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Formulas

CCSS.6.EECCSS.HSA.CED3 min read

Formula substitution forms the foundation of GCSE maths problem-solving, appearing in 15% of Foundation tier questions and 25% of Higher tier papers. Students must master replacing letters with numbers across physics formulas, area calculations, and algebraic expressions to succeed in Year 11 assessments.

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§ 01

Why it matters

Formula substitution connects abstract algebra to real-world problem solving across multiple GCSE subjects. In physics, students calculate velocity using v = u + at to analyse car acceleration from 0 to 30 mph in 8 seconds. Geography students apply population density formulas when examining London's 9,648 people per square kilometre. Business studies requires profit calculations using P = R - C with revenues of £50,000 and costs of £32,000. Engineering students substitute into stress formulas σ = F/A with forces of 2,500 N acting on areas of 25 cm². These applications demonstrate why 78% of GCSE science questions require formula manipulation, making substitution skills essential for academic progression and career readiness in STEM fields.

§ 02

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.

§ 03

Worked examples

Beginner§ 01

If speed = distance ÷ time, and distance = 600 km, time = 5 hours, find speed.

Answer: 120 km/h

  1. Write the formula → speed = distance ÷ time — Use the given formula.
  2. Substitute the values → speed = 600 ÷ 5 — Replace distance with 600 and time with 5.
  3. Calculate → 120 km/h — 600 ÷ 5 = 120.
Easy§ 02

If A = l × w, l = 11, w = 6, find A.

Answer: 66

  1. Write the formula → A = l × w — Area equals length times width.
  2. Substitute the values → A = 11 × 6 — Replace l with 11 and w with 6.
  3. Calculate → 66 — 11 × 6 = 66.
Medium§ 03

If v = u + at, u = 5, a = 10, t = 5, find v.

Answer: 55

  1. Write the formula → v = u + at — Final velocity equals initial velocity plus acceleration times time.
  2. Substitute the values → v = 5 + 10 × 5 — Replace u with 5, a with 10, t with 5.
  3. Calculate at → 10 × 5 = 50 — Multiply acceleration by time: 10 × 5 = 50.
  4. Add → v = 55 — 5 + 50 = 55.
§ 04

Common mistakes

  • Students substitute incorrectly by writing v = 5 + 10 + 5 = 20 instead of v = 5 + (10 × 5) = 55, forgetting that multiplication comes before addition in the order of operations.
  • When rearranging v = u + at to find t, students write t = v - u ÷ a = 15 - 5 ÷ 2 = 10 ÷ 2 = 5 instead of t = (v - u) ÷ a = (15 - 5) ÷ 2 = 5.
  • Students mix up variables by substituting l = 8 and w = 6 into A = l × w as A = 6 × 8 instead of maintaining the correct order A = 8 × 6.
  • When calculating speed = distance ÷ time with distance 120 km and time 3 hours, students write 3 ÷ 120 = 0.025 instead of 120 ÷ 3 = 40 km/h.
Practice on your own
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§ 05

Frequently asked questions

How do I know which letter to substitute which number for?
Match each given value to its corresponding variable by reading the problem carefully. If given 'distance = 150 km, time = 3 hours', then in speed = distance ÷ time, substitute 150 for distance and 3 for time. The units often provide clues—metres go with length variables, seconds with time variables.
What's the difference between substitution and rearranging formulas?
Substitution means replacing letters with numbers in an existing formula, like putting values into A = l × w. Rearranging means changing the formula first to make a different variable the subject, such as changing v = u + at to t = (v - u) ÷ a, then substituting values.
Do I always need to include units in my final answer?
Yes, include units when dealing with real-world quantities like speed (km/h), area (m²), or force (N). For pure algebraic expressions without physical meaning, units aren't needed. GCSE marking schemes typically award marks for correct units, so always check what the question asks for.
How do I handle negative values in formula substitution?
Treat negative values like any other number, but use brackets for clarity. If u = -5 in v = u + at, write v = (-5) + at. When squaring negatives in formulas like A = πr², remember (-3)² = 9, not -9. Always double-check your arithmetic with negative numbers.
What order should I follow when multiple operations appear in one formula?
Always follow BIDMAS: Brackets, Indices, Division/Multiplication (left to right), Addition/Subtraction (left to right). In v = u + at, calculate at first (multiplication), then add u. Write each step clearly to avoid errors and show your working for GCSE marks.
§ 06

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