# GCSE Maths: Direct And Inverse Proportion

## Introduction

As part of the GCSE Higher Maths Paper you will cover a key topic known as direct and inverse proportion

With this topic you need to establish an algebraic equation whilst establishing something known as the constant of proportionality which is generally represented by the letter k.

1. solve problems where two unknowns have a proportional on inverse proportional relationship
2. define the constant of proportionality

Direct proportion is also known as direct variation. It is simply a way of saying that there is a direct proportion between two variables when one is a multiple of another.

For instance a bag of sugar weighs 1 kg which is 2.2 pounds. So we have a multiple of 2.2.

Quite often we need to determine what this multiplying factor is and this is known as the constant of proportionality

## Deciding if it is Direct And Inverse Proportion

When you are in employment your pay is usually directly proportional to the amount of time involved. We can write this in terms of maths as follows:

\text { pay } \propto \text { time }

OR                                              \text { pay }=k \times \text { time }

The multiplying factor or constant of proportionality is represented by k.

When it comes to determining what the value of k actually is, we need to set up an equation.

Example:

The cost of writing a magazine is directly proportional to the time spent making it. If the magazine costs £30 and takes 6 hours to make find:

• The cost of a magazine that takes 4 hours to make
• The length of time needed to make a magazine only costing £40

Solution:

First we need to set up an actual equation here:

\begin{gathered} C \propto T \\ C=k T \\ 30=k 5 \\ k=6 \end{gathered}

Because we have found the value of k, our equation is C = 6T and we can use this to actually answer the questions.

1. C = 6 x 4 = £24
2. 40 = 6T 🡪 T = 40 ÷ 6 = £6.67

## Direct proportion involving powers and roots

Example:

The cost of a circular badge is proportional to the square of its radius. The cost of a badge with radius 2cm is 68 pence. Find:

Solution:

Again we need to create a equation: C \propto r^2 and this can be written as C=k r^2 \text {. }

Next we need the value of k.

68=k 2^2 \rightarrow k=68 \div 4=17C=17 r^2

a) \mathrm{r}=\mathbf{3}: C=17 \times 3^2=153 \text { pence }=\mathbf{f} 1.53

b) \mathrm{C}=\mathbf{f} 1.63=163 \text { pence: } 163=17 r^2 \rightarrow r=\sqrt{\frac{163}{17}}=\mathbf{3 . 1 0} \mathrm{cm}

The key with establishing the correct type of equation is first to know and understand what the relationship is. This you can obtain from carefully reading the question. Our expert online maths tutors are on hand to help you further with this topic or indeed any other area of GCSE Maths as well helping you succeed with the final summer exams.

1. Inverse Proportion

Inverse proportion is also known as inverse variation.

We have an inverse variation between 2 variables when one is directly proportional to the reciprocal of the other.

So as one variable increases another decreases. An example of this is speed and time. The faster you walk the less time is needed to reach your destination. The slower you walk the more time is needed to reach your destination.

That means speed is inversely proportional to time and we can write this as S \propto \frac{1}{T} OR S=\frac{k}{T}, where k is the constant of proportionality.

Example:

Q is inversely proportional to R. If Q = 9 when R = 4. Find

• Q when R = 2
• R when Q = 3

Solution:

Q \propto \frac{1}{R} OR Q=\frac{k}{R}

9=\frac{k}{4} \text { i.e. } k=36 . \text { Our equation is } Q=\frac{36}{R}

Part a) Q=\frac{36}{2}=18

Part b) 3=\frac{36}{R} \rightarrow R=12

What is important with this topic is to ensure that you are reading the question correctly and to know if you are dealing with a direct or inverse proportion question and also what the relationship is such as is one the square of the other or the cube of the other and so on?

During our half term revision courses for GCSE Maths we explore some of the more harder questions surrounding this area of maths as well as help you feel more confident for any upcoming mocks and final year 11 exams.

You should try the above direct and inverse proportions again on your own to make sure that you understand them. Always look carefully at the wording of the question as this will tell you what you are working with.

Questions involving direct and inverse proportion will always appear on the higher paper and not the foundation and the subject knowledge is also seen at A Level.

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