Direct Proportion

Statement

Two quantities are in direct proportion if they increase or decrease at the same rate. This relationship is written as:

\[ y \propto x \quad \Rightarrow \quad y = kx \]

Here, \(k\) is called the constant of proportionality. It is the multiplier that connects \(y\) to \(x\).

Why it’s true

• If doubling one quantity also doubles the other, they are directly proportional.
• The ratio \( \frac{y}{x} \) is always constant and equal to \(k\).
• This constant allows us to write an exact equation linking the two variables.

Recipe (how to use it)

1. Identify that the relationship is proportional (look for “directly proportional” or “varies directly with”).
2. Write \( y = kx \).
3. Use known values of \(x\) and \(y\) to calculate \(k\).
4. Use \( y = kx \) to find other values as required.

Spotting it

Key phrases in questions include “directly proportional to”, “varies directly with”, or “in proportion to”. Graphs of direct proportion are straight lines through the origin.

Common pairings

• Speed and distance (with constant time).
• Mass and weight (on Earth, weight = 9.8 × mass).
• Cost and quantity (with constant price per item).

Mini examples

1. Given: \( y \propto x \), and when \(x = 4, y = 20\). Find: \(k\). Answer: \(20/4 = 5\), so \( y = 5x \).
2. Given: \( y = 3x \). When \(x = 7\), Answer: \(y = 21\).

Pitfalls

• Confusing direct proportion with inverse proportion (\(y \propto 1/x\)).
• Forgetting that the graph must pass through the origin.
• Mixing up the constant \(k\) with the values of \(x\) or \(y\).

Exam strategy

• Always start by writing \( y = kx \).
• Substitute known values to find \(k\).
• Check answers make sense (if \(x\) doubles, \(y\) should double too).

Summary

Direct proportion links two quantities by a constant multiplier: \(y = kx\). It appears in graphs as a straight line through the origin and is used to model relationships in science, finance, and everyday situations.