Constant of Proportionality

Statement

The constant of proportionality is the fixed number that links two quantities when they are directly or inversely proportional.

\[ \text{Direct: } y \propto x \;\Rightarrow\; y = kx \; (k = \tfrac{y}{x}) \qquad \text{Inverse: } y \propto \tfrac{1}{x} \;\Rightarrow\; y = \tfrac{k}{x} \; (k = xy) \]

Why it’s true

• If \(y\) is directly proportional to \(x\), the ratio \(y/x\) stays constant, so \(y = kx\).
• If \(y\) is inversely proportional to \(x\), the product \(xy\) stays constant, so \(y = k/x\).
• The constant \(k\) captures the specific relationship for a given situation.

Recipe (how to use it)

1. Decide whether the relationship is direct (\(y\) increases when \(x\) increases) or inverse (\(y\) decreases when \(x\) increases).
2. Write the appropriate formula \(y = kx\) or \(y = k/x\).
3. Use any known pair of values to find \(k\) (for direct: \(k = y/x\); for inverse: \(k = xy\)).
4. Use this value of \(k\) to find other unknown values.

Spotting it

Look for phrases like “directly proportional”, “inversely proportional”, “varies as”, or “constant ratio/product”.

Common pairings

• Speed and distance (direct when time is fixed).
• Gas laws (pressure inversely proportional to volume for fixed temperature).
• Physics formulas like Hooke’s law (direct proportion between force and extension).

Mini examples

1. If \(y \propto x\) and \(y=12\) when \(x=4\), \(k = 12/4 = 3\), so \(y = 3x\).
2. If \(y \propto 1/x\) and \(y=5\) when \(x=2\), \(k = 5\times2 = 10\), so \(y = 10/x\).

Pitfalls

• Wrong type of proportion: Check carefully whether the question says direct or inverse.
• Forgetting units: The constant often carries units; include them in the final answer.
• Not using a known pair correctly: Always substitute values accurately to find \(k\).

Exam strategy

• Write down the relationship symbolically (\(y\propto x\) or \(y\propto 1/x\)) before finding \(k\).
• Show the calculation of \(k\) clearly.
• Check the final formula by substituting the known values to ensure it fits.

Summary

The constant of proportionality \(k\) is the fixed multiplier or product that defines the exact relationship between two proportional quantities. For direct proportion \(y=kx\) with \(k=y/x\); for inverse proportion \(y=k/x\) with \(k=xy\). Identifying and using \(k\) allows you to find unknown values once one pair is known.