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

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

The **direct proportion formula** is an algebraic formula which represents the directly proportional relationship between two variables.

**Direct proportion** is one type of a proportionality relationship. As one value increases, so does the other value.

The proportionality symbol is **∝**. If the variables x and y are directly proportional then we can write the relationship using the proportionality symbol.

We can also write this as a formula using the constant of proportionality, k.

k is a constant value that links the two variables.

y=kxProportional relationships can also be represented by graphs.

The following is a type of direct proportion graph.

It is a straight line graph going through the origin. It has the equation y=kx.

Direct proportion formulas can involve powers and roots. If the variables were x and y, and y is directly proportional to x^{2}, we can write the relationship using the proportionality symbol.

y\propto x^2We can also write this as a formula using the constant of proportionality k.

y=kx^2The graph of this relationship would be quadratic and have the shape of a parabola.

In order to recognise a direct proportion formula:

**Look for an equation in the form**y=kx^n.**Check there are no other terms in the equation.**

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

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

DOWNLOAD FREEWhich of these equations indicate that y\propto x?

A B C D

y=x+5 \hspace{1cm} y=\frac{5}{x} \hspace{1cm} y=1-5x \hspace{1cm} y=5x**Look for an equation in the form**y=kx^n.

We are looking for a direct proportion equation. We are looking for an equation with the variable, x^n, being multiplied by a number.

We can eliminate equation B as this has a \frac{1}{x} term, this would be inverse proportion.

2**Check there are no other terms in the equation.**

For direct proportion there can be no addition or subtraction involved in the equation.

So we can eliminate equations A and C.

The only equation which only has an x^n -term being multiplied by a number is equation D. Here the power of the x is 1.

y=5x indicates that y \propto x.

Which of these equations indicate that a direct proportion?

A B C D

y=\frac{4}{x} \hspace{1cm} y=4x^2 \hspace{1cm} y=4x-3 \hspace{1cm} y=4x+2**Look for an equation in the form ** y=kx^n.

We are looking for a direct proportion equation. We are looking for an equation with the variable, x^n, being multiplied by a number.

We can eliminate equation A as this has a \frac{4}{x} term, this would be inverse proportion.

**Check there are no other terms in the equation.**

For direct proportion there can be no addition or subtraction involved in the equation.

So we can eliminate equations C and D.

The only equation that only has an x^n -term being multiplied by a number is equation B. Here the power of the x is 2.

y=4x^2 indicates y\propto x^2.

In order to construct a direct proportion formula:

**Write down the general proportion formula.****Find the constant**\textbf{k}**, the constant of proportionality.****Write down the equation.**

y is directly proportional to x.

When y=35, \ x=5.

Find a formula for y in terms of x.

**Write down the general proportion formula.**

From the first sentence we can write down the relationship as

y\propto x.

We can then write this as a formula using the constant of proportionality k,

y=kx .

Where k is the constant of proportionality.

**Find the constant of proportionality.**

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

We are told ‘when y=35, \ x=5’.

\begin{aligned}
y&= kx \\\\
35 &= k \times 5 \\\\
k&=35 \div 5=7
\end{aligned}

**Write down the equation.**

To finish off we can write the direct proportion equation as

y=7x.

y is directly proportional to x.

When y=4, \ x=20.

Find a formula for y in terms of x.

**Write down the general proportion formula.**

From the first sentence we can write down the relationship as

y\propto x.

We can then write this as a formula using the constant of proportionality k,

y=kx .

Where k is the constant of proportionality.

**Find the constant of proportionality.**

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

We are told ‘when y=4, \ x=20’.

\begin{aligned}
y&= kx \\\\
4 &= k \times 20 \\\\
k&=4 \div 20=0.2
\end{aligned}

**Write down the equation.**

To finish off we can write the direct proportion equation as

y=0.2x \ \text{or} \ y=\frac{x}{5}.

y is directly proportional to \sqrt{x}.

When y=40, \ x=4.

Find a formula for y in terms of x.

**Write down the general proportion formula.**

From the first sentence we can write down the relationship as

y\propto \sqrt{x}.

