Proportion

Direct proportion Extended

Two quantities are directly proportional when multiplying one of them by any number multiplies the other by the same number: double one and the other doubles, divide one by 10 and the other is divided by 10.

We write $y\propto x$, read "$y$ is proportional to $x$". The symbol $\propto$ means "is proportional to". It can always be replaced by "$=k\times$", where $k$ is a fixed number called the constant of proportionality:
$$y\propto x\text{means}y=kx$$

The cost of fuel is proportional to the number of litres bought, and $k$ is the price of one litre. The graph of $y=kx$ is a straight line through the origin, as in the diagram below, where $k=2.5$. Every point on it has $\frac{y}{x}=2.5$, which is why a straight line through the origin always means direct proportion.

Proportion to a power or a root Extended

One quantity can be proportional to a power or a root of another. The syllabus includes five kinds:

• $y\propto x$: $y=kx$
• $y\propto x^2$: $y=k{x}^2$, for example the area of a shape is proportional to the square of its length when the shape stays similar;
• $y\propto x^3$: $y=k{x}^3$, for example the mass of a solid ball is proportional to the cube of its radius;
• $y\propto \sqrt{x}$: $y=k\sqrt{x}$;
• $y\propto \sqrt[3]{x}$: $y=k\sqrt[3]{x}$.

Read the wording carefully: "$y$ is proportional to the square of $x$" is $y\propto x^2$, while "$y$ is proportional to the square root of $x$" is $y\propto \sqrt{x}$. The power changes what doubling does. If $x$ doubles, then $x^2$ is multiplied by 4 and $x^3$ by 8, but $\sqrt{x}$ is multiplied by only $\sqrt{2}\approx 1.41$.

Inverse proportion Extended

Two quantities are inversely proportional when multiplying one by a number divides the other by the same number: double one and the other halves. We write
$$y\propto \frac{1}{x}\text{which means}y=\frac{k}{x}$$
Equivalently $xy=k$: the product stays the same. If 4 workers take 6 days to paint a building, then $k=24$ worker-days, so 8 workers take 3 days and 3 workers take 8 days.

Inverse proportion also comes with powers and roots: $y\propto \frac{1}{x^2}$ (the inverse square law, $y=\frac{k}{x^2}$), $y\propto \frac{1}{x^3}$, $y\propto \frac{1}{\sqrt{x}}$ and $y\propto \frac{1}{\sqrt[3]{x}}$. "Inversely proportional to the square of $x$" means $y=\frac{k}{x^2}$.

The graph of $y=\frac{k}{x}$ for positive $x$ is a curve that gets closer and closer to both axes but never touches them.

The method: find k, then use it Extended

Almost every proportion question follows the same steps:

1. Write the statement with $\propto$, for example $y\propto x^2$.
2. Replace $\propto$ by "$=k\times$": $y=k{x}^2$.
3. Substitute the pair of values the question gives you to find $k$.
4. Write the formula with the value of $k$ in it. Many questions award a mark for this formula on its own.
5. Use the formula to find the unknown quantity. If you are finding $x$, rearrange: for $y=k{x}^2$, $x=\sqrt{\frac{y}{k}}$.

Finding $k$ first is what makes the method reliable: once you have the formula, every later part is just substitution.

Using multipliers Extended

Some questions only ask what happens to one quantity when the other changes. You do not need $k$ for these: apply the change to the power or root.

• $y\propto x^2$ and $x$ is multiplied by 3: $y$ is multiplied by $3^2=9$.
• $y\propto x^3$ and $x$ is halved: $y$ is multiplied by $(\frac{1}{2}{)}^3=\frac{1}{8}$.
• $y\propto \frac{1}{x^2}$ and $x$ is doubled: $y$ is multiplied by $\frac{1}{2^2}=\frac{1}{4}$.

Percentage changes work the same way through the multiplier. If $y\propto x^2$ and $x$ increases by 10%, the multiplier for $x$ is 1.1, so the multiplier for $y$ is ${1.1}^2=1.21$, an increase of 21%, not 20%.