Constructing a formula means turning a word description of a rule into an algebraic equation. You identify the unknown quantities, choose a letter for each, translate each part of the rule into an operation, and write the result in the form "output = expression". This skill connects real-life situations to algebra.
What is the difference between an expression, a formula, and an equation?
- An expression is a collection of terms with no equals sign: 3n + 5.
- A formula has an equals sign and shows how one quantity depends on others: C = 3n + 5. There are at least two different letters (or a letter and a context).
- An equation has an equals sign and can usually be solved for a specific value: 3n + 5 = 20.
When you construct a formula, you are writing a general rule that works for any allowed value of the variable, not solving for one value.
How do you identify variables and constants?
Read the word description carefully and ask two questions:
- What changes? Each quantity that varies is a variable — give it a letter.
- What stays the same? Fixed amounts are constants — write them as numbers.
Worked example: "A plumber charges a £40 call-out fee plus £25 for each hour worked."
- What changes? Number of hours worked → call it h.
- What is the output? Total charge → call it C.
- What stays the same? £40 call-out fee, £25 per hour.
Formula: C = 40 + 25h
How do you translate word rules into algebraic operations?
Certain words map reliably onto operations.
| Word/phrase | Algebraic operation | Example |
|---|---|---|
| "more than", "plus", "added to" | + | 7 more than n → n + 7 |
| "less than", "minus", "subtract" | − | 5 less than n → n − 5 |
| "times", "multiplied by", "per" | × | 3 per ticket → 3t |
| "divided by", "shared equally" | ÷ | shared among 4 → n ÷ 4 |
| "squared", "times itself" | ² | side squared → s² |
| "twice", "double" | 2× | twice the length → 2l |
Watch out for the order in subtraction: "5 less than n" is n − 5, not 5 − n.
How do you construct a formula step by step?
Step 1: Read the problem once and underline the quantities mentioned.
Step 2: Decide which is the output (what you are finding) and which are inputs (what you are given).
Step 3: Choose a letter for each variable — use meaningful letters where possible.
Step 4: Write the formula as: output = (expression involving inputs).
Step 5: Check with a specific number to verify the formula works.
Worked example: "A rectangular swimming pool is three times as long as it is wide. Write a formula for the perimeter."
- Quantities: length, width, perimeter.
- Output: perimeter P. Input: width w (length depends on width).
- Length = 3w (three times the width).
- Perimeter of a rectangle = 2 × length + 2 × width = 2(3w) + 2w = 6w + 2w = 8w.
- Formula: P = 8w.
- Check: if w = 5 m, length = 15 m, perimeter = 2(15) + 2(5) = 30 + 10 = 40. Formula gives 8 × 5 = 40. ✓
How do you construct formulae with more than one variable?
When several quantities vary independently, include a letter for each.
Worked example: "A taxi firm charges a £2 booking fee, plus 80p per kilometre and £1.50 per minute of waiting time."
- Variables: distance d km, waiting time m minutes, total cost C in pence (or pounds — be consistent).
- Using pounds: C = 2 + 0.8d + 1.5m.
Formula: C = 2 + 0.8d + 1.5m
Check: 5 km journey, 3 minutes waiting: C = 2 + 0.8(5) + 1.5(3) = 2 + 4 + 4.50 = £10.50.
Frequently asked questions
How do I choose which letter to use for a variable?
Choose a letter that relates to the quantity where possible — h for height, t for time, C for cost, n for an unknown number, l for length. Avoid using letters that look like numbers — l (lowercase L) can look like 1, and O looks like 0. Your teacher may specify letters; always follow their guidance in an exam.
What if the formula involves a fraction?
Write fractions using the ÷ symbol or as a fraction bar. For example, if the cost is shared equally among n people, write C = Total ÷ n, or equivalently C = Total/n. In algebra, the fraction bar is the preferred notation: C = T/n.
How does constructing a formula differ from substituting into one?
Constructing means writing the formula from scratch — translating a description into algebra. Substituting means replacing the letters with given numbers and evaluating. After you construct a formula, you can substitute values to test it, then use it to answer further questions.
Is there a difference between a formula and a function?
At KS3, they are used almost interchangeably. At GCSE and beyond, a function has a specific notation (f(x) = …) and focuses on the relationship between input and output as a mapping. A formula describes a real-world relationship. The underlying algebra is the same.
For Socratic KS3 algebra practice constructing and using formulae with Professor Pi, see aitutors.me.