Formulae and Equations

Formulae and Equations

Creating formulae

mathsYou may not realise it but you often use formulae without even realising. For instance, say you were going on holiday to France and wanted to convert pounds into Euros or you need to work out the circumference of a circle for a wood work project.

For your exam you’ll be expected to know how to construct your own formulae using given data.

For example, imagine a rectangle: the longer sides are 3x and the shorter sides are x. Write down the perimeter of this rectangle as a formulae.

So, the rectangle will be made up of two 3x sides and two x sides.

The perimeter of a rectangle is all the sides added together.

Using this information we can make the following equation for the perimeter (P):

P = 3x + 3x + x + x = 8x



Substituting is where you replace a letter in a formulae with a number.

For example, a shop sells ‘n’ amount of cloth for 60p per metre. This can be written in an equation as:

C = 0.60 x n

A woman wants to buy 7 metres: how much will this cost her? Replace ‘n’ by 7:

C = 0.60 x 7 = 4.2

So 7m would cost her £4.20.


Changing the subject

To find the circumference of a circle you simply follow the equation:


However, if you know the circumference and in fact want the radius then you need to rearrange the equation to make r the subject. For that, you need to divide both sides by 2? so you then end up with:


Remember to always do to one side what you did to the other and to cancel on one side do the opposite: for instance, in this example you needed to divide as the 2? was already multiplied.


Harder Formulae (higher tier)

For the higher tier exam you’ll be dealing with harder formulae. A good starting block is to remind yourself of the opposite pairs:





For example, the volume of a sphere is V = 4/3 ? r3. Make r the subject.

First of all, multiply both sides by 3.

3V = 4 ? r3

Then divide both sides by 4?

3V/4? = r3

Then cube root each side:

3? 3V/4? = r or r = 3? 3V/4?


Index notation

Instead of using a number and a power you can use a letter. This is known as index notation.

An example where the index is 2 would be a2 or a x a.

b3 = b x b x b and so the index is 3 and so on.

Sometimes, you could get a number in front of the variable, like 3c2. This would be the equivalent of 3 x c x c.


Index laws: multiplying and subtracting

To multiply or divide indices there are a few of laws you need to follow.

Index numbers follow the same rules as indices. So:

  • – if it’s multiplication you add them
  • – if it’s division you subtract them

However, the variable in front follows the general rule of multiplication or division.

For example: 4a2 x 5a5

So, you’d add the indices: 2 + 5 = 7

And multiply the variables: 4 x 5 = 20

Therefore, final answer would be:

4a2 x 5a5 = 20a7

Index laws: adding and subtracting

With adding and subtracting you can only do so with like terms.

For instance, p and 2p are like terms or p2 and -4p2.

However, 6p and 6p2 are not like terms.

For example: what is 6d2 + 4d2 + 3d + 8d2

6d2, 4d2 and 8d2 are all like terms but 3d is not. So, the answer is:

6d2 + 4d2 + 3d + 8d2 = 3d + 18d2

You might get a question in which you have to substitute a number into an equation. You’ll be given the equation and what you should substitute in.

For example: what is the value of 5r + 4r3 when r = 3.

Simply substitute all the r’s in the equation with 3:

(5 x 3) + (4 x 33)= 15 + 36 = 51

(Remember that 33 = 3 x 3 x 3).