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Inverse Variation Formula

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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 their product remains constant. This product is also known as the proportionality constant.

If the product of two non-zero numbers provides a constant term, they are said to be in inverse variation (constant of proportionality). In other words, inverse variation occurs when one variable is directly proportional to the reciprocal of the other quantity. 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. 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.

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

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 or y = k/x, where k is the 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.

Sample problems

Question 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

Question 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

Question 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

Question 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

Question 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

Question 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

Question 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


Last Updated : 10 Jan, 2024
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