In our daily lives, we see that the variation in values of one quantity is dependent on the variation in values of another quantity. Inverse variation occurs when one variable varies inversely with respect to another. It denotes the inverse connection of two quantities. As a result, one variable is inversely proportional to another. Using an example, this post will teach you the 7th Grade Math Concept of Inverse Variations using Unitary Method of Proportion, and how to define the Inverse Variations Using Method of Proportion.
What is Inverse Variation?
Inverse variation is defined as if the product of both amounts equals a constant value regardless of their values changing. In fact, if one quantity increases the other quantity decreases. For Example, with increase in speed the time needed to cover the distance reduces.
Inverse Variation Formula
The inverse variation formula can be used to determine the connection between two inversely proportional numbers. Assume that x and y are two values that decrease when y increases and vice versa.
y = k/x is the inverse variation formula.
Time and speed are inversely proportional. The time it takes us to traverse the same distance lowers as our speed rises. When we consider speed as y and time as x, we may say that y is inversely proportional to x, which is stated formally as the inverse variation formula.
with k being the proportionality constant.
As x lowers, y increases.
As x rises, y lowers.
The symbol represents the proportionate connection between two quantities in this case.
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How to Solve Inverse Variation using Method of Proportion?
Let’s say the number of workers and the number of days they’ll need to finish a specific amount of labor are x and y, respectively.
- Make a proportional relationship.
- Convert to an equation using a proportionality constant
- Find the proportionality constant using the information provided.
- In the equation, substitute the proportionality constant.
| Numbers of Workers (x) | Number of Days Required (y) |
| 16 | 3 |
| 12 | 4 |
| 8 | 6 |
| 4 | 12 |
Is there an inverse relationship between the number of workers and the number of days?
Solution:
Let’s have a look. Keep a close eye on the table’s figures. You’ll see that the product of x and y for each row is the same.
If there are 16 people, the task will be completed in three days.
As a result, xy = 16*3 = 48.
When the number of workers is reduced, it is evident that the same task will be completed in more time.
However, the product of x and y is 12*4 = 48 in this case.
Again, the product is 48 for 8 employees in 6 days. In 12 days, the same thing happened to four workers.
As a result, the product of two inversely proportioned amounts is always equal.
Unitary Method Inverse Variation Examples with Solutions
Example 1: Assume that x and y are in inverse proportion, with y = 10 at x = 100. Using the inverse proportion rule, find the value of y for x = 200.
Solution:
To discover: y’s value.
If y = 10, x = 100.
x ∝ 1/y
x = k / y,
where k is a constant,
or k = XY
by putting x = 100 and y = 10,We obtain
k = 100×10= 1000
When x = 200,
the equation becomes 200 y = 1000
y = 1000/200 = 5.
That is, if x is increased to 200, y is reduced to 5.
Example 2: A car traveling at 60 miles per hour will take 4 hours. What would be the pace at which the same distance might be covered in five hours?
Solution:
Consider the speed parameter m and the time parameter n.
The speed diminishes as the time taken rises. As m ∝ 1/n is an inverse proportional relationship.
m = k/ n , m × n = k is the inverse proportion formula.
Time = 4 hours at a speed of 60 miles per hour,
hence k = 60 × 4 = 240.
Now we must calculate speed when time is n = 5.
m × n = k.
m × 5 = 240
m = 240/5 = 48
As a result, at 5 hours, the speed is 48 miles per hour.
Example 3: A supervisor at a construction business believes that 8 guys can accomplish a task in 40 days. How long will it take 16 workers to do the same task?
Solution:
Consider M as the number of men and D as the number of days.
M1 = 8, D1 = 40, and M2 = 16 are given.
This is an inverse proportional relationship, meaning that as the number of workers grows, so does the number of days.
M ∝ 1/D
Consider the first situation: M1 = k/D1
8 = k/40
k = 8×40 = 320
Consider the second scenario: M2 = k/ D2
16 = 320/ D2
D2 = 320/16 = 20.
The same work will take 16 guys 20 days to complete.
FAQs on Inverse Variation Unitary Method
1. In actual life, how is an inverse variation used?
There are several real-world examples of inverse variation.
- Body weight is inversely related to body acceleration.
- The power of the battery is inversely proportional to the amount of time it is utilized.
- A man’s ability to work hard is inversely proportionate to his age.
- The amount of errors in the workplace is inversely proportional to practice.
2. How should the Inverse Variation Formula be represented?
The inverse variation formula displays the connection between two variables, which is explained by the following formula:
- Determine the two variables in the following issue.
- Recognize the existence of an inverse variation. 1/y ∞ x
- Invert the percentage formula x = k/y.
3. What is the formula for K in Inverse Variation?
In an inverse variation, K denotes the proportionality constant, which remains constant regardless of the values of the provided variables. Determine the product of x and y to find k in inverse variation. The formula is y = k/x, resulting in k=xy.