Direct and Inverse Proportion

Last year you described direct proportion in words: two quantities grow together at the same rate, and dividing one by the other always gives the same number. This year you write that idea as a tidy algebraic equation.

If $y$ is directly proportional to $x$, then: $y = kx,$ where $k$ is the constant of proportionality — the fixed rate that links the two quantities. The symbol $\propto$ means "is proportional to", so $y \propto x$ is read "$y$ is proportional to $x$".

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.

The equation $y = kx$ is only useful once you know $k$. The good news: a single known pair of values is enough to find it.

The method is always the same:

1. Write the proportion equation: $y = kx$.
2. Substitute the known pair of values for $x$ and $y$.
3. Solve for $k$.
4. Write the full rule with $k$ filled in — now use it for any value.

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.

Inverse proportion also becomes an equation, but a different shape. If $y$ is inversely proportional to $x$, then: $y = \frac{k}{x},$ which can also be written $xy = k$. That second form shows the meaning clearly: $x$ and $y$ multiplied together always give the same number $k$ — the constant product.

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.

With both equations in your toolkit, real problems become a clear two-stage job: find $k$, then use the rule.

The reliable method:

1. Decide whether the situation is direct (quantities grow together) or inverse (one grows as the other shrinks).
2. Write the matching equation, $y = kx$ or $y = \dfrac{k}{x}$.
3. Substitute the given pair to find $k$.
4. Use the completed rule to answer the question.

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.

Each proportion equation has its own distinctive graph, and the shape alone tells you which relationship you are looking at.

A direct proportion graph, $y = kx$, is a straight line through the origin $(0, 0)$. Its gradient is exactly $k$ — the steeper the line, the larger the constant.

An inverse proportion graph, $y = \dfrac{k}{x}$, is a smooth curve that falls steeply then flattens, drawing close to both axes but never touching them.

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.

Writing proportion as an equation is a big step towards higher mathematics:

• Linear functions and graphs — $y = kx$ is a straight-line equation, and recognising the gradient as $k$ links the two topics directly.
• Algebra — finding $k$ by substituting and solving is the same skill you use across all of equation work.
• Science — many laws are proportion equations: distance and time at fixed speed, or how pressure and volume trade off in inverse proportion.
• In everyday life, currency conversion, scaling recipes and best-value comparisons are all proportion equations in disguise.

If proportion feels tricky later, it is usually deciding direct or inverse that needs a refresh — and then choosing $y = kx$ or $y = k/x$. Keep this guide handy, and remember: find $k$ first, and the rest is arithmetic.