Proportion

What proportion actually means

Two quantities are in direct proportion when multiplying one by a number multiplies the other by the same number. If 1 litre of paint covers 12 m${}^2$, then 3 litres cover 36 m${}^2$ and 5 litres cover 60 m${}^2$. The ratio $\frac{y}{x}$ never changes, and that fixed ratio is what makes the relationship a proportion rather than a coincidence.

Two quantities are in inverse proportion when multiplying one by a number divides the other by the same number. If 6 workers take 4 hours to paint a wall, 12 workers take 2 hours: doubling the workers halves the time, so the product $x\times y$ stays at 24. Both quantities change, but they change in opposite directions.

The two cases are the same idea with the multiplication in a different place. Direct proportion keeps a quotient constant, $\frac{y}{x}=k$; inverse proportion keeps a product constant, $xy=k$. Everything else in this topic, including the square, cube and root versions, is one of those two sentences with a power of x in it.

The proportionality symbol and the constant $k$

The symbol $\propto$ means is proportional to. So $y\propto x$ is read as 'y is proportional to x' and means $y=kx$ for some fixed number k. The symbol is not an equals sign: $y\propto x$ says the two quantities move together, while $y=kx$ gives you the exact formula you can calculate with.

The number k is the constant of proportionality. It is fixed for a given relationship, and finding it is usually the first thing a question wants. Writing $y\propto x$ is only the first line of the answer; the marks are in turning it into $y=kx$ and working out k.

The same symbol covers every power and root the syllabus lists. $y\propto x^2$ means $y=k{x}^2$, $y\propto \sqrt{x}$ means $y=k\sqrt{x}$, $y\propto x^3$ means $y=k{x}^3$, and $y\propto \sqrt[3]{x}$ means $y=k\sqrt[3]{x}$. Inverse proportion is written with a reciprocal: $y\propto \frac{1}{x}$ means $y=\frac{k}{x}$, and $y\propto \frac{1}{x^2}$ means $y=\frac{k}{x^2}$.

Two phrases mean the same thing and both turn up in questions. 'y varies directly as x' is $y\propto x$, and 'y varies inversely as x' is $y\propto \frac{1}{x}$. The word to watch for is inversely: it is the one that turns the formula upside down.

Finding the constant of proportionality from given data

Every proportion question gives you one pair of values, and that pair is what fixes k. The method has three steps: write the proportionality as an equation with k in it, substitute the pair you have been given, then solve for k. Once k is known, the formula is yours to use for any other value in the question.

Direct proportion: $y\propto x$, and $y=12$ when $x=4$. Substituting gives $12=k\times 4$, so $k=3$ and the formula is $y=3x$. Check it against the pair you started with: when $x=4$, $y=3\times 4=12$.

Inverse proportion: $y\propto \frac{1}{x}$, and $y=5$ when $x=4$. Substituting into $y=\frac{k}{x}$ gives $5=\frac{k}{4}$, so $k=20$ and the formula is $y=\frac{20}{x}$. Here k is the product of the pair: $5\times 4=20$.

Square proportion: $y\propto x^2$, and $y=36$ when $x=3$. Then $36=k\times 3^2=9k$, so $k=4$ and $y=4x^2$. The commonest slip is dividing by x instead of by $x^2$, which would give 12 instead of 4.

• $y\propto x$ with $y=12$, $x=4$: $k=\frac{12}{4}=3$, so $y=3x$
• $y\propto \frac{1}{x}$ with $y=5$, $x=4$: $k=5\times 4=20$, so $y=\frac{20}{x}$
• $y\propto x^2$ with $y=36$, $x=3$: $k=\frac{36}{9}=4$, so $y=4x^2$
• $y\propto \sqrt{x}$ with $y=14$, $x=49$: $k=\frac{14}{7}=2$, so $y=2\sqrt{x}$

Using the formula to find unknown quantities

Once the formula is written, the rest of the question is substitution. With $y=3x$: when $x=7$, $y=3\times 7=21$; and if $y=24$, then $x=\frac{24}{3}=8$. Read carefully which quantity you are given, because a question can ask for either one, and a question that gives you y is asking you to rearrange.

