Inversely Proportional - Math Steps, Examples & Questions

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Request a demo

In order to access this I need to be confident with:

Multiplying and dividing integers Multiplying and dividing fractions Multiplying and dividing decimals Inverse operations

What is a proportion in math

Math resources Ratio and proportion

Proportion

Inversely proportional

Here you will learn about inverse proportion, including what inverse proportion is and how to solve inverse proportion problems including real-life problem solving. You will also have a look at the inverse proportionality formula.

Students will first learn about inversely proportional as part of functions in high school.

What is inversely proportional?

Inversely proportional is a type of proportionality relationship. If two quantities are inversely proportional then as one quantity increases, the other decreases.

An example of inverse proportion would be the hours of work required to build a house. If more people are building the same house, the time taken to build the house reduces.

Conversely, an example of direct proportion would be that the area of a circle is directly proportional to its radius.

Inverse proportion is also known as indirect proportion or inverse variation.

Inverse proportion is applied to real-life problems such as the speed of a moving object, determining whether an item will float or sink in water, or the time taken to complete a finite task, whereas direct proportion is useful in numerous real-life situations such as exchange rates, conversion between units, and fuel prices.

To determine the value of a variable that is inversely proportional to another, you need to determine the relationship between the two inversely proportional variables and then use this to find our unknown value.

Similar to a directly proportional relationship, you need to determine the constant of proportionality, $k.$

Step-by-step guide: Directly proportional

[FREE] Ratio Check for Understanding Quiz (Grade 6 to 7)

Use this quiz to check your grade 6 to 7 students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

DOWNLOAD FREE

[FREE] Ratio Check for Understanding Quiz (Grade 6 to 7)

Use this quiz to check your grade 6 to 7 students’ understanding of ratios. 10+ questions with answers covering a range of 6th and 7th grade ratio topics to identify areas of strength and support!

DOWNLOAD FREE

Inverse proportion formula

The symbol $\propto$ is the proportionality symbol and it represents a proportional relationship between two variables. If it is inversely proportional to $x,$ you write this relationship as $y\propto\cfrac{1}{x}.$

This relationship can be described using an equivalence relationship. When $y$ is inversely proportional to $x,$ the value of $x \times y$ is a constant value.

This value is the constant of proportionality and you use the letter $k$ to denote this value. Using a formula, you have $k=xy.$

Rearranging this formula to make y the subject, you obtain the inverse proportion formula:

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

Step-by-step guide: Inverse proportion formula

Graphs representing an inverse proportion between two variables

Proportional relationships can also be represented by graphs. If you sketch a graph of the line $y=\cfrac{k}{x},$ as $x$ increases in size, $k$ is divided by a larger number, so the result is a value $y$ that gets increasingly smaller.

This gives us the curved line graph of the reciprocal function.

Note, the value of $y$ can be inversely proportional to other powers of $x$ including $x^{2}, \, x^{3},$ or even $\sqrt{x}.$ Each of these has a different algebraic and graphical representation.

Step-by-step guide: Directly proportional graphs

What is inversely proportional?

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.

How to use inverse proportion

In order to find an unknown value given an inversely proportional relationship:

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

Inversely proportional examples

Example 1: complete the table y ∝ ¹⁄ₓ

Given that $y$ is inversely proportional to $x,$ calculate the missing value of $y$ in the table below:

Write down the inverse proportion formula.

As $y\propto\cfrac{1}{x},$ you can write the formula $y=\cfrac{k}{x}.$

2Determine the value of $\textbf{k}$ .

Substituting a known pair of values $(3, \, 20),$ you can say:

\begin{aligned}20&=\cfrac{k}{3} \\\\ 20\times{3}&=k \\\\ k&=60 \end{aligned}

3Substitute $\textbf{k}$ and the known value into the inverse proportion formula.

Now you have the equation $y=\cfrac{60}{x}.$ Substituting $x=12$ into the equation to calculate the value for $y,$ you have:

y=\cfrac{60}{12}

4Solve the equation.

Dividing $60$ by $12,$ you have $y=5.$

Notice that as the value for $x$ increased, the value for $y$ decreased.

Example 2: complete the table y ∝ ¹⁄ₓ

Let $y\propto\cfrac{1}{x}.$ Calculate the value for $x$ when $y=20.$

Write down the inverse proportion formula.

As $y\propto\cfrac{1}{x}$ as stated in the question, you have:

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

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

As $x=5$ when $y=60,$ substituting these values into the formula, you get:

$\begin{aligned}60&=\cfrac{k}{5} \\\\ 60\times{5}&=k \\\\ k&=300 \end{aligned}$

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

Now you have $y=\cfrac{300}{x}.$ You need to determine the value for $x$ when $y=20$ and so, substituting this value into the equation, you get:

$20=\cfrac{300}{x}$

Solve the equation.

\begin{aligned}20\times{x}&=300 \\\\ 300\div{20}&=x \\\\ x&=15 \end{aligned}

Example 3: complete the table y ∝ ¹⁄ₓ₂

Let $y$ be inversely proportional to $x^{2}.$ Calculate the missing value $q.$

Write down the inverse proportion formula.

