Working with Proportion Equations Revision Notes | GCSE AQA Higher Maths

1. Why Proportion Equations Matter

The stopping distance of a car does not double when you double the speed — it quadruples. The brightness of a lamp does not halve when you double your distance from it — it drops to a quarter. These are not coincidences. They are proportion relationships, and once you can read the equation, you can predict exactly what happens.

At GCSE Higher, you need to do more than recognise the words "directly proportional" or "inversely proportional". You need to interpret the equation itself — read what $y = kx^2$ or $y = \frac{k}{x^3}$ is actually telling you — and use it to find missing values. That is the skill this lesson builds.

2. Direct Proportion: Reading the Equation

When two quantities are in direct proportion, one is always a fixed multiple of a power of the other. The general form is $y = kx^n$, where $k$ is the constant of proportionality and $n$ tells you the power relationship.

The most common cases you will meet are $y = kx$ (proportional to $x$ itself), $y = kx^2$ (proportional to the square of $x$), and $y = kx^3$ (proportional to the cube of $x$). Each one has a different shape when plotted against $x$, but they all share one crucial feature: when $x = 0$, $y = 0$.

Notice that the table above shows the graph of $y$ against $x$. If instead you plotted $y$ against $x^2$ for the equation $y = kx^2$, you would get a straight line through the origin with gradient $k$. Examiners sometimes ask you to identify which graph would be a straight line — that is the clue they are testing.

3. Inverse Proportion: Reading the Equation

Inverse proportion works the other way: as $x$ gets larger, $y$ gets smaller. The general form is $y = \frac{k}{x^n}$, which can also be written as $yx^n = k$. That second form is worth memorising — it tells you that the product of $y$ and $x^n$ is always the same fixed number.

The most common cases are $y = \frac{k}{x}$ (inversely proportional to $x$) and $y = \frac{k}{x^2}$ (inversely proportional to the square of $x$). A key feature of all inverse proportion graphs: the curve approaches the axes but never crosses them. This is called an asymptote, and it tells you that $y$ can never be zero and $x$ can never be zero.

The graph above shows $y = \frac{6}{x}$ (the falling curve) alongside $y = 2x$ (the straight line through the origin). You can see immediately how different the shapes are. The curve never touches the axes; the line passes straight through the origin. That visual contrast is exactly what an examiner might ask you to interpret.

4. Finding the Constant of Proportionality

Knowing the form of the equation is only half the job. To use it, you need the value of $k$. The method is always the same: substitute a known pair of values $(x, y)$ into the equation and solve for $k$. Once you have $k$, you can find any other value.

For inverse proportion, the same logic applies. From $y = \frac{k}{x^n}$, rearranging gives $k = y \cdot x^n$. Substitute the known pair, compute $k$, and you have the complete equation. Let's see both types worked through in full.

Worked example: Finding $k$ for a Direct Proportion Equation

$y$ is directly proportional to $x^2$. When $x = 5$, $y = 75$. Find the equation connecting $y$ and $x$, then find $y$ when $x = 3$.

1. $y = kx^2$ — Because $y \propto x^2$, the equation has the form $y = kx^2$. [M1 — correct form]
2. $75 = k \times 5^2$ — Replace $y$ with 75 and $x$ with 5. [M1 — correct substitution]
3. $5^2 = 5 \times 5 = 25$, so $75 = 25k$ — Always expand the power longhand to avoid errors.
4. $k = \frac{75}{25} = 3$ — Divide both sides by 25. [A1 — correct value of $k$]
5. $y = 3x^2$ — Substitute $k = 3$ back into the form. This is the equation the examiner wants.
6. $y = 3 \times 3^2 = 3 \times (3 \times 3) = 3 \times 9 = 27$ — Expand $3^2 = 9$ longhand, then multiply. [A1 — correct final answer]

Answer: $y = 3x^2$; when $x = 3$, $y = 27$

Worked example: Finding $k$ for an Inverse Proportion Equation

$y$ is inversely proportional to $x^3$. When $x = 2$, $y = 10$. Find the equation connecting $y$ and $x$, then find $x$ when $y = 80$.

