## Topic Overview

A formula is a type of equation which shows the relationship between different variables such as ‘x’ and ‘y’.

For an equation to be a formula it must have more than one variable. For example:

x + y = 7

is a formula because it shows the relationship between ‘x’ and ‘y’. Whereas:

2x – 8= 0

is not a formula because it only has one variable, ‘x’.

The subject of a formula is the single variable to which everything else in the formula is equal. The subject of a formula will usually be positioned to the left of the equals sign. For example:

x= 2y + 4z

In this example ‘x’ is the subject of the formula because everything else is equal to it.

Using the mathematical principles which are demonstrated in the worked examples below, you can rearrange a formula in order to make another variable the subject.

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems related to formulae and their subject. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Recognise the properties of a formula
- Substitute numbers into a formula
- Change the subject of a formula

Listed below are a series of summaries and worked examples to help you solidify your knowledge about how to change the subject of a formula.

## Worked Examples

**1-Using mathematical formulae**

During your GCSE maths exam, you may be required to construct mathematical formulae out of various data.

*Example*

(a) – A DVD rental company charges £0.50 per day to rent a film, plus an initial fixed price of £1.

Write a mathematical formula for the cost of renting a DVD (C) for (n) number of days.

*Solution*

(a) – The question has given you all the information you need in order to construct a formula.

The cost (C) of renting a day is the initial fixed charge (£1) plus a £0.50 charge for a non-specified number of days (n).

This information can be written as:

Cost = (Initial Fixed Price of £1) + (£0.50 charge x number of days rented )

Therefore, as a formula, this information can be written as:

C = £1 + (n x £0.50)

**2-Substituting numbers into a formula**

After you have constructed a formula, you may be asked to use it to solve various questions. These questions will often involve substituting numbers into the formula.

*Example*

(a) – Using the formula you have just constructed, work out the cost of renting a DVD for 22 days

*Solution*

(a) – To work out how much it would cost to rent a DVD for 22 days, you substitute the value 22 into the variable ‘n’ which represents ‘number of days rented’. Therefore:

C = £1 + (n x £0.50)

C = £1 + (22 x £0.50)

C = £1 + £11

C = £12

Therefore, it would cost £12 to rent a DVD for 22 days

**3-Changing the subject of a formula**

Once you have understood the basic principles of mathematical formulae, you can tackle more complex problems such as hanging the subject of a formula.

Changing the subject of a formula is very similar to using inverses to solve mathematical equations.

Therefore you must remember the following rules:

- adding and subtracting are the opposite of one another
- multiplying and dividing are the opposite of one other
- when using inverses you must perform the same method to both sides of the equation

*Example*

(a) – Make ‘x’ the subject of the formula:

Z = 4x + 4y

*Solution*

(a) – To make ‘x’ the subject of this formula, first you must move 4y to the other side of the equals sign:

Z(- 4y) = 4x + 4y (-4y)

Z -4y = 4x

Now you need to get ‘x’ on its own so you must divide both sides by 4. By doing so you have changed the subject of the formula from ‘Z’ to ‘x’ :

4x ÷4 = (Z- 4y) ÷ 4

x = (Z -4y)/ 4

*Example*

(b) – Make ‘q’ the subject of the formula :

P = 4/7 (Q – 28)

*Solution*

(b) – When you have to change the subject of a formula which includes a fraction, you must first multiply both sides of the formula by the denominator of the fraction to get rid of it:

P x 7 = 4/7 (Q – 28) x 7

7P = 4(Q – 28)

Next you must expand the brackets on the right hand side of the formula:

7P= 4Q – 112

From here, you can get 4Q on its own and divide to get Q:

7P +112 = 4Q

Q = (7P +112) / 4

*Example*

(c) – Make ‘b’ the subject of the formula:

C = b2 / 8π

*Solution*

(c) – Multiply both sides by 8π:

8πC = b2

Take square roots to get b on its own:

b = √(8πC)

*Example*

(d) – Make ‘a’ the subject of the formula:

s = ut +½at2

*Solution*

(d) – First subtract ‘ut’ from both sides:

s – ut = ½at2

Then multiply both sides by 2 to get rid of the fraction:

at2 = 2(s – ut)

Finally, divide both sides by t2 to get ‘a’ on its own:

at2 ÷ t2 = 2(s – ut) ÷ t2

a = 2(s – ut) / t2

## Exam Tips

- When changing the subject of a formula, use the rules of inverses: always remember that you must perform the same method to both sides of the equation
- Remember that adding and subtracting are the opposite of one another
- Remember multiplying and dividing are the opposite of one other
- When faced with an equation which has multiple solutions, look at each bracket individually and then solve them to find each value
- When you have to change the subject of a formula which includes a fraction, you must first multiply both sides of the formula by the denominator of the fraction to get rid of it

## Topic Summary

Once you have a strong understanding of the fundamental properties of mathematical formulae, you can begin to confidently apply this knowledge to more complex problems. When constructing and substituting values into different formulae, it is crucial that you write down ALL of your working out. This is because, even if you write down an incorrect formula, if you substitute values into it using the correct process, you can still earn vital method marks! Moreover, if you write down all of your working out, then you are more likely to notice any mistakes you have made and then correct them. Seemingly small points such as these can make be the difference between a B and an A grade!

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