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.
One move covers all of them. Write the relationship, apply the scale factor to the right-hand side, and read off what happened on the left. No values are needed anywhere, and examiners are fond of these questions for exactly that reason. They test whether you can see the structure without hiding behind arithmetic.
Scaling without a calculator
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.
GUIDED PRACTICE
A square-root scaling
For a pendulum, T2 ∝ l. The length is quadrupled. Decide what happens to the period.
Show the working
If T2 becomes four times bigger, T becomes √4 = 2 times bigger, so 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.