Key points about formulae
A A mathematical rule that links variables, by substituting the values that can be changed with words or letters, eg πΒ² + πΒ² = πΒ². A formula contains an equals (=) sign. is a mathematical rule or relationship that uses words or letters to represent amounts that can be changed (variables).
Values must be In algebra, to replace a letter with a number. into A mathematical rule that links variables, by substituting the values that can be changed with words or letters, eg πΒ² + πΒ² = πΒ². A formula contains an equals (=) sign. using the correct order of operations (A way to remember the correct order of operations when calculating the value of an expression. BIDMAS stands for Brackets, Indices, Division, Multiplication, Addition, Subtraction. ).
The The subject of a formula or equation is the variable on its own on one side of the equals sign, eg π¦ is the subject in π¦ = 3π₯ + 5. of a formula can be changed so that a different variable appears at the start. This requires ββββThe opposite of a mathematical process. For example, the inverse of Γ 5 is Γ· 5. The inverse operation undoes the original process.Β .
Refresh your understanding of this topic by looking at this guide on calculating with negative numbers and using BIDMAS.
Check your understanding
Video β Creating formulae
Watch this video to find out how to create algebraic formulae for real-life situations, using worked examples.
Creating formulae.
An algebraic formula is a bit like an equation. It has an equals sign and variables; these are letters that represent values that can be changed. It might also have a constant, which is a number that always stays the same.
Let's look at how to create formulae for real life situations.
Vina receives Β£20 from her parents for her birthday. She then earns Β£30 each week from her Saturday job. She puts all the money she receives into a savings account.
Part A of this question asks: write a formula for the amount of money, π, in pounds, Vina will have saved after π₯ weeks.
To answer this, you can use a bar model where a 20 bar represents the initial Β£20 Vina receives from her parents and separate 30 bars represent the extra Β£30 Vina earns each week from her Saturday job.
So, after 1 week, Vina will have a total of Β£20 add Β£30. Then, after 2 weeks, Vina will have a total of Β£20 add two Β£30. After 3 weeks, Β£20 add three Β£30, and so on. The number of 30 bars is equal to the number of weeks. So, after π₯ weeks there will be π₯ 30 bars.
The total amount of money Vina has saved is equal to the sum of all the bars. So, after π₯ weeks, Vina will have Β£20 add π₯ times Β£30, or 20 add 30π₯. In the question, π is used to represent this amount. So, the formula is π equals 20 add 30π₯.
Part B of the question asks you to work out how much money, in pounds, Vina would have saved after 12 weeks.
π₯ represents the number of weeks. So, substitute π₯ equals 12 into the formula. 30 times 12 is 360 and 360 add 20 is 380. So, Vina will have saved Β£380 after 12 weeks.
Let's look at another question.
Bruno adds Β£56 to his travel card. Every day he buys a Β£2 bus ticket using this money. Write a formula for how much money, π, in pounds, Bruno has left on his travel card after π₯ days.
Pause the video and have a go.
Be careful here. This time, the amount heβs spending has been taken away from the initial Β£56.
Starting with a 56 bar, after 1 day, Bruno will have 56 subtract 1 times Β£2, which is Β£54. Then, after 2 days he will have 56 subtract 2 times Β£2, or Β£52. After 3 days, 56 subtract 3 times Β£2, or Β£50, and so on. So, after π₯ days, he will have 56 subtract Β£2π₯ left.
π is how much money Bruno has left. So, the formula is π equals 56 subtract 2π₯.
Writing formulae
Formulae are used in many areas of maths and science, eg for finding the area of a shape or for calculating speed.
Formulae are also helpful in everyday life, for example working out the time needed to cook something according to its weight - if we know certain values we can create a formula to work out an unknown.
