Chapter 11.  Proportional Models

Most problems in real life involve simple proportional relationships.  In fact, you may find it surprising how easy it is to handle rates, percents, prices, probabilities, ratios, “fractions of”, and proportions when we use Unified Math®.  In proportional relationships, the expression in the Relationship column results in the same division as in the meaning of the constant of proportionality.  Keep in mind that the even the physical world around us is highly proportional as illustrated by the Newton Laws of Motion, Ideal Gas Laws, Weights on a Balance, etc.

In the above table, we identify (Step 1) the quantities (x, 50, and 150) and measure (Step 2) them by assigning meaning expressions. 

Formulate:  50 = 150/x

Solve: 

50*x = 150

x = 3

We relate (Step 3) the quantities by observing that the meaning “mi/hr” in the first row involves proportionality and we divide the “mi” in the meaning of quantity q3  by the “hr” in the meaning of quantity q1.

We formulate (Step 4) an equation from the first line in the table by equating the expression in the Quantity column with the expression in the Relationship column:

            50 = 150 / x          or rearranged as         50 x = 150

We then solve (Step 5) this equation for the variable, x = 3.

Formulate:  x/100 = 40/200

Solve:  x = 80

The problem asks for “what percent” so the variable x combines with 100 in the value column to create a “x per 100” or in other words “x percent”.    This results in the formula:  x/100 = 40/200  where we solve and get x = 20.

Formulate:  0.25 = 50 / x

Solve: 

0.25 * x = 50

x = 200

Formulate:  p = 300 / 1000

Solve:  x = .3

The ration of 2 to 5 parts (sometimes written as 2:5) in this problem indicates a proportional relationship between the amount of vinegar to the amount of oil.  The constant of proportionality comes from dividing the numbers in the ratio.  When we construct the meaning of the constant of proportionality we must put the meaning of the numerator number in the numerator of the meaning, and similarly for the denominator.  On constructing the expression in the relationship column, we again make sure that the meanings line up (the x in the numerator corresponds to its meaning  [oz~Volume(Vinegar)] in the numerator and the 20 in the denominator corresponds to its meaning [oz~Volume(Oil)] in the denominator.

Formulate:  2/5 = x / 20 

Solve:  x = (2/5)*20 = 8

The fraction 2/7 indicates a proportional relationship between the instances of contracts and the instances of appliances.  This fraction determines the constant of proportionality with a meaning having the numerator 2 associated with "contracts" and the 7 associated with "appliances".  On constructing the expression in the relationship column, we again make sure that the meanings line up with the x in the numerator corresponds to its meaning "contracts" in the numerator and the 140 in the denominator corresponds to its meaning "appliances" in the denominator.

Formulate:  2/7 = x / 140 

Solve:  x = (2/7)*140 = 40

Formulate:  2/3  = 48 /  x

Solve: 

2/3 * x  = 48 

x = 72

The meaning of 2 is not just "miles", but it's definition involves the 3.  Similarly, the meaning of 3 depends on 2.  The words "for every" mean the same as "per" so this reduces to a rate-type of problem. 

The characters depicted in red indicate some abbreviations that would still completely determine the model.  The user can use any abbreviations they desire as long as they remain consistent and avoid ambiguity.

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Copyright © 2004  Dr. Ranel E. Erickson