4.1 Forms of Proportional Relationships

Learning Objectives:

You will understand relationships that are proportional and nonproportional using the formula y= kx.
You will understand that a proportional relationship can be graphed with a line through the origin whose slope is the unit rate.

Introduction

Manuel was reading all of the medieval books on knights. Well after he finished reading the series, he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.

Rafael had finished 9 of the 12 books. It took Manuel three - fourths of the books in the same time. Are Rafael and Manuel reading at the same rate?

To figure this out, you will need to write two proportions and figure out if they are equal.

Guided Learning

A ratio represents a comparison between two quantities. We can write ratios in fraction form, using a colon or using the word “to”.

We also learned that equivalent ratios are two ratios that are equal. The numbers in the ratios may not be the same, but the comparison of quantities is the same.

Equivalent ratios are directly related to proportions.

What is a proportion?

A proportion states that two ratios are equivalent. Here is an example of a proportion.

$\frac{1}{2} = \frac{2}{4}$

This proportion shows that the ratios $\frac{1}{2}$ and $\frac{2}{4}$ are equivalent. In other words, a proportion is made up of two equivalent ratios.

In the situation above, we knew all of the parts of the two ratios that made up the proportion. Sometimes, we will know three of the numbers, but not four of them. When this happens, we have to use a variable and solve for the missing number.

Look at this proportion.

$\frac{1}{2} = \frac{n}{12}$

Notice that the first term of the second ratio––its numerator––is a variable. Suppose we wanted to find the value of this variable. We could do that by using proportional reasoning.

Proportional reasoning is when we figure out a missing value in a proportion by thinking about the relationship between the numbers in the two ratios.

Example A

Use proportional reasoning to solve for $n: \ \frac{1}{2} = \frac{n}{12}$ .

To figure this out, we need to figure out a relationship between either numerators or denominators. The proportion does not show the relationship between the first terms in the ratios––the numerators of the fractions. However, we can determine the relationship between the second terms in the ratios––the denominators of the fractions.

We can ask ourselves: “What number, when multiplied by 2, results in 12?”

Since $2 \times 6 = 12$ , we can multiply both the numerator and the denominator of $\frac{1}{2}$ by 6 to find the value of .

$\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{n}{12}$

This shows that when the second term (the denominator) of the ratio is 12, the first term (the numerator) is 6.

The value of is 6.

Example B

Introduction to Proportions

When we write proportions, we have to be comparing items with the same units. Let’s try one together!

Each spring you plant 60 carrot seeds in your vegetable garden. Not all of the carrots survive until the end of the season because they are eaten by a rabbit that also lives in your garden. Therefore, by the end of the season you usually harvest around 45 carrots for eating. The following year you notice that the rabbit in your garden has had babies. Do you think the amount of carrots will increase or decrease with the increased number of rabbits in your garden?

One week you go to the store and buy three apples. The following week you go to the store and buy eight apples. Assuming that the prices of the items remain the same for both weeks, which week do you expect to spend more money on groceries? How do you know this is true?

For linear equations, those with the form, y = kx, the y and x variables are said to be directly proportional to one another, and k is a proportionality constant which has the units of y divided by the units of x. For example, if x is the pounds of beef in a package at the butcher shop and y is the cost of the package in dollars, then k is the cost per pound. This equation shows that as the value of x increases, the value of y also increases. For each unit change in x, the value of y will change by k. If y is graphed against x, the straight line will have a slope of k. If we take five packages of beef from the meat compartment and plot the cost (y-axis) versus the weight (x-axis), then all of the five points should fall on a straight line with the slope being the cost per pound as shown in the plot below.

Weight (lbs)	Cost (dollars)
0.5	3.495
1	6.99
3	20.97
4.5	31.455
6	41.94

Example C

Engineers involved in the design of irrigation systems need to know the relationship between evaporation rates (E in mm/day) and air temperature (T, °C). Assume that for a given geographic location, the following equation applies:

E = 0.25 T

What does this equation indicate? If E were plotted against T, what type of plot would result?

The equation indicated that values of E can be estimated for given values of T. The equation also indicated that E changes by 0.25 mm/day for each 1°C change in T. Therefore, E would increase with an increase in T, which makes these two variables DIRECTLY proportional. Since E has units of mm/day and T has units of °C, then the constant, 0.25, has units of mm/(day-°C) or mm/day/°C. At a temperature of 25°C, the evaporation rate is 6.25 mm/day while at 30°C, the rate is 7.5 mm/day. Because Equation 1 is linear, it appears as a straight line graph as shown in the plot below. Evaporation will change by (7.5-6.25) = 1.25 mm/day for 5°C change in temperature, which corresponds to change of 0.25 mm/day for each unit change in T.

