Rearranging Formulae Quizzes
Visual overview of Rearranging Formulae.
Introduction
Rearranging formulae means manipulating an equation to make a different variable the subject. It’s essential across GCSE Maths, physics, and everyday problem solving. By mastering inverse operations and tidy algebraic steps, you can switch subjects, substitute values accurately, and apply formulas flexibly.
Example: area of a rectangle \(A=l\times w\). If \(A\) and \(l\) are known, then \(w=\dfrac{A}{l}\).
Core Concepts
Identifying the Subject
The subject is the variable you want alone on one side. Use inverse operations to isolate it.
Example
- Formula: \(A=l\times w\). Make \(w\) the subject.
- Divide by \(l\): \(w=\dfrac{A}{l}\).
Inverse Operations
- Addition ↔ Subtraction
- Multiplication ↔ Division
- Powers ↔ Roots
- Example: \(y+3=10\Rightarrow y=10-3=7\)
Multiplication & Division
Move between numerator/denominator and opposite sides by multiplying or dividing.
- \(F=m\times a\) → \(m=\dfrac{F}{a}\), \(a=\dfrac{F}{m}\)
Addition & Subtraction
- \(v=u+at\) → \(u=v-at\)
- Make \(t\) the subject: \(v-u=at\Rightarrow t=\dfrac{v-u}{a}\)
Formulas with Brackets
Expand or divide to remove brackets.
- \(A=2(l+w)\) → \(\dfrac{A}{2}=l+w\) → \(w=\dfrac{A}{2}-l\)
Formulas with Fractions
Clear denominators first.
- \(P=\dfrac{F}{A}\). Make \(F\) the subject: multiply by \(A\) → \(F=PA\).
Formulas with Powers
Use roots to undo powers (or logs at higher levels).
- \(A=\pi r^2\) → \(\dfrac{A}{\pi}=r^2\) → \(r=\sqrt{\dfrac{A}{\pi}}\)
Multiple Variables
Work step by step, moving terms away from the desired subject.
- \(y=mx+c\) → \(y-c=mx\) → \(x=\dfrac{y-c}{m}\)
Checking Your Rearrangement
Substitute sample numbers to verify.
- From \(F=ma\), \(m=\dfrac{F}{a}\). If \(F=20\), \(a=5\) → \(m=4\). Check: \(4\times5=20\) ✓
Real-Life Applications
- Physics: speed, distance, time; force, mass, acceleration.
- Geometry: area, perimeter, volume rearrangements.
- Finance: interest \(I=Prt\) for any variable.
- Engineering: density \(=\dfrac{m}{V}\), pressure \(=\dfrac{F}{A}\).
Worked Examples
Example 1 (Foundation): Subtraction
Formula \(v=u+at\). Make \(u\) the subject: \(u=v-at\).
Example 2 (Foundation): Division
\(F=ma\). Make \(a\) the subject: \(a=\dfrac{F}{m}\).
Example 3 (Higher): Brackets
\(A=2(l+w)\). Make \(w\): \(\dfrac{A}{2}=l+w\Rightarrow w=\dfrac{A}{2}-l\).
Example 4 (Higher): Fraction
\(P=\dfrac{F}{A}\). Make \(F\): \(F=PA\).
Example 5 (Higher): Power
\(A=\pi r^2\). Make \(r\): \(r=\sqrt{\dfrac{A}{\pi}}\).
Example 6 (Higher): Multiple variables
\(y=mx+c\). Make \(x\): \(x=\dfrac{y-c}{m}\).
Example 7 (Higher): Negative coefficient
\(-2x+5=y\). Make \(x\): \(-2x=y-5\Rightarrow x=\dfrac{5-y}{2}\).
Example 8 (Real life): Distance–speed–time
\(d=st\). Make \(t\): \(t=\dfrac{d}{s}\).
Example 9 (Real life): Simple interest
\(I=Prt\). Make \(r\): \(r=\dfrac{I}{Pt}\).
Example 10 (Higher): Fraction + brackets
\(V=\dfrac{1}{3}\pi r^2 h\). Make \(h\): \(3V=\pi r^2h\Rightarrow h=\dfrac{3V}{\pi r^2}\).
Common Mistakes
- Not doing the same operation to both sides.
- Dropping signs when moving terms.
- Forgetting to clear denominators before isolating.
- Expanding/contracting brackets incorrectly.
- Leaving the subject on both sides (not fully isolated).
Applications
- Physics: \(v=u+at\), \(F=ma\), \(p=\dfrac{F}{A}\), \(s=\dfrac{d}{t}\).
- Geometry: \(A=\pi r^2\), \(V=\pi r^2h\), \(P=2(l+w)\).
- Finance: \(I=Prt\) (solve for any variable).
- Engineering/Science: density \(\rho=\dfrac{m}{V}\), power \(P=\dfrac{W}{t}\).
Strategies & Tips
- Decide the subject first; circle it.
- Undo operations in reverse order using inverses.
- Add brackets where needed to keep terms together.
- Clear fractions early by multiplying through.
- Write one neat step per line; then verify with numbers.
Summary / Call-to-Action
Rearranging formulae unlocks flexible problem solving. By isolating any chosen variable using inverse operations—while handling brackets, fractions, and powers—you can adapt formulas to fit the question quickly and reliably.
- Practise turning common formulas into each possible subject.
- Check your work by substitution.
- Mix brackets, fractions, and powers for higher-level fluency.