Direct variation is a sort of proportionality in which one quantity changes directly in response to a change in another number. This means that if one quantity increases, the other quantity will grow proportionally. Similarly, if one number diminishes, the other amount lowers as well. Because direct variation is a linear connection, the graph will be straight. Let us discuss the 7th Grade Math Concept situations of direct variation in this article with related solved examples.
Direct Variation Definition
If two quantities rise or decrease by the same factor, they are said to follow a direct variation. As a result, a rise in one quantity causes an increase in the other, while a drop in one causes a reduction in the other. In other words, if the first item’s ratio to the second quantity is a constant term, the quantities have been shown to be directly proportional to each other. The coefficient or constant of proportionality is the name given to this constant value. Furthermore, if two values vary directly, one will be a constant multiple of the other.
Direct Variation Formula
The direct variation formula connects two numbers by establishing a mathematical connection in which one of the variables is a constant multiple of the other. The sign ” ∞” is used to denote two quantities that are directly proportional to each other or are in direct variation. If two quantities, x, and y, are in direct variation, they may be stated as follows: x = y
The direct variation formula is shown below when the proportionality sign is removed.
How can you Illustrate a Situation that Involves Direct Variation?
Let’s go down the direct variation formula with the aid of a simple example. Assume that y changes straight with x and that y = 60 when x = 8. When x = 120, what will be the value of y?
The provided values are y1 = 60, x1 = 8, x2 = 120, and y2 =? We get the following equation using the direct variation formula.
y1/x1 = y2/x2
60/8 =y2 / 120 = y2 / 120
y2= 900
Therefore the value of y when x = 120 is 900.
Do Refer:
Direct Variation Situations
- More articles equal more money spent on purchases.
- Fewer articles equal less money spent on purchases.
- When there are more men at work, more work gets done.
- When there are fewer men at work, less work gets done.
- Higher money borrowed means more interest to pay.
- When you borrow less money, you pay less interest.
- More speed equals more distance traveled in a given amount of time.
- Less speed equals less distance traveled in a given amount of time.
- More hours worked equals more work completed.
- Work will be done less if working hours are reduced.
Important Points about Direct Variation
- A proportionality connection in which two quantities follow a direct relationship is known as a direct variation. This means that a rise (or decrease) in one quantity causes an equal increase (or decrease) in another.
- The direct variation equation is a two-variable linear equation defined by y = kx, where k is the constant proportionality.
- In a coordinate plane, a straight line represents direct variation.
- In a straight variation, the ratio of two quantities is a constant.
Direct Variation Real Life Examples with Solutions
The simple examples below will help you grasp the concepts of direct variation.
Example 1:
C = 2r or C = d is the formula for finding the circumference of a circle. Here, r represents the radius and d represents the diameter. This is an illustration of a direct variant. As a result, the circumference of a circle and its associated diameter vary directly with being the proportionality constant.
Example 2:
The number of iron boxes produced is proportional to the number of iron blocks. For 40 boxes, 160 iron blocks are required. How many iron bricks are required to make a box?
Solution:
In the above problem, y = 120 denotes the number of iron blocks required for 30 boxes, and x = 30 denotes the number of boxes. The number of iron blocks required to construct a box is k. In this case, we apply the direct variation formula y = kx.
120 = 30 k = 120/30 k = 4
As a result, a box requires four iron blocks.
Example 3:
Find the proportionality constant if x = 60 and y = 60 follow a straight variation.
Solution:
Since x and y are in direct proportion, y = kx or k = y / x. k = 60 / 30 = 2
k = 2 is the answer.
Example 4:
Let x and y be in direct variation, with x equal to 7 and y equal to 28. Then determine the direct variation equation.
Solution:
Because x and y have a straight variation, y = kx or k = y / x.
k = 28/7
Therefore by substituting the value of k in the equation,
We get, y = 4 x
Thus, the direct variation equation will be y = 4x.
Example 5:
A motorcycle can ride 180 kilometers on 60 liters of fuel. How far can it travel on 7 liters of fuel?
Solution:
This is a direct variation condition.
Less gas means less distance traveled.
Distance traveled with 50 liters of fuel = 180 km
Distance covered per liter of fuel = 180/60 km
Distance travelled in 9 litres of fuel = 280/40 * 7 kilometres = 21 km
FAQs on Situations of Direct Variation
1. What exactly is Direct Variation?
Direct variation occurs when one quantity directly fluctuates with regard to another quantity. This means that if one quantity rises or falls, the other quantity rises or falls correspondingly.
2. What is an example of a Real-Life Direct Variation?
A car’s speed and distance traveled are two examples of direct variation. When the speed increases, so do the distance covered in a given amount of time. Similarly, when the speed of the vehicle lowers, so does the distance traveled in that time frame.
3. How to Deal with Direct Variation?
To answer questions about direct variation, use the formula y = kx. If the proportionality constant must be calculated, y is divided by x to obtain the result. If k is provided and either x or y must be found, these values can be swapped in the aforementioned equation to obtain the unknown value.
4. How to Determine Whether an Equation Is a Direct Variation?
A direct variation does not exist if an equation does not take the form y = kx. For example, y = 4x + 5 is not of the type y = kx; hence it is not a direct variant. y = 1.6x, on the other hand, is of the type y = kx and represents a straight variation.