Inverse Proportion Formula - Math Steps, Examples & Questions

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Inverse proportion formula

Here you will learn about inverse proportion formulas, including what the inverse proportion formulas are and how to interpret them.

Students will first learn about inverse proportion formulas as part of functions in high school.

What is the inverse proportion formula?

The inverse proportion formula is an algebraic formula that represents the relationship between two inversely proportional variables, meaning that as one variable increases, the other decreases.

If the variables were $x$ and $y$ where $y$ is inversely proportional to $x,$ you can write the relationship using the proportionality symbol as

$y\propto\cfrac{1}{x}$

If $y$ is inversely proportional to $x,$ the constant of proportionality $k=xy.$

Rearranging this equation to make $y$ the subject, you have

$y=\cfrac{k}{x}$

An inverse proportion formula can involve exponents and roots. For example, If the variables were $x$ and $y,$ and $y$ is inversely proportional to $x^{2},$ you can write the relationship using the proportionality symbol as

$y\propto\cfrac{1}{x^2}$

This can be written as an equation using the constant of proportionality, $k,$

$y=\cfrac{k}{x^2}$

To recognize when two variables are inversely proportional to one another, the product of the two variables will be a constant $k.$

Inverse proportion is a relationship between two variables, it is sometimes known as indirect proportion or inverse variation. If two quantities are inversely proportional, then as one quantity increases, the other decreases.

What is the inverse proportion formula?

Common Core State Standards

How does this relate to high school math?

High School – Functions – Interpreting Functions (HS.F.IF.B.4)
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

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How to recognize when two variables are inversely proportional

In order to recognize when two variables are inversely proportional to one another:

The product of the two variables will equal a constant.

Inverse proportion formula examples

Example 1: recognizing an inverse proportion

Which of these equations indicate that $y\propto\cfrac{1}{x}?$

The product of the two variables will equal a constant.

For $A,$ you have the two variables $x$ and $y.$

If you multiply both sides by $x$ , you have $xy=\cfrac{x^2}{a}, xy$ is not a constant as $\cfrac{x^2}{a}$ will change, depending on the value for $x.$

So $x$ and $y$ are not inversely proportional for $A.$

For $B,$ you have the two variables $x$ and $y.$ If you multiply both sides by $x,$ you get $xy=2.$

As the product of $x$ and $y$ always equals $2$ (a constant value), $x$ is inversely proportional to $y$ for $B.$

For $C,$ you have the two variables $y$ and $x^{2}.$

If you multiply both sides of the equation by $x^{2},$ you have $x^{2}y=x^{4}$ which is not a constant as $x^{4}$ will change as the value of $x$ changes. This means that for $C, y$ is not inversely proportional to $x^{2}.$

For $D,$ you have the two variables $y$ and $x.$ If you multiply both sides by $x,$ you have $xy=2x^{2}$ and so these are not inversely proportional to one another.

$B$ shows an inverse proportion.

Example 2: recognizing an inverse proportion

Which of these equations indicate that $y$ is inversely proportional to $x?$

The product of the two variables will equal a constant.

For $A,$ you have the two variables $x^{3}$ and $y.$

If you multiply both sides by $x^{3}$ , you have $x^{3}y=x^{6}, x^{3}y$ is not a constant as $x^{6}$ will change, depending on the value for $x.$

So $x$ and $y$ are not inversely proportional for $A.$

For $B,$ you have the two variables $x$ and $y.$ If you multiply both sides by $x,$ you have $xy=\cfrac{x^2}{3}.$

This means that $xy$ is not a constant as $\cfrac{x^2}{3}$ will change, depending on the value for $x.$ So $x$ and $y$ are not inversely proportional for $B.$

For $C,$ you have the two variables $x$ and $y.$

Multiplying both sides by $x,$ you have $xy=3x^{2}$ which is not a constant value and so $y$ is not inversely proportional to $x$ for $C.$

For $D,$ you have the two variables $x$ and $y.$

Multiplying both sides by $x,$ you have $xy=3$ which is a constant and so $y$ is inversely proportional to $x$ for $D.$

$D$ shows an inverse proportion.

How to use the inverse proportion formula

In order to use the inverse proportion formula:

Write down the inverse proportion formula.
Determine the value of $\textbf{k}.$
Substitute $\textbf{k}$ into the inverse proportion formula.

Inverse proportion formula examples

Example 3: determine the inverse proportion formula when y ∝ ⅟x

Let $y$ be inversely proportional to $x.$ When $y=5, x=6.$ By calculating the constant of proportionality, determine a formula for $y$ in terms of $x.$

Write down the inverse proportion formula.

