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Substitution Powers and roots Direct proportion Rearranging equationsThis topic is relevant for:

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

There are also inverse proportion formula* *worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if youβre still stuck.

The **inverse proportion formula** is an algebraic formula which represents the inverse proportion relationship between two variables.

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

y\propto \frac{1}{x}.If y is inversely proportional to x, the constant of proportionality k=xy.

Rearranging this equation to make y the subject, we have

y=\frac{k}{x}.An inverse proportion formula can involve **powers and roots**.

For example, if the variables were x and y and y is inversely proportional to x^2 we can write the relationship using the proportionality symbol as

y\propto \frac{1}{x^2}.This can be written as an equation using the constant of proportionality, k,

y=\frac{k}{x^2}.To recognise 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.

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

**The product of the two variables must always equal a constant.**

Get your free inverse proportion formula worksheet of 20+ inverse proportion questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free inverse proportion formula worksheet of 20+ inverse proportion questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWhich of these equations indicate that y\propto \frac{1}{x}?

ABC D

y=\frac{x}{2} \quad \quad y=\frac{2}{x} \quad \quad y=x^2 \quad \quad y=2x**The product of the two variables must always equal a constant**

For A, we have the two variables x and y. If we multiply both sides by x so we have xy=\frac{x^2}{a}, \ xy is not a constant as \frac{x^2}{a} will change, depending on the value for x. So x and y are not inversely proportional for A.

For B, we have the two variables x and y. If we multiply both sides by x, we 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, we have the two variables y and x^{2}. If we multiply both sides of the equation by x^{2}, we 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, we have the two variables y and x. If we multiply both sides by x, we 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?

ABC D

y=x^3 \quad \quad y=\frac{x}{3} \quad \quad y=3x \quad \quad y=\frac{3}{x}**The product of the two variables must always equal a constant**

For A, we have the two variables x^{3} and y. If we multiply both sides by x^{3} so we 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, we have the two variables x and y. If we multiply both sides by x, we have xy=\frac{x^{2}}{3}. This means that xy is not a constant as \frac{x^{2}}{3} will change, depending on the value for x. So x and y are not inversely proportional for B.

For C, we have the two variables x and y. Multiplying both sides by x, we have xy=3x^2 which is not a constant value and so y is not inversely proportional to x for C.

For D, we have the two variables x and y. Multiplying both sides by x, we 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, we can write y\propto\frac{1}{x} and so we have the formula y=\frac{k}{x} where k is the constant of proportionality.

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

We can find the constant of proportionality by substituting in the values we are given.

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

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

As k=30, we have the equation

y=\frac{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\frac{1}{x^{3}} and so y=\frac{k}{x^{3}}.

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

As y=4 when x=2,

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

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

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

y=\frac{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 we can write down y\propto \frac{1}{x^2}.

We can therefore state

y=\frac{k}{x^2}

where k is the constant of proportionality.

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

We can find the constant of proportionality by substituting in the values we are given.

Here x=2 and y=10 .

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

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

As k=40 we can write down the equation

y=\frac{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\frac{1}{\sqrt{a}} and so b=\frac{k}{\sqrt{a}} .

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

When a=4, \ b=1 .

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

Note that we 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, we have the inverse proportion formula

b=\frac{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\frac{1}{t} and so p=\frac{k}{t}.

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

When p=6, \ t=48 and so

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

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

Now that k=288, we have the inverse proportion formula

p=\frac{288}{t}.

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

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

**Direct proportion and inverse proportion**

You will need to learn which formula is for which type of proportionality.

Direct proportion Inverse proportion

\hspace{0.5cm} y\propto x \hspace{3.5cm} y\propto \frac{1}{x}

\hspace{0.5cm} y=kx \hspace{3.35cm} y=\frac{k}{x}

**Check if there are any powers or roots involved**

Most inverse proportional formulae just involve x, but they can involve powers such as x^2 or roots such as \sqrt{x}.

**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=\frac{5}{2x}=\frac{5}{2}\times\frac{1}{x}=2.5\times\frac{1}{x}=\frac{2.5}{x} and so k=2.5.

**Inverse proportion** **formula** is part of our series of lessons to support revision on **proportion**. You may find it helpful to start with the main proportion lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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

y=\frac{2}{x}

y=5x

xy=8

y=\frac{1}{3x}

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

2. Which of these equations indicate y\propto\frac{1}{x}?

y=5x

y=\frac{5}{x}

y=\frac{x}{5}

y=5x^2

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

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

y=\frac{10}{x}

y=10x^2

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

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

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

For y=\frac{10}{x^{2}}, multiplying both sides by x^{2}, we have x^{2}y=10 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=\frac{4}{x}

y=\frac{x}{4}

y=\frac{16}{x}

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

As y=2 when x=8, \ y=\frac{k}{x}

\begin{aligned} 2&= \frac{k}{8} \\\\ k&=2\times 8=16 \end{aligned}

So the equation is y=\frac{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=\frac{50}{x}

y=50x

y=\frac{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\frac{1}{x} and so y=\frac{k}{x}.

As y=10 when x=5,

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

So the equation is y=\frac{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=\frac{6}{\sqrt{x}}

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

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

y=6\sqrt{x}

As y\propto\frac{1}{\sqrt{x}}, \ y=\frac{k}{\sqrt{x}}

As y=2 when x=9,

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

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

1. Let y be inversely proportional to x, where k is a constant.

Identify the correct equation.

Equation AEquation B Equation C Equation D

\hspace{.5cm} y=\frac{k}{x} \hspace{1.5cm} y=kx \hspace{1.4cm} y=k-x \hspace{1.4cm} y=\frac{x}{k}

**(1 mark)**

Show answer

Equation A or y=\frac{k}{x} .

**(1)**

2. Let y be inversely proportional to x, When y=9, \ x=3.

(a) Find the equation for y in terms of x.

(b) Find the value of x when y=10.

**(5 marks)**

Show answer

(a)

y=\frac{k}{x}

**(1)**

**(1)**

**(1)**

(b)

10=\frac{27}{x} or x=27\div 10 .

**(1)**

**(1)**

3. The number of workers (w) building a pyramid is inversely proportional to the time taken to build the pyramid (t).

It takes 9 men, 5 years to build a pyramid.

How many workers would be needed to build the same pyramid in 3 years?

**(3 marks)**

Show answer

w=\frac{k}{t} or 9=\frac{k}{5}

**(1)**

**(1)**

**(1)**

You have now learned how to:

- Solve problems involving inverse proportion, including algebraic representations

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