How to Find Constant of Proportionality?

The value of the constant of proportionality depends on the type of relationship we have between the two quantities. In this step-by-step guide, you learn more about the constant of proportionality and how to find it.

How to Find Constant of Proportionality?

When two variables are directly or indirectly proportional to each other, then their relationship can be described as \(y=kx\),  where \(k\) is known as the constant of proportionality.

Step by step guide to finding constant of proportionality

The constant of proportionality is the constant value of the ratio between two proportional quantities. Two varying quantities are said to be in a relation of proportionality when their ratio or product yields a constant. The constant value of proportionality depends on the type of proportion between the two given quantities:

  1. Direct variation: The equation of direct proportionality is \(y = kx\), which shows that as \(x\) increases, \(y\) also increases at the same rate. Example: the cost of each item \((y)\) is directly proportional to the number of items \((x)\) purchased, expressed as \(y ∝ x\).
  2. Inverse variation: The indirect proportionality equation is \(y= \frac{k}{x}\), which shows that as \(y\) increases, \(x\) decreases and vice versa. Example: the speed of a moving vehicle \((y)\) inversely varies as the time taken \((x)\) to travel a certain distance, expressed as \(y ∝ \frac{1}{x}\).

In both cases, \(k\) is constant. The value of this constant is called the coefficient of proportionality. The constant of proportionality is also known as the unit rate.

Why do we use the constant of proportionality?

We use the proportionality constant in mathematics to calculate the rate of change, and at the same time determine whether the change is direct or inverse.

If we want to graph an image of the Taj Mahal by sitting in front of it on a piece of paper by looking at the real image in front of us, we must maintain a proportional relationship between the dimensions of length, height, and width of the building. We must identify the proportionality constant to achieve the desired result. Based on this, we can draw the monument with proportional measurements. For example, if the height of the dome is \(3\) meters, then in our drawing we can represent the same dome with a height of \(3\) inches. In the same way, we can draw other parts. In such scenarios, we use the proportionality constant.

Working with the proportional relationships allows a person to solve many real-life problems like:

  • Adjusting the ratio of ingredients in the recipe
  • Quantifying chance like finding odds and probability of events
  • Scaling a diagram for design and architectural applications
  • Finding percent increase or percent decrease for price mark-ups
  • Discount products based on unit rate

How to solve the constant of proportionality?

We apply our knowledge of direct and inverse changes, identify them, and then determine the constant of proportionality, thereby finding solutions to our problems.

For example: Find the constant of proportionality, if \(y=28\) and \(x=4\) and \(y ∝ x\).

Solution: 

We know that \(y\) varies proportionally with \(x\). We can write the equation of the proportional relationship as \(y = kx\). Substitute the given \(x\) and \(y\) values, and solve for \(k\).

\(28 = k (4)\)

\(k = 28 ÷ 4 = 7\)

Therefore, the constant of proportionality is \(7\).

Identifying the constant of proportionality

We will learn how to identify the constant of proportionality (unit rate) in tables or graphs. Check the table below and determine if the relationship is proportional and find the constant of proportionality.

We infer that as the number of days increases, so does the number of articles written. Here we recognize that it is a direct proportion. We apply the equation \(y= kx\). To find the proportionality constant, we determine the ratio between the number of articles and the number of days. We must evaluate \(k =\frac{y}{x}\)

\(k=\frac{y}{x} =\frac{3}{1}= \frac{9}{3}= \frac{15}{5}=\frac{18}{6} = 3\)

From the result of the \(y\) and \(x\) ratios for the given values, it can be seen that a value is obtained for all cases. The Constant of Proportionality is \(3\). 

If we plot the values from the table above, we see that the straight line passing through the origin shows a proportional relationship. The constant of proportionality under the direct proportion condition is the slope of the line when plotted for two proportional constants \(x\) and \(y\) on a graph.

Find Constant of Proportionality – Example 1:

Examine the given table and determine if the relationship is proportional. If yes, determine the constant of proportionality. 

Solution : 

First, find the ratio of \(x\) and \(y\) for all the given values. 

\(\frac{2 }{1} = 2\)

\(\frac{4}{2} = 2\)

\(\frac{8}{4} = 2\)

When we take the ratio of \(x\) and \(y\) for all the given values, we get equal values for all the ratios. So, the relationship given in the table is proportional.   

Exercises for Finding Constant of Proportionality

Examine each table and determine if the relationship is proportional.

This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{Not}\)
  2. \(\color{blue}{Yes}\)

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