A5: Rearranging Formulae
Use standard mathematical formulae; rearrange formulae to change the subject
📋 Key Concepts
Subject of a formula: The variable that is on its own on one side of the equals sign.
Changing the subject: Rearrange the formula to make a different variable the subject.
Example: In A = lb, A is the subject
We can rearrange to make l the subject: l = A⁄b
We can rearrange to make l the subject: l = A⁄b
📝 Basic Rearranging
Method: Use inverse operations to isolate the new subject. Do the same to both sides.
Example 1
Rearrange v = u + at to make t the subject.
Solution:
Step 1: Subtract u from both sides: v - u = at
Step 2: Divide both sides by a: v - u⁄a = t
Answer: t = v - u⁄a
Example 2
Rearrange A = πr² to make r the subject.
Solution:
Step 1: Divide both sides by π: A⁄π = r²
Step 2: Take the square root: √(A⁄π) = r
Answer: r = √(A⁄π)
📝 Rearranging with Multiplication
Rule: If the new subject is multiplied by something, divide both sides by that factor.
Example 3
Rearrange A = lb to make b the subject.
Solution:
Divide both sides by l: A⁄l = b
Answer: b = A⁄l
Example 4
Rearrange s = d⁄t to make d the subject.
Solution:
Multiply both sides by t: st = d
Answer: d = st
📝 Rearranging with Addition/Subtraction
Rule: If something is added to the new subject, subtract it from both sides.
Example 5
Rearrange y = mx + c to make x the subject.
Solution:
Step 1: Subtract c from both sides: y - c = mx
Step 2: Divide both sides by m: y - c⁄m = x
Answer: x = y - c⁄m
📝 Rearranging with Powers
Rule: If the new subject is squared, take the square root. If it's inside a square root, square both sides.
Example 6
Rearrange A = ½bh to make h the subject.
Solution:
Step 1: Multiply both sides by 2: 2A = bh
Step 2: Divide both sides by b: 2A⁄b = h
Answer: h = 2A⁄b
Example 7
Rearrange E = ½mv² to make v the subject.
Solution:
Step 1: Multiply both sides by 2: 2E = mv²
Step 2: Divide both sides by m: 2E⁄m = v²
Step 3: Take square root: v = √(2E⁄m)
📝 Rearranging with the Variable in Two Places
Rule: Collect all terms with the new subject on one side, then factorise.
Example 8
Rearrange ax + b = cx + d to make x the subject.
Solution:
Step 1: Subtract cx from both sides: ax - cx + b = d
Step 2: Subtract b from both sides: ax - cx = d - b
Step 3: Factorise left side: x(a - c) = d - b
Step 4: Divide by (a - c): x = d - b⁄a - c
❓ Practice Questions
Q1: Rearrange a = 3b + 2 to make b the subject.
Q2: Rearrange F = ma to make a the subject.
Q3: Rearrange C = 2πr to make r the subject.
Q4: Rearrange v² = u² + 2as to make s the subject.
Q5: Rearrange P = 2(l + w) to make w the subject.
Q6: Rearrange ax - 3 = bx + 5 to make x the subject.
✅ Answers
- b = a - 2⁄3
- a = F⁄m
- r = C⁄2π
- s = v² - u²⁄2a
- w = P⁄2 - l
- x = 8⁄a - b
🎯 Exam Tips
- Always do the same to both sides
- Work backwards through BIDMAS to isolate the subject
- If the variable appears twice, factorise to collect terms
- Check your answer by substituting values
- Remember: inverse of squaring is square rooting
- Be careful with negative signs
🧠 Problem-Solving Strategies
Problem-Solving
Work backwards through BIDMAS to isolate the new subject. Deal with additions/subtractions first, then multiplications/divisions, then powers/roots. If the variable appears twice, collect all terms with it on one side and factorise.
Multi-Step Problem
The kinetic energy formula is E = ½mv². (a) Rearrange to make v the subject. (b) Find v when E = 100 and m = 8.
Solution:
(a) Step 1: 2E = mv². Step 2: v² = 2E⁄m. Step 3: v = √(2E⁄m)
(b) v = √(200⁄8) = √25 = 5 m/s
⚠️ Common Errors
Watch Out!
1. Wrong: Rearranging v = u + at to make t: t = v - u/a Correct: t = v - u⁄a (the whole of v - u is divided by a)
2. Wrong: From A = πr², rearranging to r = A⁄π Correct: r² = A⁄π, so r = √(A⁄π) (don't forget the square root)
3. Wrong: Rearranging ax + b = cx + d by moving ax and d: ax - d = cx Correct: ax - cx = d - b, then factorise: x(a - c) = d - b
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The formula for the surface area of a cylinder (excluding ends) is A = 2πrh. (a) Rearrange to make h the subject. (b) Rearrange to make r the subject. (c) A can has A = 314 cm² and r = 5 cm. Find h. (Use π = 3.14)
(a) h = A⁄2πr
(b) r = A⁄2πh
(c) h = 314⁄2 × 3.14 × 5 = 314⁄31.4 = 10 cm
Mark scheme: (a) 1 mark. (b) 1 mark. (c) 2 marks for substitution, 1 mark for correct answer, 1 mark for units.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
The stopping distance formula is d = v²⁄2a where d = distance, v = initial speed, a = deceleration.
(a) Rearrange to make v the subject.
(b) If a car decelerates at 8 m/s² and stops in 25 m, what was its initial speed?
(c) Explain why doubling the speed more than doubles the stopping distance.
Answers: (a) v = √(2ad). (b) v = √(2 × 8 × 25) = √400 = 20 m/s. (c) Because v is squared in the formula, doubling v gives (2v)² = 4v², so stopping distance is 4 times as large.
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