Direct and Inverse Proportion

In the table, $y \div x$ is always $3$, so $k = 3$ and the rule is $y = 3x$. The power of the equation is that it works both ways: give it any $x$ and it returns $y$, or rearrange it to find $x$ from $y$. One short formula replaces the whole table.

• Direct proportion is written as the equation y = kx.
• k is the constant of proportionality — a fixed number.
• The symbol ∝ means 'is proportional to'.
• The equation gives any value, in either direction.

Suppose $y$ is directly proportional to $x$, and $y = 20$ when $x = 4$.

Substituting gives $20 = k \times 4$, so $k = 20 \div 4 = 5$. The complete rule is $y = 5x$.

Now any question is quick. When $x = 9$, $y = 5 \times 9 = 45$. When $y = 35$, rearrange: $x = 35 \div 5 = 7$. Finding $k$ first turns every later step into simple arithmetic.

• Substitute one known pair into y = kx.
• Solve the resulting equation to find k.
• Write the full rule with k filled in.
• Use the rule for any value, in either direction.

In the table $x \times y$ is always $24$, so $k = 24$ and the rule is $y = \dfrac{24}{x}$.

To find $k$, the steps mirror direct proportion: substitute a known pair into $y = \dfrac{k}{x}$, or simply multiply the pair together since $k = xy$. If $y = 6$ when $x = 5$, then $k = 5 \times 6 = 30$, and the rule is $y = \dfrac{30}{x}$. When $x = 10$, $y = 30 \div 10 = 3$ — bigger $x$, smaller $y$, exactly as inverse proportion should behave.

• Inverse proportion is y = k/x, which is the same as xy = k.
• k is the constant product of the two quantities.
• Find k by multiplying a known pair: k = xy.
• Larger x always gives smaller y, and the other way round.

Example: $3$ painters finish a job in $8$ days. How long would $6$ painters take? More painters means less time, so this is inverse proportion. The constant product is $k = 3 \times 8 = 24$ painter-days. For $6$ painters: time $= 24 \div 6 = 4$ days.

Always finish with a sense check. Six painters took fewer days than three — fewer days for more workers is exactly right for inverse proportion.

• First decide whether the problem is direct or inverse.
• Write the matching equation and substitute to find k.
• Use the completed rule to answer the question.
• Sense check: does the answer move the right way?

Why does the direct line go through the origin? Because $y = kx$ gives $y = 0$ when $x = 0$ — no $x$ means no $y$. Why does the inverse curve never reach the axes? Because $\dfrac{k}{x}$ can never equal exactly zero, and you cannot divide by $x = 0$.

To sketch a graph, plot two or three points from the equation and join them with the right shape — a ruled straight line for direct, a smooth freehand curve for inverse.

• Direct proportion: a straight line through the origin, gradient k.
• Inverse proportion: a smooth falling curve.
• The inverse curve approaches but never touches the axes.
• Sketch by plotting a few points, then joining with the right shape.

• y = kx connects directly to straight-line graphs and gradient.
• Finding k uses the same substitute-and-solve skill as algebra.
• Many science laws are proportion equations.
• Currency, recipes and best-value problems all use proportion.