Temperature (oC)	Evaporation Rate (mm/day)
20	5
25	6.25
30	7.5
35	8.75

Another way of looking looking at this is by solving the equation y = kx for the proportionality constant k. This gives the equation k = y/x. Since you know that k is a constant, which means that it will always be the same number, you can see that in order for the fraction y/x to stay the same you would either have to increase y with an increase in x or decrease y with a decrease in x.

Example D

If k = 6 in the equation y = k, then, y = 6x. Solve this equation for the k value and find x and y values that makes this equation true. Then find another set of x and y values that also make this equation true. What do you notice about the relationship between the two sets of x and y values?

y= 6x goes to 6= y/x

If y equals 6 and x equals 1 then this equation would be true: 6= 6/1

If y equals 12 and x equals 2 then this equation would also be true: 6 = 12/2

Therefore, the relationship between these two equations is that as the y value increases from 6 in the first equation to 12 in the second equation, the x value also increases from 1 in the first equation to 2 in the second equation. The fact that one variable increases with the increase of the other variable is what makes the x and y values DIRECTLY proportional.

Practice Set:

The cost of land is $7,500 per acre. Write an equation that shows the total cost of the land (C) for a given acreage (A). Is the relationship between the total cost and the area of a piece of property directly proportional? Explain why.

Example E

Inversely Proportional Relationships

For equations of the form y = k/x, the y and x variables are said to be inversely proportional to one another, and k is again a proportionality constant. This means that as the value of x increases, the value of y decreases. If y is graphed against x, the slope is not constant; the graph will be curved with a negative slope. This equation is sometimes written as y = kx -1 where the -1 exponent indicates that y and x are inversely proportional.

Ecologists are interested in relationship between the size of fish (and other aquatic life) versus the population density of the fish i.e., the number of fish per unit volume of the pond. As the number of fish increases, the food supply per fish decreases and so the average size of the fish decreases. In one study, the maximum length of one-year old haddock (L in cm) was related to a dimensionless density index (D) by: L = 122/D for 2 < D < 5

What does this relationship indicate?

At a density index of 2, the average length is expected to be 61 cm. At a density index if 5, and average length of 24.4 cm is expected. Limits of 2 and 5 are placed on the index because the equation might give unrealistic lengths for values of D less than 2 or greater than 5.

Another way of looking at this is by solving the equation y = k/x for the proportionality constant k. This will give you the equation k = yx. Since you know that k is a constant, which means that it will always be the same number, you can see that for the value of k to stay the same, the value of y must increase when the value of x decreases. The opposite is also true, namely that y will decrease when x is increased.

Example F

If the value of k = 6 in the equation y = k/x, then, y = 6/x. Solve this equation for the k value and find x and y values that makes this equation true. Then find another set of x and y values that also make this equation true. What do you notice about the relationship between the two sets of x and y values?

y = 6/x goes to 6 = yx.

If y equals 6 and x equals 1 then this equation would be true: 6 = 6×1

If y equals 3 and x equals 2 then this equation would also be true: 6 = 3×2

Therefore, the relationship between these two equations is that, as the y value decreases from 6 in the first equation to 3 in the second equation, the x value increases from 1 in the first equation to 2 in the second equation. The fact that one variable increases with the decrease of the other variable is what makes the x and y values INVERSELY proportional.

Lesson Review

If an equation shows a directly proportional relationship, the x and y values will be directly proportional.
A proportion must have an equal sign between the comparison of two ratios.

Direct Variation

Direct variation is when one variable changes the other changes in proportion to the first.

Inversely Proportional

Inversely proportional relationships are in the form y=k/x. As the one of the values increases, the

other decreases.

Proportional

A proportion is a statement in which two ratios are equal.

Ratio

A ratio represents a comparison between two quantities. We can write ratios in fraction form, using

a colon or using the word "to."

Equivalent Ratio

Equivalent ratios are two ratios that are equal.

Proportional Relationship

A proportional relationship between two quantities is one in which the two quantities vary directly

with one another. If one item is doubled, the other, related item is also doubled.

Video Resources

Introduction to Proportions