Maths bridge › Proportional reasoning
Proportional reasoning
Direct, inverse and square relationships, and answering doubling questions without a calculator.
Reading a relationship
y ∝ x means doubling x doubles y. y ∝ x2 means doubling x quadruples y. y ∝ 1/x means doubling x halves y, and the inverse square, y ∝ 1/x2, means doubling x cuts y to a quarter. Every one of these is a one-line answer to a question that looks like it wants a full calculation.
The move is always the same: write the relationship, apply the scale factor to the right-hand side, and read off what happens to the left. No values needed, which is exactly why examiners love these questions: they test whether you see the structure.
Worked and handed over
WORKED EXAMPLE
An inverse-square jump
The gravitational force between two masses follows F ∝ 1/r2. The separation triples. What happens to the force?
Scale the right side: r becomes 3r, so 1/r2 becomes 1/(3r)2 = 1/9r2.
The force falls to one ninth of its value. No masses, no constant, no calculator.
YOUR TURN
A square-root scaling
For a pendulum, T2 ∝ l. The length is quadrupled. Decide what happens to the period before opening the working.
Show the working
If T2 becomes four times bigger, T becomes √4 = 2 times bigger: the period doubles.
The trap is answering ×4. Quadrupling belongs to T2; the question asked about T.
Where physics leans on this: Newton's law of gravitation · The inverse-square law for gamma · Pendulums and springs. All eight skills: the maths bridge.