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205 LearnersLast updated on September 30, 2025
Direct Variation
Direct variation is also known as direct proportionality. It is the relationship where one variable changes consistently with the other variable. We use direct variation to calculate discount rates or the total cost of products. In this topic, we will learn more about direct variation and its applications.
What is Direct Variation?
In direct variation, the increase or decrease in one variable directly affects the other. The ratio between two directly proportional quantities will always remain the same. Direct variation involves two variables, where one variable (y) depends on the independent variable (x).
The symbol we use to denote the direct proportionality is ∝. Since the variable y is directly proportional to x, we can express it mathematically as y ∝ x.
Direct Variation Formula
The direct variation formula connects two quantities by defining a mathematical connection where one variable is a constant multiple of the other.
It can be written as:
y = kx.
Here, (y, x) represents the variables, k is the constant (fixed value), and the equation indicates that y is directly proportional to x.
Importance of Direct Variation
Direct variation is essential for predicting how a specific quantity will change in relation to another. It is an important concept in various sectors; as one quantity increases or decreases, the other changes in the same way.
We can apply direct variation in real-life situations, such as calculating the distance and speed or the quantity and cost. This concept is also used in physics and economics. If a relationship is directly proportional, problem-solving becomes easier by using the formula, y = kx.
Properties of Direct Variation
To understand the concept of direct variation, children should understand its unique properties.
Let’s look into a few:
- The direct variation connects two variables in a relationship where an increase or decrease in one variable results in a proportional change in the other.
- To show the relation between two variables, we use the formula y= kx (where y and x are variables and k is the constant of proportionality).
- The symbol ∝ is used to denote the proportionality. For example: Number of candies ∝ total cost.
Tips and Tricks for Direct Variation
Learning direct variation helps children solve many real-life problems efficiently. Here, we will look at a few tips and tricks that help you understand the concept better:
- Always check if the given equation can be expressed as y = kx, which means it represents direct variation.
- Make sure that the ratio y/x is always constant for different values of x and y.
- It can be confirmed by plotting the equation on a graph, which will give a straight line that crosses its point of origin (0,0).
- The fastest way to identify non-direct variation is by checking the formula to see if it includes an additional term, such as y = kx + c.
Common Mistakes and How to Avoid Them in Direct Variation
Students often confuse direct variation with other relationships or misapply the constant of proportionality. In this section, we will learn a few common mistakes and the ways to avoid them.
Mistake 1
Using the Incorrect Constant of Proportionality (k)
Always remember the formula (k = y/x) to find the constant of proportionality. For example, if x = 2 and y = 10, then k = 10/2 = 5; the correct equation is y = 5x.
Mistake 2
Misunderstanding Real-life Situations
The scorecard of a child is not always proportional to the hours he/she spends learning because such situations don’t follow y = kx. Always check if direct variation applies in each situation.
Mistake 3
Using Inconsistent Units
Ensure that you convert the units before the calculation.
Mistake 4
Trying to Determine 'k' Without Sufficient Information
It is important to understand that we require the values of x and y to determine k.
For example: If x = 5 and y = 10; k = y/x = 10/5 = 2. Therefore, y = 2x.
Mistake 5
Applying Direct Variation for a Non-Linear Relationship
A curved or non-linear relationship does not follow the equation y = kx. Direct variation is represented by a straight line and follows the equation y = kx.
Solved Examples of Direct Variation
Problem 1
Let's assume that Sara’s savings from photography is directly proportional to her working hours. What would her savings be for 20 hours if she saved $200 for 10 hours of work?
Sara will save $400 after 20 hours of her work.
Explanation
Assume y = savings and x = number of hours worked. Since her savings is directly proportional to her working hours:
y = kx
Substituting the values, x = 20 hours and y = $200:
200 = k × 20
k = 200/10
= 20
We will now use k = 20 to find her savings for 20 hours of work.
y = 20 × 20 = 400
Therefore, Sara will save $400 after 20 hours of her work.
Problem 2
Assume that y and x are directly proportional. When y is 60, x equals 20. Calculate the value of x when the value of y becomes 100.
The value of x when y = 100 is 33.33 (approx).
Explanation
Here, we use the formula y = kx
60 = k × 20
k = 60 / 20 = 3
Now, we find the value of x when y = 100:
Substituting k = 3 and y = 100,
100 = 3 × x
x = 100/3 ≈ 33.33
Therefore, the value of x when y = 100 is approximately 33.33.
Problem 3
Leona bakes 50 cookies using 5 cups of sugar. Calculate the number of cups of sugar she would require for 75 cookies, given that the number of cups of sugar is directly proportional to the number of cookies she baked.
Leona would require 7.5 cups of sugar to bake 75 cookies.
Explanation
Assume, y = amount of sugar; x = number of cookies
y = kx (since the amount of sugar \(\propto
\) the number of cookies).
Now, we find k using x = 50 cookies and y = 5 cups of sugar:
5 = k × 50
k = 5/50 = 0.1
To find the number of cups of sugar for 75 cookies, multiply the value of k by 75.
So, y = 0.1 × 75 = 7.5
Therefore, Leona would require 7.5 cups of sugar to bake 75 cookies.
Problem 4
Suppose the value of y changes proportionally with x. Given that y = 15 when x = 5, calculate the value of y when x becomes 10.
When x = 10, the value of y will be 30.
Explanation
From the question, we understand that y \(\propto\) x
So we use the formula, y = kx
Substitute the given values:
15 = k × 5
k = 15/5 = 3
When x = 10:
y = 3 × 10 (substituting k = 3)
y = 30
Therefore, when x = 10, the value of y will be 30.
Problem 5
Let's say a and b are directly proportional. Given that a = 36 and b = 6, express their relationship in the form of a direct variation equation.
We can express their relationship as: a = 6b
Explanation
Since a and b are directly proportional,
a = kb
Where k is the proportionality constant.
Substitute a = 36 and b = 6
36 = k × 6
k = 36 / 6 = 6
Therefore, we can express it as a = 6b
FAQs on Direct Variation
1.What do you mean by direct variation?
2.How can we identify a direct variation?
3.When does the constant of proportionality have a negative value?
4.What is the difference between direct variation and inverse variation?
5.Give one real-life example of direct variation.
6.Do we use direct variation for non-linear relationships?
7.How does the graph of direct variation appear?
8.How can we identify non-direct variation?
9.Suggest the easiest way to determine the constant of proportionality.
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Dr. Sarita Ghanshyam Tiwari
About the Author
Dr. Sarita Tiwari is a passionate educator specializing in Commercial Math, Vedic Math, and Abacus, with a mission to make numbers magical for young learners. With 8+ years of teaching experience and a Ph.D. in Business Economics, she blends academic rigo
Fun Fact
: She believes math is like music—once you understand the rhythm, everything just flows!
