glass hold? 0.52 44. Paul weighs 90,250 grams. What is Paul s weight in kilograms? 90.25 3.1 Properties of Multiplication and Division 147