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Inverse proportionality is a concept that can be contrasted with direct proportionality. Consider two variables that are considered to be "inversely proportional." When all other variables remain constant, the magnitude or absolute value of one inversely proportional variable drops as the other grows, while their product (the constant of proportionality k) remains constant.

For example, the time taken for a journey is inversely proportional to travel speed.

Inverse proportionality occurs when one quantity decreases while the other increases, or vice versa. If one quantity's magnitude or absolute value declines while the further increases, its product remains the same. This product is also termed as the constant of proportionality.

If one number is directly proportional to the reciprocal of the other, the two quantities follow inverse variation. This indicates that increasing one quantity causes the other to decrease while decreasing one quantity causes the other to rise.

**Note the following examples:**

The statement "y varies inversely as x means that when x increases, y decreases by the same factor. In other words, the expression xy is constant:

xy = k

Where, k is the constant of variation.

You can also express the relationship between x and y as:

y = k/x

Where, k is the constant of variation.

Since k is constant, you can find k given any point by multiplying the x-coordinate by the y-coordinate.

**Example 1:** If y varies inversely as x, and y = 6 when x = 4/3, write an equation describing this inverse variation.

k = 4/3 (6) = 8

xy = 8 or y = 8/x

**Example 2: **If y varies inversely as x, and the constant of variation is k = 5/2, what is y when x = 10?

xy = 5/2

10y = 5/2

y= 5/2 x 1/10 = ½ x ½ = ¼

k is constant. Thus, given any two points (x1, y1) and (x2, y2) which satisfy the inverse variation, x1y1 = k and x2y2 = k. Consequently, x1y1 = x2y2 for any two points that satisfy the inverse variation.

**Example 3: **If y varies inversely as x, and y = 10 when x = 6, what is y when x = 15?

x1y1 = x2y2

6(10) = 15y

60 = 15y

y = 4

Thus, when x = 6, y = 4.

If (x1,y1) and (x2,y2) are solutions of an inverse variation, then x1y1=k and x2y2=k .

Substitute x1y1 for k .

x1y1=x2y2 or x1x2=y2y1

The equation x1y1=x2y2 is called the product rule for inverse variations.

Key Ideas of Inverse Variation

â— You may say that y varies inversely with x if y is expressed as the product of some constant number k and the reciprocal of x.

â— However, the value of k cannot equal 0.

â— Isolating k on one side, it becomes clear that k is the fixed product of x and y. That means, multiplying x and y always yields a constant output of k.

K = xy

The product of variables x and y is constant for all pairs of data. You can claim that k = 24, is the constant of variation.

Writing the equation of inverse proportionality,

In an inverse variation, as one number increases, the other decreases. This is also called inverse proportion.

An example would be the relationship between time spent wasting time in class and your grade on the midterm. The more you waste time, the lower your score on the test.

If you wanted to give this one an equation, it would state:

y = k/x, where x and y are the two quantities, and k is still the constant of proportionality, telling how much one varies when the other changes.

So the bigger the value of x, the smaller the value of y will be. That's inverse variation: as one goes up, the other goes down.

When two variables change in inverse proportion, it is called indirect variation. In indirect variation, one variable is constant times inverse of another. If one variable increases, the other will decrease vis a vis. This means that the variables change in the same ratio but inversely.

The general equation for an inverse variation is y = k1/x or r xy = k which is constant.

So the product of two variables is a constant for inverse variation.

The fundamental counting principle tells you that the number of total possible outcomes in a sample space with multiple events is given by the product of individual events, as long as the events are pairwise independent. You can check whether the events are pairwise independent by considering whether or not a specific outcome of one event changes the number of possible outcomes from any other event.

Fundamental Counting Principle

If you have a ways of doing event 1, b ways of doing event 2, and c ways of event 3, then you can find the total number of outcomes by multiplying:

a x b x c

As per the inverse variation formula, if any variable x is inversely proportional to another variable y, then the variables x and y are represented by the formula:

xy = k or y=k/x

where k is any constant value.

For two quantities with inverse variation, as one quantity increases, the other quantity decreases.

The inverse variation is represented by x = k/y or xy = k

In order for the table to have an inverse variation characteristic, the product for all pairs of xx and yy in the data set must be the same.

When two quantities are related to each other inversely, i.e., when an increase in one quantity brings a decrease in the other and vice versa then they are said to be in inverse proportion. In inverse proportion, the product of the given two quantities is equal to a constant value.

Some word problems require the use of inverse variation. Here are the ways to solve inverse variation word problems.

â— Understand the problem.

â— Write the formula.

â— Identify the known values and substitute in the formula.

â— Solve for the unknown.

The concept of inverse proportion thus states the relationship between two variables when their product is equal to a constant value. While the value of one variable increases, the other decreases, so their product remains unchanged.

Save this post. Understand the vital concepts, comprehend the formulae, take inspiration from examples to nail your inverse variation calculations like never before!

**Ans: **The mathematical relationship between two variables which can be expressed by an equation in which the product of two variables is equal to a constant.

Now to solve the inverse variation problems,

â— Write the variation equation: y = k/x or k = xy.

â— Substitute in for the given values and find the value of k.

â— Rewrite the variation equation: y = k/x with the known value of k.

â— Substitute the remaining values and find the unknown.

**Ans: **To solve a basic inverse variation problem, the steps are simple -

â— Write the variation equation: y = k/x or k = xy.

â— Substitute in for the given values and find the value of k.

â— Rewrite the variation equation: y = k/x with the known value of k.

â— Substitute the remaining values and find the unknown.

**Ans: **Since k is constant, we can find k given any point by multiplying the x-coordinate by the y-coordinate. For example, if y varies inversely as x, and x = 5 when y = 2, then the constant of variation is k = xy = 5(2) = 10.

**Ans: **Direct variation means when one quantity changes, the other quantity also changes in direct proportion. Inverse variation is exactly opposite to this.

For direct variation, use the equation y = kx, where k is the constant of proportionality. For inverse variation, use the equation y = k/x, again, with k as the constant of proportionality.

**Ans: **For direct variation, use the equation y = kx, where k is the constant of proportionality. For inverse variation, use the equation y = k/x, again, with k as the constant of proportionality.

**Inverse Variation Equations and Ordered Pairs**

The inverse variation is represented by x = k/y or xy = k.

Ordered pairs - (x1, y1) and (x2, y2), x1y1 = x2y2

**Ans: **Mentioned below are two real-life examples of inverse proportion -

â— If you increase the speed of the car, then the time taken to reach the destination decreases.

â— The brightness of the sunlight decreases as the distance from the sun increases.

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