As $y\propto\cfrac{1}{x^2}$ , you have the formula $y=\cfrac{k}{x^{2}}.$

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

Using the table, you know that $y=4$ when $x=2.$

Note, you could also use the pair of values $y=16$ when $x=1.$

As $y=\cfrac{k}{x^{2}},$ substituting the values for $y=4$ and $x=2,$ you have:

$\begin{aligned}4&=\cfrac{k}{2^{2}} \\\\ 4&=\cfrac{k}{4} \\\\ 4\times{4}&=k \\\\ k&=16 \end{aligned}$

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

Now you have $y=\cfrac{16}{x^{2}}.$ You need to determine the value of $y=q$ when $x=4$ and so, substituting these into the equation, you have:

$q=\cfrac{16}{4^{2}}$

Solve the equation.

q=\cfrac{16}{16}=1

Example 4: inverse proportion word problem y ∝ ¹⁄ₓ

$2$ workers paint a fence in $9$ hours.

How long would it take $6$ workers to paint the same fence?

Write down the inverse proportion formula.

As the number of workers $x$ increases, the time $y$ taken to paint the same fence would decrease and so this is an inverse proportion of the form $y\propto\cfrac{1}{x}.$

This means that you have the formula $y=\cfrac{k}{x}.$

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

As you know $x=2$ when $y=9,$ substituting these values into the formula, you get:

$\begin{aligned}9&=\cfrac{k}{2} \\\\ 9\times{2}&=k \\\\ k&=18 \end{aligned}$

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

Now you have the equation $y=\cfrac{18}{x}.$ As you want to know the value for $y$ when $x=6,$ you substitute $x=6$ into the equation to get:

$y=\cfrac{18}{6}$

Solve the equation.

As $18$ divided by $6$ is equal to $3,$ you have the solution $y=3.$

Example 5: inverse proportion word problem y ∝ ¹⁄ₓ₂

Let $n$ represent the number of dogs and $d$ represent the number of days. A bag of biscuits feeds $4$ dogs for $12$ days. Given that $d\propto\cfrac{1}{n^{2}},$ how many days would the same bag feed $5$ dogs?

Write down the inverse proportion formula.

As $d\propto\cfrac{1}{n^{2}},$ you can state the inverse proportion formula:

$d=\cfrac{k}{n^{2}}$

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

As $d=12$ when $n=4,$ substituting these into the formula, you have:

$\begin{aligned}12&=\cfrac{k}{4^{2}} \\\\ 12&=\cfrac{k}{16} \\\\ 12\times{16}&=k \\\\ k&=192 \end{aligned}$

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

Now $d=\cfrac{192}{n^{2}}.$ Now you have $5$ dogs, you can substitute $n=5$ into the formula to calculate the number of days of food they have:

$d=\cfrac{192}{5^{2}}$

Solve the equation.

$d=\cfrac{192}{25}=7.68$ days.

Example 6: inverse proportion word problem y ∝ ¹⁄ₓ₃

A cube of cheddar cheese has a density $d=1.075\text{g/cm}^{3}.$ The side length of the cube is $x=1.5cm.$ Another cube of cheese has a density $d=1.072\text{g/cm}^{3}.$ What is the side length of the second cube of cheese? Write your answer to the nearest thousandth.

Write down the inverse proportion formula.

The density of an object is equal to its mass divided by its volume.

As you know the mass and the side length of the first cube, you can determine the density using the inverse proportion formula $d=\cfrac{k}{x^{3}}$ where $k$ is the constant of proportionality.

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

As $d=1.075$ when $x=1.5,$ substituting these values into the formula, you get:

$\begin{aligned}1.075&=\cfrac{k}{1.5^{3}} \\\\ 1.075&=\cfrac{k}{3.375} \\\\ 1.075\times{3.375}&=k \\\\ k=3.628125&=\cfrac{1161}{320} \end{aligned}$

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

Now you have the equation $d=\cfrac{3.628125}{x^{3}}.$ As the second cube has a density $d=1.072\text{g/cm}^{3},$ substituting this into the equation, you get:

$1.072=\cfrac{3.628125}{x^{3}}$

Solve the equation.

\begin{aligned}1.072\times{x}^{3}&=3.628125 \\\\ x^{3}&=3.628125\div{1.072} \\\\ x^{3}&=3.384444963… \\\\ x&=\sqrt[3]{3.384444963…} \\\\ x&=1.50139795… \\\\ x&=1.501 \, cm\text{ (nearest thousandth)} \end{aligned}

Teaching tips for inversely proportional

Incorporate real-world examples on proportion worksheets such as speed and travel time. For instance, if a car’s speed increases, the time it takes to cover a fixed distance decreases. Relating this concept to everyday scenarios helps students grasp the idea intuitively.

Help students distinguish between direct and inverse proportion. Use comparative examples (for example, in direct proportion, both quantities increase together, while in inverse proportion, one increases as the other decreases).
This comparison can clarify the difference.

Point out where inverse proportionality appears in other areas, such as physics (example, pressure and volume of a gas in Boyle’s Law), economics (supply and demand), and biology (population size and food availability).

