High Impact Tutoring Built By Math Experts

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

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

Substitution Exponents Square rootWhat is a proportion in math

Directly proportional Rearranging equationsHere 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.

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.

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.

Use this quiz to check your grade 6 to 7 studentsβ 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 FREEUse this quiz to check your grade 6 to 7 studentsβ 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 FREEIn order to recognize when two variables are inversely proportional to one another:

**The product of the two variables will equal a constant.**

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.

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.

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.**

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}.

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}}.

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}

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}}.

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}

- 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.

**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

1. Which of these equations does **not** indicate y\propto\cfrac{1}{x}?

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.

2. Which of these equations indicateΒ y\propto\cfrac{1}{x}?

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.

3. Which of these equations indicate y\propto\cfrac{1}{x^{2}}?

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

4. y is inversely proportional to x.

When y=2, x=8.

Find a formula for y in terms of x.

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}.

5. y is inversely proportional to x.

When y=10, x=5.

Find a formula for y in terms of 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}Β

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

6. y is inversely proportional to \sqrt{x}.

When y=2, x=9.

Find a formula for y in terms of \sqrt{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}}.

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.

“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.

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) .

- Converting fractions decimals and percents
- Percent
- Compound measures

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.

Find out how we can help your students achieve success with our math tutoring programs.

x
####
[FREE] Common Core Practice Tests (3rd to 8th Grade)

Download free

Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.

Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!