? LESSN. Proportional and Nonproportional Situations ESSENTIAL QUESTIN How can ou distinguish between proportional and nonproportional situations? Distinguish Between Proportional and Nonproportional Situations Using a Graph If a relationship is nonlinear, it is nonproportional. If it is linear, it ma be either proportional or nonproportional. When the graph of the linear relationship contains the origin, the relationship is proportional. Proportionalit Distinguish between proportional and nonproportional situations using tables, graphs, and equations in the form = k and = m + b, where b 0. Math n the Spot m.hrw.com EXAMPLE 1 The graph shows the sales ta charged based on the amount spent at a video game store in a particular cit. Does the graph show a linear relationship? Is the relationship proportional or nonproportional? The graph shows a linear proportional relationship because it is a line that contains the origin. Sales ta ($) 6.0.8 3.6. 1. 0 0 60 80 100 Amount spent ($) Houghton Mifflin Harcourt Publishing Compan YUR TURN Determine if each of the following graphs represents a proportional or nonproportional relationship. 1. 10 8 6. 0 16 1 8 Math Talk Mathematical Processes What do the slope and the -intercept of the graph represent in this situation? 6 8 10 8 16 3 0 nline Assessment and Intervention m.hrw.com Lesson. 107
Math n the Spot m.hrw.com Distinguish Between Proportional and Nonproportional Situations Using an Equation If an equation is not a linear equation, it represents a nonproportional relationship. A linear equation of the form = m + b ma represent either a proportional (b = 0) or nonproportional (b 0) relationship. EXAMPLE The number of ears since Keith graduated from middle school can be represented b the equation = a - 1, where is the number of ears and a is his age. Is the relationship between the number of ears since Keith graduated and his age proportional or nonproportional? = a - 1 The equation is in the form = m + b, with a being used as the variable instead of. The value of m is 1, and the value of b is -1. Since b is not 0, the relationship between the number of ears since Keith graduated and his age is nonproportional. Reflect 3. Communicate Mathematical Ideas In a proportional relationship, the ratio _ is constant. Show that this ratio is not constant for the equation = a - 1.. What If? Suppose another equation represents Keith s age in months given his age in ears a. Is this relationship proportional? Eplain. YUR TURN Determine if each of the following equations represents a proportional or nonproportional relationship. 5. d = 65t 6. p = 0.1s + 000 Houghton Mifflin Harcourt Publishing Compan nline Assessment and Intervention m.hrw.com 7. n = 50-3p 8. 36 = 1d 108 Unit
Distinguish Between Proportional and Nonproportional Situations Using a Table If there is not a constant rate of change in the data displaed in a table, then the table represents a nonlinear nonproportional relationship. Math n the Spot m.hrw.com A linear relationship represented b a table is a proportional relationship when the quotient of each pair of numbers is constant. therwise, the linear relationship is nonproportional. EXAMPLE 3 Houghton Mifflin Harcourt Publishing Compan Image Credits: Jupiter Images/ Hemera Technologies/Gett Images The values in the table represent the numbers of U.S. dollars three tourists traded for Meican pesos. The relationship is linear. Is the relationship proportional or nonproportional? U.S. Dollars Traded Meican Pesos Received 130 1,690 55 3,315 505 6,565 1,690 130 = 169 13 = 13 3,315 55 = 1 17 = 13 6,565 505 = 1313 101 = 13 The ratio of pesos received to dollars traded is constant at 13 Meican pesos per U.S. dollar. This is a proportional relationship. YUR TURN Determine if the linear relationship represented b each table is a proportional or nonproportional relationship. 9. 30 8 90 1 150 Simplif the ratios to compare the pesos received to the dollars traded. 10. 5 1 0 8 65 13 Math Talk Mathematical Processes How could ou confirm that the values in the table have a linear relationship? nline Assessment and Intervention m.hrw.com Lesson. 109
