Name: ______Date: ______
Lessons 6-2 and 7-4: Proportional and Nonproportional Relationships and Direct Variation
Key Vocabulary:
· Proportional – two quantities are proportional if they have a ______ratio or rate
· Nonproportional – relationships in which the ratios or rates are ______.
Proportional Nonproportional
How do I know which number is my x, and which number is my y?
1) x is usually the ______number in a ______table.
2) x is usually the number on the ______in a ______table.
3) x is the ______variable, and y is the ______variable.
* An independent variable stands alone and is not changed by the other variable.
* A dependent variable depends on the other variable. The independent variable causes a change in the
dependent variable.
Examples:
Determine whether the set of numbers in each of the tables below is proportional. Explain why or why not.
1) 2)
Coffee (pounds) / 1 / 2 / 3 / 4Cost (dollars) / 3 / 6 / 9 / 12
Time (hours) / 1 / 2 / 3 / 4
Distance (miles) / 50 / 70 / 90 / 110
3) 4)
Number of Spiders / 1 / 2 / 3 / 4Number of Legs / 8 / 16 / 24 / 32
5) 6)
Years / 1 / 2 / 3 / 4Population (100,000) / 1.3 / 2.1 / 3.3 / 5.2
7) 8)
Cups of Rice / 1 / 2 / 2.5 / 3Cups of Water / 1.5 / 3 / 3.75 / 4.5
Miles Driven / 1 / 2 / 6 / 9
Toll Fare / $1.07 / $1.14 / $1.42 / $1.63
For each of the following problems, complete the table. Then determine whether the pattern is proportional. If the relationship is proportional, write an equation.
9) Aliya earns $7 per hour babysitting for her neighbors.
10) Antonio’s Pizzeria charges $10 for a large pizza, plus $1.50 for each additional topping.
· Direct Variation – a type of linear equation that describes a constant rate of change and a proportional relationship; this type of graph always passes through the ______.
Equation: ______
k =constant of proportionality, which is also known as the ______.
What does a direct variation (proportional relationship) graph look like?
Proportional Nonproportional Proportional Nonproportional
Exercises:
Suppose y varies directly as x. For each of the following, write a direct variation equation. Then, use the direct variation equation to find the missing variable.
11) If y = 28 when x = 7, find x when y = 52. 12) If y = 6 when x = -18, find y when x = -2.
13) If y = 9 when x = -3, find x when y = 15. 14) If y = 2.5 when x = 0.5, find y when x = 20.
15) Suppose your pay varies directly as the number of hours you work. Your pay for 7.5 hours is $45.
a. Write a direct variation equation relating your b. Find your pay if you work 30 hours.
pay to the hours worked.
16) The cost of bananas varies directly with their weight. Miguel bought 3 ½ pounds of bananas for $1.12.
a. Write a direct variation equation relating the b. Find the cost of 4 ¼ pounds of bananas.
cost of the bananas to their weight.
17) The cost for 2 ½ pounds of ground beef is $7.20.
a. Write an equation relating cost to the number of pounds of ground beef.
b. How much is 4 pounds of ground beef?
Additional Examples
18) A school is repainting some of its classrooms. 19) On a cross-country road trip, a family drives
Each classroom is repainted with 5.5 gallons of 240 miles each day. Write and solve an
paint. Write and solve an equation to determine equation to determine how far the family has
the number of gallons of paint the school must traveled after 4 days.
purchase if they repaint18 classrooms.
20) The circumference of a circle with a diameter of 4 meters is about 12.56 meters. A circle’s circumference
is proportional to its diameter. y x
a. Write an equation relating the circumference of a circle to its diameter.
Equation: ______
b. What would be the circumference of a circle with a 6-inch diameter?
21) Sue burned 284 calories from walking 4 miles. 22) A store is having a sale where all jeans are
Write an equation relating the number of calories off the sale price. Write an equation
burned to the number of miles walked. How relating the sale price to the regular price.
many calories would she burn from walking 7 miles? How much would a pair of $35 jeans be on
sale?