Formulas
A formula is a mathematical rule that shows the relationship between different quantities using letters called variables. When specific values are given, these letters can be replaced with numbers through a process called substitution. This technique allows precise calculations across countless real-world scenarios.
Why it matters
Formulas power everything from calculating travel times to determining compound interest on savings accounts. The speed formula helps drivers work out journey times: travelling 300 miles at 60 mph takes exactly 5 hours. Engineers use formulas to design bridges, calculating loads and stresses with precise measurements. In finance, the compound interest formula A = P(1 + r)t determines how £1000 invested at 5% annual interest becomes £1276 after 5 years. Medical professionals use formulas for drug dosages based on patient weight, whilst meteorologists apply temperature conversion formulas when Celsius readings need converting to Fahrenheit. Students encounter formulas throughout GCSE mathematics, from basic area calculations like A = πr² for circles to complex physics equations involving acceleration and velocity. These mathematical relationships form the foundation for A-level sciences and university engineering courses.
How to solve formulas
Substitution into Formulas
- Identify which variable each value replaces.
- Substitute (replace) the letters with the given numbers.
- Follow order of operations (PEMDAS) to evaluate.
- Include units in your final answer if applicable.
Example: A = πr². If r = 4: A = π(16) ≈ 50.3.
Worked examples
If speed = distance ÷ time, and distance = 425 km, time = 5 hours, find speed.
Answer: 85 km/h
- Write the formula → speed = distance ÷ time — Use the given formula.
- Substitute the values → speed = 425 ÷ 5 — Replace distance with 425 and time with 5.
- Calculate → 85 km/h — 425 ÷ 5 = 85.
If A = l × w, l = 6, w = 7, find A.
Answer: 42
- Write the formula → A = l × w — Area equals length times width.
- Substitute the values → A = 6 × 7 — Replace l with 6 and w with 7.
- Calculate → 42 — 6 × 7 = 42.
If v = u + at, u = 5, a = 5, t = 4, find v.
Answer: 25
- Write the formula → v = u + at — Final velocity equals initial velocity plus acceleration times time.
- Substitute the values → v = 5 + 5 × 4 — Replace u with 5, a with 5, t with 4.
- Calculate at → 5 × 4 = 20 — Multiply acceleration by time: 5 × 4 = 20.
- Add → v = 25 — 5 + 20 = 25.
Common mistakes
- Forgetting the order of operations when multiple operations appear: calculating v = 3 + 2 × 4 as (3 + 2) × 4 = 20 instead of 3 + 8 = 11.
- Substituting incorrectly by placing numbers in wrong positions: using A = l × w with l = 5, w = 3 but writing A = 3 × 5 = 15 instead of A = 5 × 3 = 15 (same result but wrong method).
- Missing units in final answers: calculating speed = 240 ÷ 4 = 60 but writing just 60 instead of 60 km/h.
- Confusing variables when similar letters appear: in v² = u² + 2as, mixing up initial velocity u = 10 with final velocity v when substituting values.