Proportional Reasoning

A garden hose is 2 cm in diameter and delivers a steady stream of water. Your neighbor’s hose is 4 cm in diameter - double yours. You might guess their hose delivers twice as much water. But according to Poiseuille’s Law, flow rate is proportional to the fourth power of the radius. Double the radius means $2^{4}$ = 16 times the flow. Not twice. Sixteen times. If you only thought “double,” you just missed the question.

Proportional reasoning is the single most useful math skill on the MCAT. At least a third of physics and chemistry questions boil down to “if this variable changes, what happens to that variable?” You do not need to calculate anything - you need to see the relationship in the formula and trace the change through.

The Four Fundamental Proportionalities

1. Direct Proportion: y = kx

If x doubles, y doubles. If x triples, y triples. The ratio y/x is constant.

Graph: Straight line through the origin.

MCAT examples:

• Ohm’s Law: V = IR (double I at constant R, V doubles)
• Hooke’s Law: F = kx (double displacement, double force)
• Weight: W = mg (double mass, double weight)

2. Inverse Proportion: y = k/x

If x doubles, y halves. If x triples, y becomes one-third. The product xy is constant.

Graph: Hyperbola (curve that never touches either axis).

MCAT examples:

• Boyle’s Law: PV = constant, so P = k/V (double volume, halve pressure)
• Frequency and wavelength: c = f x λ, so f = c/λ (double wavelength, halve frequency)

3. Square Proportion: $y = kx^2$

If x doubles, y quadruples ($2^{2}$ = 4). If x triples, y increases 9-fold ($3^{2}$ = 9).

Graph: Parabola opening upward.

MCAT examples:

• Kinetic energy: $KE = \dfrac{1}{2}mv^2$ (double velocity, quadruple KE)
• Power dissipated: $P = I^2 R$ (double current, quadruple power)
• Centripetal acceleration: $a = v^2/r$ (at constant r, double v, quadruple a)

4. Inverse Square Proportion: $y = k/x^2$

If x doubles, y becomes one-quarter (1/$2^{2}$). If x triples, y becomes one-ninth (1/$3^{2}$).

Graph: Steeper hyperbola that drops off quickly.

MCAT examples:

• Gravitational force: $F = GMm/r^2$ (double distance, quarter the force)
• Coulomb’s Law: $F = kq_1q_2/r^2$ (double distance, quarter the force)
• Sound intensity: $I$ proportional to $1/r^2$ (double distance from source, quarter the intensity)

The Substitution Method

For any “what happens when” question:

1. Write the relevant equation
2. Identify which variable changes and how
3. Substitute the change (replace x with 2x, 3x, x/2, etc.)
4. Simplify and read off the factor

Example: Coulomb’s Law states $F = kq_1q_2/r^2$. If the distance between two charges is tripled, what happens to the force?

Replace r with 3r: $F_{new} = kq_1q_2/(3r)^2 = kq_1q_2/(9r^2) = F/9$

The force decreases to one-ninth of its original value.

High-Yield Proportional Relationships