1. $y = \frac{k}{x^3}$ — Because $y \propto \frac{1}{x^3}$, the equation has the form $y = \frac{k}{x^3}$. [M1 — correct form]
2. $10 = \frac{k}{2^3}$ — Replace $y$ with 10 and $x$ with 2. [M1 — correct substitution]
3. $2^3 = 2 \times 2 \times 2 = 8$, so $10 = \frac{k}{8}$ — Expand the cube longhand: $2 \times 2 = 4$, then $4 \times 2 = 8$.
4. $k = 10 \times 8 = 80$ — Multiply both sides by 8. [A1 — correct value of $k$]
5. $y = \frac{80}{x^3}$ — Substitute $k = 80$ back into the form.
6. $80 = \frac{80}{x^3}$ $x^3 = \frac{80}{80} = 1$ $x = \sqrt[3]{1} = 1$ — Rearrange to find $x^3$, then take the cube root. [A1 — correct final answer]

Answer: $y = \frac{80}{x^3}$; when $y = 80$, $x = 1$

5. Interpreting Proportion Equations in Context

Examiners do not always hand you the equation on a plate. Sometimes you are given a table of values and asked to decide whether the relationship is direct or inverse proportion, and if so, which power is involved. The strategy is to test what stays constant.

• For direct proportion $y = kx^n$: check whether $\frac{y}{x^n}$ is constant across all rows of the table.
• For inverse proportion $y = \frac{k}{x^n}$: check whether $y \times x^n$ is constant across all rows of the table.
• If neither ratio nor product is constant, the relationship is not a simple proportion of that power.

Worked example: Identifying the Proportion Type from a Table

The table shows values of $x$ and $y$. Determine whether $y$ is directly or inversely proportional to a power of $x$, and write the equation. $x$: 1, 2, 4 $y$: 12, 3, 0.75

1. $\frac{y}{x}$: $\frac{12}{1} = 12$, $\frac{3}{2} = 1.5$, $\frac{0.75}{4} = 0.1875$ — Not constant, so $y$ is not directly proportional to $x$.
2. $\frac{y}{x^2}$: $\frac{12}{1} = 12$, $\frac{3}{4} = 0.75$, $\frac{0.75}{16} = 0.047$ — Not constant, so $y$ is not directly proportional to $x^2$.
3. $y \times x$: $12 \times 1 = 12$, $3 \times 2 = 6$, $0.75 \times 4 = 3$ — Not constant, so $y$ is not inversely proportional to $x$.
4. $y \times x^2$: $12 \times 1 = 12$, $3 \times 4 = 12$, $0.75 \times 16 = 12$ — Constant value of 12 across all rows. [M1 — correct identification]
5. $y = \frac{12}{x^2}$ — $k = 12$ and $n = 2$, so $y = \frac{k}{x^2}$. [A1 — correct equation]

Answer: $y$ is inversely proportional to $x^2$; the equation is $y = \frac{12}{x^2}$

6. Visualising the Difference: Graphs of Proportion

Being able to sketch or interpret a proportion graph is a separate skill from solving the algebra. Examiners regularly ask you to match an equation to a graph shape, or to describe what a graph tells you about the relationship. The key features to look for are: does the graph pass through the origin, does it curve upward or downward, and does it have asymptotes?

The blue line ($y = 2x$) and the green curve ($y = 0.5x^2$) both pass through the origin — that is the hallmark of direct proportion. The red dashed curve ($y = \frac{4}{x}$) never touches either axis. If you double $x$ on the red curve, $y$ halves. If you double $x$ on the green curve, $y$ quadruples. That contrast is exactly what the examiner wants you to articulate.

7. Formula Reference and Exam Technique

The most mark-winning habit in proportion questions is writing the general form first, before substituting any numbers. Examiners award a method mark for the correct form $y = kx^n$ or $y = \frac{k}{x^n}$ even if you make an arithmetic slip later. Never skip straight to substituting numbers without writing the form.