Make sure students understand that an inversely proportional graph will not be a straight line but a curve, where as one variable increases, the other decreases. You can have students plot simple inverse relationships, such as $y=\cfrac{1}{x},$ to observe the shape.

Easy mistakes to make

Making assumptions
Whenever you solve word problems for inverse proportion you assume that everything has the same rate. For example, if the question involves the number of people working, you assume all the workers work at the same rate.

Confusing the constant of proportionality
For direct proportion, the constant of proportionality $k$ is the ratio of the two variables such as $k=y\div{x}.$ For inverse proportion, $k$ is the product of the two variables, such as $k=xy.$

Allowing the graph of an inversely proportional relationship to cross either axis
The graph of any inversely proportional relationship cannot cross either axis. This is because, if you use the example $y\propto\cfrac{1}{x},$ if $x=0,$ the value for $y$ is undefined as you cannot divide a number by $0.$

If $y=0,$ then the value for the constant of proportionality must be $0$ or $x=0,$ but as was just stated, $x$ ≠ $0$ and so the relationship between $x$ and $y$ cannot exist for these two values as $k=0.$

Writing time using an incorrect decimal
Time is used in some inverse proportion word problems. If an answer is $3.1$ you may be tempted to write it as $3$ hours $10$ minutes, but it would be $3$ hours $6$ minutes. (Remember there are $60$ minutes in an hour).

Graphs of the reciprocal function
The standard graph for $y\propto{x}$ is a straight line graph that passes through the origin with the gradient $k.$ The standard graph for $y\propto\cfrac{1}{x}$ is a curved line that does not cross either axis.

Direct variation equation
Direct variation
Direct and inverse variation

Practice inverse proportion questions

160

40

20

10

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

k=2\times{40}=80

y=\cfrac{80}{x}

When $x=8, \, y=\cfrac{80}{8}=10.$

45

3.87

6.708

15

$n=\cfrac{k}{m^2}$ and so $k=n m^2.$

k=3^2 \times 40=9 \times 40=360

n=\cfrac{360}{m^2}

When $n=8,$

\begin{aligned}& 8=\cfrac{360}{m^2} \\\\ & m^2=\cfrac{360}{8}=45 \\\\ & m=\sqrt{45}=6.708 \end{aligned}

1.25

$3.9$ (nearest tenth)

20

2

$b=\cfrac{k}{\sqrt{a}}$ and so $k=b\sqrt{a}.$

k=5\times{\sqrt{4}}=5\times{2}=10

b=\cfrac{10}{\sqrt{a}}

When $a=25, \, b=\cfrac{10}{\sqrt{25}}=\cfrac{10}{5}=2.$

$20$ days

$24$ days

$12$ days

$108$ days

Let $p$ represent the number of people and $d$ represent the number of days of building, then

$d=\cfrac{k}{p}$ and so $k=dp.$

k=36\times{20}=720

d=\cfrac{720}{p}

When $p=60, \, d=\cfrac{720}{60}=12.$

$60$ minutes

$20$ minutes

$80$ minutes

$10$ minutes

Let $s$ represent speed and $t$ represent time taken, then

$s=\cfrac{k}{t}$ and so $k=st.$

k=5\times{40}=200

s=\cfrac{200}{t}

When $s=10,$

\begin{aligned}10&=\cfrac{200}{t} \\\\ 10\times{t}&=200 \\\\ t&=200\div{10} \\\\ t&=20 \end{aligned}

Note, the speed-distance-time formula is a known relationship (speed = distance ÷ time).

For a constant distance, if the time taken to reach the destination increases, the speed must have decreased, or vice versa.

16

7

80

259

You have $c=\cfrac{k}{b^{2}}$ and so $k=b^{2}c.$

k=12^{2}\times{2.7}=388.8

c=\cfrac{388.8}{b^2}

When $c=1.5,$

\begin{aligned}1.5&=\cfrac{388.8}{b^{2}} \\\\ 1.5\times{b^{2}}&=388.8 \\\\ b^{2}&=388.8\div{1.5} \\\\ b^{2}&=259.2 \\\\ b&=\sqrt{259.2} \\\\ b&=16.09968944… \\\\ b&=16\text{ bags} \end{aligned}

Inversely proportional FAQs

What does inversely proportional mean?

Inversely proportional means that as one quantity increases, the other decreases in such a way that their product remains constant.

What is the formula for inversely proportional relationships?

The formula is $x \times y=k,$ where $x$ and $y$ are the two variables and $k$ is the constant. You can also express this as $y=\cfrac{k}{x}.$

How is an inverse proportion graph different from a direct proportion graph?

A graph of an inverse proportion forms a curve that approaches the axes but never touches them (a hyperbola), while a direct proportion graph is a straight line passing through the origin.

How do inversely proportional relationships differ from directly proportional ones?

In directly proportional relationships, both quantities increase or decrease together, whereas in inversely proportional relationships, as one quantity increases, the other decreases.

The next lessons are

Converting fractions decimals and percents
Percents
Compound measures

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.