As $y$ is inversely proportional to $x,$ you can write $y\propto{1}/{x}$ and so you have the formula $y=\cfrac{k}{x}$ where $k$ is the constant of proportionality.

Determine the value of $\textbf{k}.$

You can find the constant of proportionality by substituting in the values you are given.

$\begin{aligned}y&=\cfrac{k}{x}\\\\5&=\cfrac{k}{6}\\\\k&=5\times{6}=30\end{aligned}$

Substitute $\textbf{k}$ into the inverse proportion formula.

As $k=30,$ you have the equation $y=\cfrac{30}{x}.$

Example 4: determine the inverse proportion equation when y ∝ ⅟x³

$y$ is inversely proportional to $x^{3}.$

When $y=4, x=2.$

Find a formula for $y$ in terms of $x.$

Write down the inverse proportion formula.

$y\propto\cfrac{1}{x^{3}}$ and so $y=\cfrac{k}{x^{3}}.$

Determine the value of $\textbf{k}.$

As $y=4$ when $x=2,$

$\begin{aligned}4&=\cfrac{k}{2^{3}}\\\\4&=\cfrac{k}{8}\\\\4\times{8}&=k\\\\k&=32\end{aligned}$

Substitute $\textbf{k}$ into the inverse proportion formula.

As $y=\cfrac{k}{x^{3}}$ and $k=32,$

$y=\cfrac{32}{x^{3}}.$

Example 5: constructing an inverse proportion formula when y ∝ ⅟x²

Let $y$ be inversely proportional to $x^{2}.$

When $y=10, x=2.$

Find a formula for $y$ in terms of $x.$

Write down the inverse proportion formula.

From the first sentence you can write down $y\propto \cfrac{1}{x^2}.$

You can therefore state $y=\cfrac{k}{x^2},$ where $k$ is the constant of proportionality.

Determine the value of $\textbf{k}.$

You can find the constant of proportionality by substituting in the values you are given. Here $x=2$ and $y=10.$

$\begin{aligned}y&=\cfrac{k}{x^2}\\\\10 &=\cfrac{k}{2^2}\\\\10&=\cfrac{k}{4}\\\\k&=10\times{4}=40\end{aligned}$

Substitute $\textbf{k}$ into the inverse proportion formula.

As $k=40,$ you can write down the equation

$y=\cfrac{40}{x^2}$

Example 6: constructing an inverse proportion equation from a table

Use the information in the table below to determine the formula, given that $b$ is inversely proportional to the square root of $a.$

Write down the inverse proportion formula.

As stated in the question, $b\propto\cfrac{1}{\sqrt{a}}$ and so $b=\cfrac{k}{\sqrt{a}}.$

Determine the value of $\textbf{k}.$

When $a=4, b=1$

$\begin{aligned}1&=\cfrac{k}{\sqrt{4}}\\\\1&=\cfrac{k}{2}\\\\1\times{2}&=k\\\\k&=2\end{aligned}$

Note that you could have used any pair of values for $a$ and $b$ from the table.

Substitute $\textbf{k}$ into the inverse proportion formula.

Now that $k=2,$ you have the inverse proportion formula

$b=\cfrac{2}{\sqrt{a}}.$

Example 7: constructing and solving an inverse proportion equation from a word problem

The number of people setting tables in a restaurant $(p)$ is inversely proportional to the time taken to set all of the tables $(t).$ It takes $6$ people $48$ minutes to set up all of the tables in the restaurant. How many minutes would it take to set up all of the tables with $10$ people?

Write down the inverse proportion formula.

As stated in the question, $p\propto\cfrac{1}{t}$ and so $p=\cfrac{k}{t}.$

Determine the value of $\textbf{k}.$

When $p=6, t=48$ and so

$\begin{aligned}6&=\cfrac{k}{48}\\\\6\times{48}&=k\\\\k&=288\end{aligned}$

Substitute $\textbf{k}$ into the inverse proportion formula.

Now that $k=288,$ you have the inverse proportion formula $p=\cfrac{288}{t}.$

There is the extra condition that you need to determine the number of minutes it would take $10$ people to set up the tables. To do this, you need to substitute $p=10$ into the inverse proportion formula, and solve for $t.$

$\begin{aligned}10&=\cfrac{288}{t}\\\\10\times{t}&=288\\\\t&=\cfrac{288}{10}=28.8\text{ minutes}\end{aligned}$

Teaching tips for inverse proportion formula

Use examples of inverse proportion that involve real-life situations involving speed and time (for example, if your speed increases while driving, you take less time) or work and time (for example, a higher number of workers can complete a task in a fewer number of days).

