Proportional Relationships (situations)

Proportional Relationships (situations)

Recall: A proportion is an equality between two ratios or two rates. If the ratio of a to b is equal to the ratio of c to d, then...

The following situations are proportional: Proportions

Recall: extremes and means A proportion consists of four terms: o The 1st and 4th terms are the extremes o The 2nd and 3rd terms are the means

Proportional Situations Situations involving two variables whose values result in equivalent ratios or rates are called proportional situations. Generally, there are two visuals that we use to represent proportional situations: 1) Table of Values 2) Graphs

Are They Proportional? - Using the cross-products method Ex: The cross-products can be used to check for proportionality. If Extreme x Extreme = Mean x Mean, then the situation is proportional. 3 = 15 4 x 15 = 60 4 20 3 x 20 = 60 So! This situation is proportional!

Direct proportionality A situation that involves equivalent (equal) ratios or rates is called a direct proportional situation. In the table of values of a direct proportional situation, the numbers in the first row (or column) and the second row (or column) form sequences of proportional numbers. Ex: table of values x 0 5 10 20 40 y 0 37.5 75 150 300 x 7.5 x y 0 0 5 37.5 10 75 20 150 40 300 x 7.5

Proportions using a Table of Values In a table of values where x is the first variable and y is the second variable: o The values of y are obtained by multiplying the values of x by a same number, known as the proportionality coefficient. o You get the proportionality coefficient by dividing the values of y / x in the table of values. x 0 5 10 20 40 y 0 37.5 75 150 300 x 7.5 Proportionality coefficient = 37.5 / 5 = 7.5

Proportions using a Table of Values o In the table of values, the x:y ratio is constant and is known as the proportionality ratio. o If one of the variables is zero, then so is the other. x 4 x 0 2 3 5 8 x y 0 8 12 20 32 4x Proportionality Ratio 1:4 1:4 1:4 1:4 1:4 To determine y, we simply multiply each x value by the proportionality coefficient, which, in this case is 4.

Graphing direct proportionality

Graphing direct proportionality A direct proportional situation is represented graphically in the Cartesian plane by a straight line passing through the origin. The rule for a direct proportional situation is: y = ax where a represents the coefficient of proportionality. origin 0

Practice: direct proportionality A physical fitness club charges a monthly membership rate. Number of months 1 2 3 4 5 6 Cost ($) 32 64 96 128 160 192 a) Is the rate cost ($) number of months constant? b) Find the proportionality coefficient. d) If x represents the number of months and y the cost, find the rule of this situation.

Number of months 1 2 3 4 5 6 Cost ($) 32 64 96 128 160 192 Graph this situation on the Cartesian plane:

Proportions using Graphs

Inversely Proportional Situations In an inverse proportional situation, the product (multiplication) of the independent variable and the dependent variable remains constant. In the table of values of an inversely proportional situation, the product xy is constant. An inverse proportional situation is represented graphically by a curve that gradually approaches the axes. When x increases, y decreases; when x decreases, y increases. The rule of an inverse proportional situation is y = a x

Examples of inverse situations: Inversely Proportional Situations ospeed and travel time: the faster you go, the shorter the time opainting and time: As the number of painters increases, time remaining goes down obrightness and distance: the further your are from the light, the less bright it becomes

Ex: Table of Values of an inverse situation Inversely Proportional Situations x y xy 1 50 50 2 25 50 4 12.5 50 50 1 50 xy x times y xy remains constant!!

Ex: Graph for an inversely proportional relationship y = 1/x Inversely where a = 1 Proportional Situations

Practice: inverse situation An architect decides to hire employees to re-do the sidewalk of a public park. This job will require 10 hours of work. The table below shows you the time required to complete the sidewalk and the number of employees hired. Number of employees 1 5 10 20 40 50 100 Time (h) 100 20 10 5 2.5 2 1

Practice: inverse situation cont. a) Is this a direct proportional situation? Justify your answer: b) Complete the following sentence: As the number of employees increases, the time required to complete the sidewalk. c) What do you know about the product of the variable x with the variable y? e) Determine the rule of this situation:

Practice: inverse situation cont. Graph this situation in the Cartesian plane: