# Direct Proportion Formula

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.

## What is the direct proportion formula?

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.

y\propto x

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=kx

Proportional 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^2

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

y=kx^2

The graph of this relationship would be quadratic and have the shape of a parabola.

## How to recognise a direct proportion formula

In order to recognise a direct proportion formula:

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

## Direct proportion formula examples

### Example 1: recognise the direct proportion formula

Which 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

1. 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.

2Check 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.

### Example 2: recognise the direct proportion formula

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.

Check there are no other terms in the equation.

## How to construct a direct proportion formula

In order to construct a direct proportion formula:

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

## Constructing direct proportion formula examples

### Example 3: constructing a direct proportion 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.

Find the constant of proportionality.

Write down the equation.

### Example 4: constructing a direct proportion equation

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.

Find the constant of proportionality.

Write down the equation.

### Example 5: constructing a direct proportion equation involving a power or root

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.

Find the constant of proportionality.

Write down the equation.

### Common misconceptions

• Confusing direct proportion and inverse proportion

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

Direct proportionInverse 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

## Related lessons on direct proportion formula

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:

### Practice direct proportion formula questions

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.

### Direct proportion formula GCSE questions

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)

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)

(a)

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

(1)

y=3x

(1)

(b)

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

(1)

x=\frac{50}{3}=16\frac{2}{3}

(1)

3. m is directly proportional to p .

When p=5, \ m=20.

Find m when p=3.

(3 marks)

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

(1)

k=4 \ or \ m=4p

(1)

m=12

(1)

## Learning checklist

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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