We can then write this as a formula using the constant of proportionality k,

y=k\sqrt{x} .

Where k is the constant of proportionality.

**Find the constant of proportionality.**

We can find this unknown value, the constant of proportionality by substituting in the corresponding values we are given.

We are told ‘when y=40, \ x=4’.

\begin{aligned}
y&= k\sqrt{x} \\\\
40 &= k\times \sqrt{4} \\\\
40 &= k\times 2 \\\\
k&=40 \div 2=20
\end{aligned}

**Write down the equation.**

To finish off we can write the direct proportion equation as

y=20\sqrt{x}.

**Confusing direct proportion and inverse proportion**

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

Direct proportion | Inverse proportion |
---|---|

y\propto x^n | y\propto \frac{1}{x^n} |

y=kx^n | y=\frac{k}{x^n} |

**Not recognising and applying any powers or roots**

Most direct proportional formulas just involve x, but they can involve powers such as x^{2} or roots such as \sqrt{x} . \sqrt{x} is the same as x^{\frac{1}{2}} .

**Assuming the constant of proportionality can only be an integer**

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

For example,

y=\frac{x}{2}

Even though there is a fraction, the constant of proportionality is \frac{1}{2} or 0.5.

y=\frac{x}{2}=\frac{1}{2} \times x=0.5 \times x

**Direct 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 x?

y=2x

y=x+2

y=\frac{x}{2}

y=20x

For direct proportion there can be no addition or subtraction involved in the equation. The incorrect equation is y=x+2.

2. Which of these equations indicate y \propto x?

y=x+5

y=5x

y=\frac{5}{x}

y=5x^2

For direct proportion there can be no addition or subtraction involved in the equation. The correct equation is y=5x.

3. Which of these equations indicates direct proportion?

y=4-10x

y=10x+4

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

y=10x^3

For direct proportion there can be no addition or subtraction involved in the equation. The correct equation is y=10x^3.

4. y is directly proportional to x.

When y=24, \ x=8.

Find a formula for y in terms of x.

y=4x

y=\frac{3}{x}

y=\frac{x}{4}

y=3x

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

\begin{aligned} y&\propto x \\\\ y&= kx \\\\ 24 &= k\times 8 \\\\ k&=24\div 8=3 \end{aligned}

So, the equation is

y=3x.

5. y is directly proportional to x.

When y=10, \ x=5.

Find a formula for y in terms of x.

y=50x

y=2x

y=\frac{x}{2}

y=\frac{2}{x}

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

\begin{aligned} y&\propto x \\\\ y&= kx \\\\ 10&= k\times 5 \\\\ k&=10\div 5=2 \end{aligned}

So, the equation is

y=2x.

6. y is directly proportional to x^{2}.

When y=18, \ x=3.

Find a formula for y in terms of x^{2}.

y=6x

y=2x^2

y=6x^2

y=3x

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

\begin{aligned} y&\propto x^2 \\\\ y&= kx^2 \\\\ 18&= k\times 3^2 \\\\ 18&= 9k \\\\ k&=18\div 3=2 \end{aligned}

So, the equation is

y=2x^2.

1. y is directly proportional to x and k is a constant.

Identify the correct equation.

Equation A – y=\frac{k}{x}

Equation B – y=kx

Equation C – y=k-x

Equation D – y=\frac{x}{k}

**(1 mark)**

Show answer

Equation **B**

**(1)**

2. y is proportional to x.

When y=36, \ x=12.

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

(b) Find the value of x when y=50.

**(4 marks)**

Show answer

(a)

\begin{aligned} y&=kx \\\\ 36&=k \times 12 \\\\ k&=36 \div 12=3 \end{aligned}

**(1)**

**(1)**

(b)

50=3x \ so \ x=50 \div 3

**(1)**

**(1)**

3. m is directly proportional to p .

When p=5, \ m=20.

Find m when p=3.

**(3 marks)**

Show answer

m=kp \ or \ 20=k \times 5

**(1)**

k=4 \ or \ m=4p

**(1)**

**(1)**

You have now learned how to:

- Recognise a direct proportion formula
- Construct a direct proportion equation
- Model situations by translating them into algebraic formulae

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