Math n the Spot m.hrw.com Comparing Proportional and Nonproportional Situations You can use what ou have learned about proportional and nonproportional relationships to compare similar real-world situations that are given using different representations. EXAMPLE Math Talk Mathematical Processes How might graphing the equation for Arena A help ou to compare the situations? A A laser tag league has the choice of two arenas for a tournament. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Arena A Arena B = 5 Arena A s equation has the form = m + b, where b = 0. So, Arena A s charges are proportional. The hourl rate, $5, is greater than Arena B s, but there is no additional fee. Total cost ($) 500 00 300 00 100 0.5 1.0 1.5.0 Hours Arena B s graph is a line that does not include the origin. So, Arena B s charges are nonproportional. Arena B has a $50 initial fee but its hourl rate, $00, is lower. B Jessika is remodeling and has the choice of two painters. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Painter A = $5 Painter A s equation has the form = m + b, where b = 0. So, Painter A s charges are proportional. The hourl rate, $5, is greater than Painter B s, but there is no additional fee. Painter B 0 1 3 0 55 90 15 Painter B s table is a nonproportional relationship because the ratio of to is not constant. Because the table contains the ordered pair (0, 0), Painter B charges an initial fee of $0, but the hourl rate, $35, is less than Painter A s. Houghton Mifflin Harcourt Publishing Compan 110 Unit
YUR TURN 11. Compare and contrast the following two situations. Test-Prep Center A The cost for Test-Prep Center A is given b c = 0h, where c is the cost in dollars and h is the number of hours ou attend. Test-Prep Center B Test-Prep Center B charges $5 per hour to attend, but ou have a $100 coupon that ou can use to reduce the cost. nline Assessment and Intervention m.hrw.com Guided Practice Determine if each relationship is a proportional or nonproportional situation. Eplain our reasoning. (Eample 1, Eample, Eample ) 1. 50. 30 0 30 18 0 1 10 6 8 10 6 6 1 18 30 Houghton Mifflin Harcourt Publishing Compan Look at the origin. 3. q = p + 1_ Compare the equation with = m + b.. v = 1 10 u Lesson. 111
The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation. (Eample 3, Eample ) 5. 6. 3 1 9 36 6 8 1 8 58 10 Find the quotient of and. 7. The values in the table represent the numbers of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in another wa. (Eample ) Number of Households that Watched TV Show TV Show Rating 15,000,000 1 0,000,000 16 5,000,000 0? ESSENTIAL QUESTIN CHECK-IN 8. How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional linear relationships? Houghton Mifflin Harcourt Publishing Compan 11 Unit
Name Class Date. Independent Practice m.hrw.com nline Assessment and Intervention 9. The graph shows the weight of a cross-countr team s beverage cooler based on how much sports drink it contains. a. Is the relationship proportional or nonproportional? Eplain. b. Identif and interpret the slope and the -intercept. Weight (lb) 0 16 1 8 8 1 16 0 Sports drink (cups) For 10-11, tell if the relationship between a rider s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Eplain. 10. The elevator paused for 10 seconds after ou stepped on before beginning to rise at a constant rate of 8 feet per second. height above floor height above floor Houghton Mifflin Harcourt Publishing Compan 11. Your height, h, in feet above the first floor on the escalator is given b h = 0.75t, where t is the time in seconds. 1. Analze Relationships Compare and contrast the two graphs. Graph A = 1_ 3 6-6 - - Graph B = _ 6-6 - - Lesson. 113
13. Represent Real-World Problems Describe a real-world situation where the relationship is linear and nonproportional. FCUS N HIGHER RDER THINKING Work Area 1. Mathematical Reasoning Suppose ou know the slope of a linear relationship and one of the points that its graph passes through. How can ou determine if the relationship is proportional or nonproportional? 15. Multiple Representations An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and Kelvin, respectivel. Equation A F = _ 9 5 C + 3 Equation B K = C + 73.15 Table C Degrees Celsius kelvins 8 81.15 15 88.15 36 309.15 a. Is the relationship between kelvins and degrees Celsius proportional? Justif our answer in two different was. b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Wh or wh not? Houghton Mifflin Harcourt Publishing Compan 11 Unit