Do you think that ratio and proportionality are similar? You are not! Any relationship that is constantly in the same ratio is referred to asproportionality. For example, the amount of mangoes in a tree is proportional to the number of trees in the yard. The number of mangoes present in each tree is the ratio of proportionality. Before we dive deep into the subject, let's learn about proportions.
The mathematical comparison of two numbers is called proportion. How do you represent proportionality? The symbols "::" or "=" represent proportions. Proportion says when two sets of supplied numbers increase or decrease in the same ratio, the ratios are directly proportional to each other. When two ratios are equal, they are said to be proportional. For example, the time a bus travels at 60 km/h corresponds to the time it takes the same bus to travel 360 km in 6 hours. This can be viewed as 60 km/h = 360 km / 6. (Both the left and right sides of the expression are the same).
Now that we have a basic idea of proportionality and also have revised the concept of proportion, let's explore the constant of proportionality, inverse proportionality, and the law of multiple proportions.
constant of proportionality
When two or more parameters are directly or indirectly proportional to each other, their relationship is expressed as a = kb or a = k/b, where k indicates how the two variables are related. This k is known as the constant of proportionality.
The constant of proportionality is the ratio of two proportional values to a constant value. Two variable values have a proportional relationship if either their ratio or product gives a constant. The value of the constant of proportionality is determined by the ratio between the two quantities given. This relationship can be of two types: direct variation and inverse variation.
Direct variant:The standard direct variation equation is a = kb, which means that if b increases, a increases, and if b decreases, a decreases. The best example of direct proportionality is product cost. Costs increase when demand increases and vice versa. It is expressed as a ∝ b.
Reverse variation:The indirect proportionality equation is b = k/a, which means that as "b" increases, "a" decreases, and the converse is also true. This is best illustrated using the example of a moving vehicle. The speed of a moving vehicle (b) is proportional to its time to reach a given distance (a), represented as b ∝ 1/a.
In both cases k is the same. The coefficient of proportionality is the value of this constant. The unit theorem is another name for the constant of proportionality.
Application of the constant of proportionality?
Have you ever imagined the application of proportionality in real life? In mathematics, we use the constant of proportionality to calculate the rate of change and at the same time determine whether we are dealing with direct or inverse variation. Suppose the price for 4 oranges is $40. We automatically remember that an orange costs 5 dollars. The constant of proportionality for the price of oranges was determined to be 5.
If we want to draw an image of the Statue of Liberty while sitting in front of it on a sheet of paper and looking at the real monument in front of us, we need to keep the length, height, and width of the structure proportional. To get the desired result, we must first determine the constant of proportionality. Based on this we can then draw the monument with proportional measurements. For example, if the statue is 93 meters tall, we need to draw it about 93 millimeters tall to replicate it on our drawing sheet. In the same way we draw each component that can be done by applying the constant of proportionality.
Dealing with proportional relationships will help you address a variety of real-world issues, including:
- Changing the proportions of ingredients in a recipe
- For example, finding odds and probabilities of events is a method of quantifying chance.
- Scaling a graphic for architectural and drawing purposes
- Calculation of price surcharges in the form of a percentage increase or percentage reduction
- Product discounts based on unit price
Law of Multiple Proportions
According to the law of multiple proportions, the combination of 1 element with the definite quantity of the second element is always an integer. This law has many applications in physics and chemistry.
For example, the mass of oxygen in carbon dioxide (CO2) is 32, and carbon monoxide (CO) is 16. Thus, the mass of oxygen is in the ratio 32:16, or simplified 2:1.
Find the constant of proportionality?
Determining the constant of proportionality is very easy. Previously we learned what inverse and direct variation are. We will apply this knowledge to find the value of the constant of proportionality "k".
For direct share:To find the value of "k" in a direct proportionality, use the formula b = ka. From this formula k = b/a.
For indirect share:To determine the value of "k" in indirect proportionality, use the formula b = k/a. From this formula, k = ba.
Let's learn how to determine "k" with some examples:
Example 1:The variables "a" and "b" are directly proportional. Find the value of the constant of proportionality when a = 7 and b = 49,
Solution:Suppose b = 49 and a = 7
"b" is directly proportional to "a". Therefore we use the formula b = ka.
k = b/a
k = 49/7
k = 7.
So the constant of proportionality for this question is 7.
Example 2:6 students can complete the task in 3 hours. The teacher added three more students to the group to complete the task early. Find out how much time it will take for the new group to complete the task.
Solution:Assuming that six students can complete the task in Case 1 in 3 hours.
In case 2, the number of students increases by 3. The total number of students is now 6 + 3 = 9.
Suppose the number of students increases and the number of hours to complete the work decreases. Therefore we can say that this is an indirect relationship.
We use the relation b = k/a → k = ab
For case 1
k = 6 x 3 = 18
For case 2
k = ab
18 = 9 x b
b = 18/9
Thus, 9 students need 18/9 = 2 hours to complete the given work.
How do you find the constant of proportionality?
So far we have learned how to find the constant of proportionality in problems. But what if tables and graphs are provided to us and we are asked to find the constant of proportionality simply by looking at it? Well don't worry. This section explains how to identify the constant of proportionality in tables and graphs.
| number of hours = a | 1 | 2 | 5 | 7 |
| Number of words written = b | 200 | 400 | 1000 | 1400 |
From the table above you can deduce that as the number of hours increases, so does the number of words. So it's a direct connection. We can say that the number of words written ∝ is the number of hours. So to find “k” we use b = ka.
k = b/a → 200/1 or 400/2 or 1000/5 and so on.
k = 200
So the constant of proportionality is 200.
If you plot the values from the table above, you get a straight line passing through the origin. This indicates a proportional relationship between the two values as a direct variation. TheTiltthe line drawn for two constants of proportionality a and b on a graph is the constant of proportionality under the direct proportion condition.
If you have indirect variation for the values in a table, you won't get a straight-line plot. The inverse variation graphs are curvy, like a parabola or hyperbola. Usually an inverse variation chart has the starting point with the maximum value falling to the minimum value and vice versa.
Note:To determine whether the two quantities are proportional, we need to calculate the ratio of the two numbers for each of the given values. They have a proportional relationship when their ratios are equivalent. The relationship between the ratios is not proportional unless they are all equal.
frequently asked Questions
1. How do you explain proportionality?
Ans.Proportionality is a concept that explains how two variables are related to each other. For example, if you have a variable A and a variable B, and A changes by 10%, B should change by 10%. In other words, the change in A is proportional to the change in B.
2. What is the proportionality constant?
Ans.The constant of proportionality is the value that determines the slope of a linear graph. The constant of proportionality is a constant because it stays the same throughout the chart.
3. What is proportionality for?
Ans.The purpose of proportionality is to indicate that two quantities or variables are related in a linear fashion. If one quantity doubles, the other also doubles; If one of the variables decreases to 1/10 of its previous value, so does the other.
4. How do you find the proportionality?
Ans.To find proportionality, you need to divide both sides of the equation by the same number. This makes it easier to solve for one of the variables in the equation.
5. What is the constant of proportionality in a graph?
Ans.A constant of proportionality is a constant factor that does not change the slope of a line. A graph can have more than one constant of proportionality, and each is represented by a straight line.