🔢 Proportionality Power-Up: Master Direct, Inverse & Powers!

Proportionality links variables through simple yet powerful relationships, allowing us to predict one quantity when another changes. At the O Level Extended paper, understanding **direct**, **inverse**, and **power** proportionality is essential for algebra, physics, and real-life problem solving.

Core Ideas at a Glance

**Direct proportionality** means

y \propto x

y = kx

. **Inverse proportionality** gives

y \propto \tfrac{1}{x}

y = \tfrac{k}{x}

. When a power is involved,

y \propto x^n

(where

n

can be positive, negative, or fractional). Always start by writing an equation with an unknown constant

k

Identify the type of proportionality from wording or data.
Write $y = kx$ , $y = \tfrac{k}{x}$ , or $y = kx^n$ as appropriate.
Use given values to solve for $k$ .
Substitute $k$ back to find the required unknown.

Worked Example: Squared Success

Problem: A quantity

y

is directly proportional to the square of

x

. When

x = 3

y = 27

. Find

y

when

x = 5

Step 1: Set up the model. Since $y \propto x^2$ , write
$y = kx^2.$
Step 2: Find $k$ using the given pair $(x, y) = (3, 27)$ . \[27 = k(3)^2 \Rightarrow 27 = 9k \Rightarrow k = 3.\]
Step 3: Substitute $k$ and $x = 5$ to get
$y = 3(5)^2 = 3 \times 25 = 75.$

Practice makes perfect—try forming equations quickly and solving for the constant confidently. With consistent drills, proportionality questions will become instinctive!