 ### Constant of Proportionality: How to Solve it?

The relationship between two variables can be expressed as y = kx or y = k/x if both variables are directly or indirectly proportional to one another. In this case, k determines how the two variables are related. It is known as the constant of proportionality.

## What is the Constant of Proportionality?

A constant of proportionality is the ratio between two proportional quantities. Two varying quantities are said to be in a relation of proportionality when either their ratio or their product yields a constant. Depending on the type of proportion between the two given quantities: direct variation or inverse variation, the constant of proportionality has a different value.

• Direct Variation: The equation for direct proportionality is y = kx, which shows as x increases, y also increases at the same rate. Example: The cost per item(y) is directly related to the number of items(x) purchased, y ∝ x.
• Inverse Variation: The indirect proportionality equation is y = k/x, which indicates that as y increases, x decreases and vice versa. Example: The speed of a moving vehicle (y) inversely varies with the amount of time necessary (x) to cover a certain distance, expressed as y ∝ 1/x

In both cases, k is constant. This constant is referred to as the coefficient of proportionality. The unit rate is also known as this constant.

## Why Do We Use The Constant of Proportionality?

In mathematics, the constant of proportionality is used to calculate the rate of change. We also determine if it is a direct variation or inverse variation that we are dealing with. Let’s say the cost of 2 apples is \$20. That means the cost of 1 apple is \$10. The constant of proportionality for the price of an apple is 2.

Sitting in front of the Taj Mahal and drawing a picture of it on paper, we should maintain a proportional relationship between the building’s length, height, and width. To accomplish this, we must identify the constant of proportionality. Then we can draw the monument with proportional measurements. For instance, if the height of the dome is 2 meters then in our drawing we can represent the same dome with a height of 2 inches. Similarly, we can draw other parts. In such scenarios, we use the constant of proportionality.

The use of proportional relationships allows one to solve many real-life problems, such as:

1. Adjusting a recipe’s ratio of ingredients
2. Based on odds and probabilities
3. For drafting and architectural purposes
4. Calculating price mark-ups
5. Discounts on products based on unit rate

## How to Solve The Constant of Proportionality?

In order to solve our problems, we apply our knowledge of the direct and inverse variations, identify them, and then determine the constant of proportionality.

Example 1: Find the constant of proportionality for y=24 and x=3 and y ∝ x.

Solution: Y varies proportionally with x. Y = Kx is the equation of the proportional relationship. Solve for k using the given x and y values.

24 = k (3)

k = 24 ÷ 3 = 8

Therefore, the constant of proportionality is 8.

Example 2: It takes four workers three hours to complete the task. When 2 more workers are hired, how long will it take them to complete the work?

Solution: Let x1 equal the number of workers in case 1 = 4

x2 = No. of workers in case 2 = 6

y1 = number of hours in case 1 = 3

Number of hours in case 2 = To be found

The completion time will decrease if the number of workers is increased. We find that the number of workers is inversely proportional to the time taken, (y1 = k/x1) ⇒ 3 = k  / 4⇒ k = 12

Again, to find the number of hours, (y2 = k/x2) ⇒ y2 = 12/6 =  2 hours.

## FAQs

Q: What is the Constant of Proportionality called by another name?

A: The unit rate is another name for the constant of proportionality in mathematics.

Q: What is the Constant of Proportionality in a graph?

A: In a graph, the straight line that passes through the origin is the constant of proportionality.

Q: Why do we Use the Constant of Proportionality?

A: We use the constant of proportionality in mathematics to determine the nature of proportionality, whether it is the direct proportion or indirect proportion. The constant of proportionality aids in solving equations involving ratios and proportions.

Q: The constant of proportionality of 12/6 is what?

A: In direct proportionality, we use k=y/x to find the constant of proportionality. If y = 12 and x = 6, then k = 12/6 = 2.