With an inverse formula the work is the same, only the division moves. From $y=\frac{20}{x}$: when $x=8$, $y=\frac{20}{8}=2.5$; and if $y=40$, then $40=\frac{20}{x}$, so $x=\frac{20}{40}=0.5$. A bigger y with a smaller x is exactly what inverse proportion should give, so the answer is a check on the method.

There is a shortcut worth knowing, because it saves the algebra when k is awkward. For direct proportion the quotient never changes, so $\frac{y_1}{x_1}=\frac{y_2}{x_2}$. For inverse proportion the product never changes, so $x_1{y}_1=x_2{y}_2$. Both are the formulas $y=kx$ and $xy=k$ written for two different pairs of values.

The square and cube versions have their own shortcuts: for direct square proportion $\frac{y_1}{x_1^2}=\frac{y_2}{x_2^2}$, and for inverse square proportion $x_1^2{y}_1=x_2^2{y}_2$. If you cannot remember which one applies, go back to the three-step method and find k. It is slower, but it is never wrong.

• $y=3x$: $x=7$ gives $y=21$; $y=24$ gives $x=8$
• $y=\frac{20}{x}$: $x=8$ gives $y=2.5$; $y=40$ gives $x=0.5$
• $y=4x^2$: $x=5$ gives $y=100$; $y=144$ gives $x=6$
• $y=2\sqrt{x}$: $x=81$ gives $y=18$; $y=30$ gives $x=225$

Graphs of proportional relationships

Direct proportion gives a straight line through the origin. The graph of $y=kx$ is a straight line with gradient k that passes through $(0,0)$, because when $x=0$, $y=k\times 0=0$. The gradient is the constant of proportionality, so a line through $(1,3)$ tells you $k=3$ and the formula is $y=3x$.

A straight line that does not pass through the origin is not direct proportion. The graph of $y=2x+1$ is straight, but at $x=0$ it gives $y=1$, and doubling x does not double y: $x=1$ gives $y=3$ while $x=2$ gives $y=5$. The right word for that graph is linear, not proportional.

Inverse proportion gives a curve with a reciprocal shape. $y=\frac{k}{x}$ falls steeply at first and then flattens, and it never touches either axis: as x becomes large, y gets close to zero without reaching it, and as x gets close to zero, y becomes very large. The axes are called asymptotes, which simply means lines the curve approaches but never meets.

The square and root versions curve as well. $y=k{x}^2$ is a curve through the origin that gets steeper as x increases, and $y=k\sqrt{x}$ is a curve through the origin that flattens as x increases. Only the linear version is a straight line, so a straight line is the quickest way to recognise plain direct proportion.

• Direct: straight line through $(0,0)$ with gradient $k$
• Inverse: reciprocal curve that never touches either axis
• Direct square: curve through $(0,0)$ that steepens
• Direct square root: curve through $(0,0)$ that flattens
• A straight line that misses the origin is not a proportional relationship

Deciding which proportion you have

When a question gives you a table of values instead of a formula, test for the constant. Divide each y by the matching x: if $\frac{y}{x}$ is the same every time, y is directly proportional to x. Multiply each pair instead: if $xy$ is the same every time, y is inversely proportional to x.

Direct check: $x=2,3,5,8$ with $y=6,9,15,24$ gives $\frac{6}{2}=\frac{9}{3}=\frac{15}{5}=\frac{24}{8}=3$, so $y=3x$. Inverse check: $x=2,3,4,6$ with $y=12,8,6,4$ gives $2\times 12=3\times 8=4\times 6=6\times 4=24$, so $y=\frac{24}{x}$.

If the quotient is not constant but $\frac{y}{x^2}$ is, the relationship is square proportion. The same test works for $\frac{y}{x^3}$, $\frac{y}{\sqrt{x}}$ and $\frac{y}{\sqrt[3]{x}}$: work out the quantity that would be constant if that proportion were true, and see whether it really is.

The wording of the question is usually the faster clue. 'y is proportional to the square root of x' is $y\propto \sqrt{x}$, and 'the volume is inversely proportional to the pressure' is $V\propto \frac{1}{p}$. Whenever a power or a root is involved, the constant k goes in front of the whole expression, so it is $y=k\sqrt{x}$ and not $y=\sqrt{kx}$.