For example, the formula to work out the area of a triangle, π¨, in terms of its base, π, and its height, π, isβ:β
π¨ = \(\frac{bh}{2}\)
π¨ is the The subject of a formula or equation is the variable on its own on one side of the equals sign, eg π¦ is the subject in π¦ = 3π₯ + 5. of the formula. π and π are A quantity that can take on a range of values, often represented by a letter, eg π, π₯, π¦, π§, π‘ β¦ etc. because they can change depending on the values of the base and height. To calculate the area, In algebra, to replace a letter with a number. π and π with values.
Follow the working out below
GCSE exam-style questions
- ββA taxi company charges a one-off charge of Β£6.50, and then 95p per mile. βββ
Write a formula for βπΆβ, the total cost of a journey in Β£, in terms of the number of miles, π.
Β Β Β Β Β Β
πͺ = 6Β·5ββββ + 0Β·95πββ
The units must all be in pounds. The total cost in pounds, βββββπΆβ, is equal to the one-off charge plus Β£0.95 for each mile.
Β£6.50 can be written as 6Β·5ββββ. There is no need to write the Β£ symbol in the formula because βπΆ is the cost in pounds.
Remember to include ββπΆβ =β. A formula must have a subject and an equals sign.
- A factory fills cat litter into different sized bags. Each bag contains 800 g, 1Β·4 kg or 2 kg.
Write a formula for π, the total mass of the cat litter filled in kg, in terms of the number of small bags (βπ ), medium bags (βπβ) and large (βπβ) bags.
π» = 0Β·8π + 1Β·4π + 2π
The mass of all the small bags is the number of small bags multiplied by 0Β·8, and similarly for the medium and large bags.
The total mass, π, is equal to the three expressions added together.
The formula must include βπ =β at the start.
Substituting into expressions and formulae
An expression is a mathematical sentence expressed either numerically or symbolically, made up of one or more An element within an algebraic sentence, eg π₯ or 5π¦ or 3πΒ².Β Elements (terms) are separated by + or β signs..
A formula is a mathematical rule or relationship that uses words or letters to represent amounts that can be changed, known as variables.
Substitution means putting numbers in place of letters to calculate the value of an A mathematical sentence expressed either numerically or symbolically made up of one or more terms, eg 8 + 2, or 6π₯, or 5π₯Β² + 3π¦, or 3πππ. . The expression could be part of a A mathematical rule that links variables, by substituting the values that can be changed with words or letters, eg πΒ² + πΒ² = πΒ². A formula contains an equals (=) sign..
The correct order of operations, An acronym listing the order in which each part of a calculation must be worked out. Brackets, Indices (or pOwers), Division and Multiplication, and Addition and Subtraction.Β (or BODMAS), must be followed.
For example, values containing Indices are powers. For example, 3 to the power of 2, written 3Β². The singular for indices is 'index'. must be worked out before multiplication.
Follow the working out below
GCSE exam-style questions
- Find the value of this expression when π₯ = β5.
Β Β Β Β Β
Β Β Β Β Β
74
Substitute β5 in place of π₯.
Put a bracket around β5 to show it is β5 being squared and not just 5.
ββ3. Follow BIDMAS and calculate the value with indices firstβ:
β5 Γ β5 = 25, so replace (β5)Β² with 25.
Next do the multiplication. Replace 3 Γ 25 with 75.
Then do the subtraction to get 74.
- This formula is used to convert Celsius (πΆ) to Fahrenheit (πΉ).
What is the temperature in Fahrenheit if it is 20Β°C?
68Β°F
Substitute 20 in place of πΆ in the formula.
Calculate the value of the top of the fraction first:
- 9β βΓ β20 = 180.
The fraction line means βdivided byβ so divide the answer by 5:
- 180 Γ· 5 = 36.
Finally, do the addition. Add 32 to your answer to get πΉ = 68. The formula says that when the temperature in Celsius is 20Β°C, it is 68Β°F in Fahrenheit.
Game - Substitution
Use your substitution skills to solve these puzzles from Divided Islands.