Create inverse proportion graphs to explain the concept of inverse proportionality visually. This helps students grasp the idea that when $x$ increases, $y$ decreases and vice versa.

Facilitate discussions around the results from proportion worksheets and other proportion questions presented in class. Encourage students to explain their thought processes and reasoning to deepen their understanding of the concept of inverse proportionality.

Easy mistakes to make

Not distinguishing between direct and inverse proportion
You will need to learn which formula is for which type of proportionality.

Example of direct proportion
$y\propto x$
$y=kx$

Example of inverse proportion
$y\propto \cfrac{1}{x}$

$y=\cfrac{k}{x}$

Not checking if there are any exponents or roots involved
Most inverse proportional formulae just involve $x,$ but they can involve exponents such as $x^{2}$ or roots such as $\sqrt{x}.$

Misunderstanding the constant of proportionality
The constant of proportionality can be an integer (a whole number), but they can also be decimals or fractions.

For example, $y=\cfrac{5}{2x}=\cfrac{5}{2}\times\cfrac{1}{x}=2.5\times\cfrac{1}{x}=\cfrac{2.5}{x}$ and so $k=2.5.$

Direct variation equation
Direct variation
Direct and inverse variation
Inversely proportional
Directly proportional graph

Practice inverse proportion formula questions

y=\cfrac{2}{x}

y=5x

xy=8

y=\cfrac{1}{3x}

The incorrect equation is $y=5x$ as $xy=5x^{2}$ which is not a constant.

y=5x

y=\cfrac{5}{x}

y=\cfrac{x}{5}

y=5x^{2}

For $y\propto\cfrac{1}{x}.$ When $y=\cfrac{5}{x}, xy=5$ which is a constant and so this is the correct solution.

y=\cfrac{10}{x}

y=10x^2

y=\cfrac{10}{x^{2}}

x^{2}y=\cfrac{10}{x}

When $y\propto\cfrac{1}{x^{2}}, x^{2}y=k.$

For $y=\cfrac{10}{x^{2}},$ multiplying both sides by $x^{2},$ you have $x^{2}y=10$ and so this is the correct solution.

y=16x

y=\cfrac{4}{x}

y=\cfrac{x}{4}

y=\cfrac{16}{x}

$y\propto\cfrac{1}{x}$ and so $y=\cfrac{k}{x}.$

As $y=2$ when $x=8,$

\begin{aligned} y&=\cfrac{k}{x}\\\\ 2&=\cfrac{k}{8}\\\\ k&=2\times{8}=16\\\\ \end{aligned}

So the equation is $y=\cfrac{16}{x}.$

y=2x

y=\cfrac{50}{x}

y=50x

y=\cfrac{2}{x}

Write down the relationship between $y$ and $x$ and substitute the given values to find $k,$ the constant of proportionality.

As $y\propto\cfrac{1}{x}$ and so $y=\cfrac{k}{x}.$

As $y=10$ when $x=5,$

\begin{aligned} 10&=\cfrac{k}{5}\\\\ k&=10\times{5}=50 \end{aligned}

So the equation is $y=\cfrac{50}{x}.$

y=\cfrac{6}{\sqrt{x}}

y=\cfrac{6}{x^2}

y=\cfrac{3}{\sqrt{x}}

y=\cfrac{2}{x}

As $y\propto\cfrac{1}{\sqrt{x}}, \; y=\cfrac{k}{\sqrt{x}}.$

As $y=2$ when $x=9,$

\begin{aligned} 2&=\cfrac{k}{\sqrt{9}}\\\\ 2&=\cfrac{k}{3}\\\\ k&=2\times{3}=6 \end{aligned}

So the equation is $y=\cfrac{6}{\sqrt{x}}.$

Inverse proportion formula FAQs

What is the inverse proportion formula?

The inverse proportion formula is expressed as $y=\cfrac{k}{x}$ where $k$ is a constant. This indicates that as one variable increases, the other decreases, keeping the product of the two variables constant.

What does inversely proportional mean?

“Inversely proportional” means that as one variable increases, the other variable decreases in such a way that the product of the two variables remains constant.

What is the difference between inverse proportion and direct proportion?

In inverse proportion, as one variable increases, the other decreases, keeping the product constant $(y=\cfrac{k}{x})$ . In direct proportion, as one variable increases, the other also increases resulting in a constant ratio $(y=kx)$ .

The next lessons are

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