Proportion

Direct Proportion

• **Direct proportion** means as one variable increases, the other increases by the same factor; the ratio stays constant.
• Symbol: ∝, e.g. y ∝ x means y is directly proportional to x.
• General form: $y = kx$, where k is the **constant of proportionality**.
• The graph of $y = kx$ is a straight line through the origin with gradient k.
• For powers/roots: y ∝ $x^{2}$ ⇒ $y =$ kx²; y ∝ $\sqrt{x}$ ⇒ $y =$ k√x; y ∝ $x^{3}$ ⇒ $y =$ kx³; y ∝ ∛x ⇒ $y =$ k∛x.
• Steps: (1) Write formula in terms of k, (2) Substitute given values to find k, (3) Rewrite equation with k, (4) Use equation to find unknown.

Inverse Proportion

• **Inverse proportion** means as one variable increases, the other decreases by the same factor.
• y is inversely proportional to x means y ∝ 1/x, so $y = \frac{k}{x}$.
• The graph of $y = \frac{k}{x}$ is a hyperbola (curved, never touching axes).
• For powers/roots: y ∝ $\frac{1}{x}^{2}$ ⇒ $y =$ k/x²; y ∝ $1/\sqrt{x}$ ⇒ $y =$ k/√x; y ∝ $\frac{1}{x}^{3}$ ⇒ $y =$ k/x³; y ∝ 1/∛x ⇒ $y =$ k/∛x.
• Steps: (1) Write formula in terms of k, (2) Substitute given values to find k, (3) Rewrite equation with k, (4) Use equation to find unknown.

Finding the Constant of Proportionality

• For direct proportion: substitute given (x, y) into $y =$ kxⁿ (or root) and solve for k.
• For inverse proportion: substitute given (x, y) into $y =$ k/xⁿ (or root) and solve for k.
• Always rewrite the full equation after finding k before using it for other values.
• In harder questions, you must set up the proportion yourself from the wording.

Direct Proportion with Powers and Roots

• y ∝ $x^{2}: y = kx^{2}$. Example: if $y=18$ when $x=3$, then $k=2$, so $y=2x^{2}$.
• y ∝ $\sqrt{x}: y = k\sqrt{x}$. Example: if $y=6$ when $x=9$, then $k=2$, so $y=2\sqrt{x}$.
• y ∝ $(x-1)^{2}: y = k(x-1)^{2}$. Example: if $y=4$ when $x=5$, then $k=1$, so $y=(x-1)^{2}$.
• y ∝ ∛(x+3): $y =$ k∛(x+3). Example: if $y=\frac{2}{3}$ when $x=5$, then $k=\frac{1}{3}$, so y=(1/3)∛(x+3).

Inverse Proportion with Powers and Roots

• y ∝ $\frac{1}{x}^{2}: y = \frac{k}{x}^{2}$. Example: if $y=7.5$ when $x=4$, then $k=120$, so $y=\frac{120}{x}^{2}$.
• y ∝ $1/\sqrt{x}: y = k/\sqrt{x}$. Example: if $y=7$ when $x=2.25$, then $k=10.5$, so $y=10.5/\sqrt{x}$.
• y ∝ $1/(x+1)^{2}: y = k/(x+1)^{2}$. Example: if $t=5$ when $x=2$, then $k=45$, so $t=45/(x+1)^{2}$.
• y ∝ $1/(x+3)^{2}: y = k/(x+3)^{2}$. Example: if $y=8$ when $x=2$, then $k=200$, so $y=200/(x+3)^{2}$.

Solving Proportion Problems Step by Step

• **Step 1:** Identify variables and write proportion statement (e.g., y ∝ $x^{2})$.
• **Step 2:** Convert to equation using $k (e.g., y = kx^{2})$.
• **Step 3:** Substitute given pair to find k.
• **Step 4:** Write final equation $(e.g., y = 2x^{2})$.
• **Step 5:** Use equation to find required value or solve for variable.

Common Exam Question Types

• Given a direct or inverse proportion statement and one pair of values, find another value.
• Find the formula connecting two variables (e.g., d in terms of t).
• Percentage change problems: e.g., if y is increased by 21%, find % increase in x when x ∝ $\sqrt{y}$.
• Inverse proportion with linear expressions: e.g., y ∝ $1/(x+2)^{2}$.
• Finding minimum whole number (e.g., number of people) from inverse proportion.

Graphs of Proportion

• Direct proportion $y = kx$: straight line through origin, gradient k.
• Direct proportion $y = kx^{2}$: parabola through origin.
• Direct proportion $y = k\sqrt{x}$: increasing curve, concave down.
• Inverse proportion $y = \frac{k}{x}$: hyperbola, decreasing, asymptotic to axes.
• Inverse proportion $y = \frac{k}{x}^{2}$: similar hyperbola but steeper decrease.

Direct Proportion Graph (y = kx)

Inverse Proportion Graph (y = k/x)

Direct Proportion y = kx² Graph

Inverse Proportion y = k/x² Graph