The unitary method (find one, scale up) breaks down once a relationship involves a square, cube, or root rather than a straight multiple — e.g. "$y$ is proportional to the cube of $x$" can't be solved by simple scaling. Higher Tier proportion questions are built specifically around these non-linear relationships, so you need one universal algebraic method that works for all of them.

Replace the $\propto$ symbol with "$=k$" (direct) or "$=k\div$" (inverse), where $k$ is the constant of proportionality:

• $y\propto x \Rightarrow y=kx$
• $y\propto \dfrac1x \Rightarrow y=\dfrac kx$

1. Write the statement: translate the wording exactly, e.g. "$y$ is inversely proportional to the square of $x$" → $y\propto \dfrac1{x^2}$.
2. Form the equation: $y=\dfrac{k}{x^2}$.
3. Find $k$: substitute the one given pair of values and solve.
4. Write the final formula and solve: substitute $k$ back in, then use the formula for whatever the question actually asks.

Worked example: $y\propto x^2$; $y=27$ when $x=3$. Find $y$ when $x=11$.

$27=k(3^2)=9k \Rightarrow k=3$, so $y=3x^2$. When $x=11$: $y=3(121)=363$.

Combined problems link two separate proportion statements through a shared variable — solve each with the 4-step method, then substitute one formula into the other.

Worked exam-style question: $x\propto w^2$, and $y\propto \dfrac1{x^3}$. Given $x=48$ when $w=4$, and $y=14$ when $x=2$, find $y$ in terms of $w$.

1. $x=k_1w^2 \Rightarrow 48=k_1(16)\Rightarrow k_1=3$, so $x=3w^2$.
2. $y=\dfrac{k_2}{x^3}\Rightarrow 14=\dfrac{k_2}{8}\Rightarrow k_2=112$, so $y=\dfrac{112}{x^3}$.
3. Substitute $x=3w^2$: $y=\dfrac{112}{(3w^2)^3}=\dfrac{112}{27w^6}$.

Ask: if one quantity increases, does the other increase too (direct) or decrease (inverse)? More cleaners means less time to finish a job — that's inverse, so $xy=k$: $5$ cleaners taking $4.5$ hours gives $k=5\times4.5=22.5$ "cleaner-hours", so $3$ cleaners would take $22.5\div3=7.5$ hours.

• Writing $y=kx$ when the question says "square", "cube", or "square root" — always write the proportionality statement first.
• Confusing direct and inverse — do the "increase/decrease" context check before setting up the equation.
• Solving for the wrong variable at the end — re-read exactly what value the question wants once you have the formula.
• Mixing inconsistent units (e.g. km/h with minutes) before substituting.