Inverse proportionality is a sort of proportionality in which one quantity declines while the other grows, or vice versa. This means that if the magnitude or absolute value of one item declines, the magnitude or absolute value of the other quantity grows, and the product remains constant. This product is also known as the proportionality constant.
In this article, we will learn about Inverse Variation definition, Inverse Variation formula, examples and others in detail.
Table of Content
What is Inverse Variation?
If the product of two non-zero numbers provides a constant term, they are said to be in inverse variation. In other words, inverse variation occurs when one variable is directly proportional to the reciprocal of the other quantity. his type of relationship can be described by the equation: xy = k
This indicates that a rise in one quantity causes a reduction in the other, and a drop in one causes an increase in the other.
Inverse Variation Example
Some examples of inverse variation are:
- Suppose x and y are in inverse variation, if x = 20 and y = 10, their product is 200. If x decreases to 10 then y increases to 20 to keep the product of 200 constant.
- Speed and Travel Time: Time required to travel a fixed distance varies inversely with the speed of travel.
- Intensity of Light and Distance: Intensity of light from a source varies inversely with the square of the distance from the source.
- Gravitational Force: Gravitational force between two objects varies inversely with the square of the distance between them.
Inverse Variation Formula
When two quantities, x and y, follow inverse variation, they are expressed as follows:
xy = k
Here, k is the proportionality constant. Furthermore, x ≠ 0 and y ≠ 0.
Derivation of Inverse Variation Formula
Proportionality is denoted by the symbol "∝"
When two quantities, x and y, exhibit inverse variation, they are expressed as x ∝ 1/y or y ∝ 1/x
A constant or proportionality coefficient must be included to transform this expression into an equation. As a result, the formula for inverse variation becomes as below:
x = k/y
y = k/x
where,
- k is Proportionality Constant
Rearranging the terms in either of the equations, we get
⇒ xy = k
This derives the inverse variation formula.
Product Rule for Inverse Variation
Let us take two quantities x1 and y1 inversely proportional to each other. The required relation is,
=> x1y1 = k ⇢ (1)
For another two quantities x2 and y2 inversely proportional to each other, the required relation is,
=> x2y2 = k ⇢ (2)
Using (1) and (2),
x1y1 = x2y2
This is known as the product rule for inverse variation.
Inverse Variation Graph
A rectangular hyperbola is the graph of an inverse variation. If two quantities, x and y, are in inverse variation, their product is equal to a constant k. Because neither x nor y can be 0, the graph never crosses the x- or y-axis.
The following is the graph of an inverse variation xy = k:
Let's look at examples to better comprehend the concept of the inverse variation formula.
Difference Between Direct Variation and Inverse Variation
Difference between direct variation and inverse variation is added in the table below:
| Direct Variation | Inverse Variation |
|---|---|
| Direct Variation equation is y = kx | Inverse Variation equation is y = k/x |
| Graph is a straight line through the origin | Graph is a Hyperbola |
| Graph is Linear | Graph is Non-linear |
| In direct variation, y increases if k > 0 and decreases if k < 0 y decreases if k > 0 and increases if k < 0 | In inverse variation, y decreases if x increases y increases if x decreases |
| Example: Distance traveled at constant speed: d = vt | Example: Speed and travel time: v = d/t |
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Examples on Inverse Variation
Example 1: Suppose x and y are in an inverse proportion such that, when x = 4, then y = 18. Find the value of y when x = 6.
Solution:
Find the constant of proportionality for x = 4 and y = 18.
k = xy = 4 (18) = 72
Find y for x = 6 and k = 72.
y = k/x
= 72/6 = 12
Example 2: Suppose x and y are in an inverse proportion such that, when x = 1, then y = 6. Find the value of y when x = 3.
Solution:
Find the constant of proportionality for x = 1 and y = 6.
k = xy = 1 (6) = 6
Find y for x = 3 and k = 6.
y = k/x
= 6/3 = 2
Example 3: Suppose x and y are in an inverse proportion such that, when x = 6, then y = 16. Find the value of y when x = 8.
Solution:
Find the constant of proportionality for x = 6 and y = 16.
k = xy = 6 (16) = 96
Find y for x = 8 and k = 96.
y = k/x
= 96/8 = 12
Example 4: Suppose x and y are in an inverse proportion such that, when x = 1, then y = 2. Find the value of y when x = 4.
Solution:
Find the constant of proportionality for x = 1 and y = 2.
k = xy = 1 (2) = 2
Find y for x = 4 and k = 2.
y = k/x
= 2/4 = 1/2
Example 5: Suppose x and y are in an inverse proportion such that, when x = 6, then y = 36. Find the value of y when x = 18.
Solution:
Find the constant of proportionality for x = 6 and y = 36.
k = xy = 6 (36) = 216
Find y for x = 18 and k = 216.
y = k/x
= 216/18 = 12
Example 6: Suppose x and y are in an inverse proportion such that, when x = 2, then y = 9. Find the value of y when x = 3.
Solution:
Find the constant of proportionality for x = 2 and y = 9.
k = xy = 2 (9) = 18
Find y for x = 3 and k = 18.
y = k/x
= 18/3 = 6
Example 7: Suppose x and y are in an inverse proportion such that, when x = 30, then y = 8. Find the value of y when x = 40.
Solution:
Find the constant of proportionality for x = 30 and y = 8.
k = xy = 30 (8) = 240
Find y for x = 40 and k = 240.
y = k/x
= 240/40 = 6