A ratio $A:B$ compares two parts to each other. A fraction compares one part to the total. To switch between them, find the total number of parts, $T = A + B$, then put each part over that total: the fraction for $A$ is $\dfrac{A}{T}$ and for $B$ is $\dfrac{B}{T}$.

For a ratio of cement : sand : gravel $= 3:5:2$, the total is $3+5+2=10$, so sand makes up $\dfrac{5}{10} = \dfrac{1}{2}$ of the mix.

The two uses of $A:B$ look similar but mean different things, and mixing them up is the main source of lost marks:

• Sharing a total (additive): the denominator is the sum of the parts, $A+B$.
• Finding a rate (multiplicative): the ratio becomes the quotient $\dfrac{A}{B}$ itself — this is used when $A$ and $B$ are in different units (e.g. distance ÷ time = speed, or one currency ÷ another = exchange rate).

Before calculating anything, decide which situation you're in: "Am I splitting a whole into parts?" (add) or "Am I comparing two different quantities to get a rate?" (divide one by the other).

When two quantities are in direct proportion, their relationship is a straight line through the origin, $y = kx$, where $k$ is the constant of proportionality — exactly the unit rate from the paragraph above.

Here $k$ is also the gradient of the line: any two values that satisfy the same ratio (e.g. $10:15$ and its simplified form $2:3$) sit on the same line through $(0,0)$, since $k = \frac{10}{15} = \frac{2}{3}$ is unchanged.

If you're given an equation like $2y = 10x$, rearrange it first: dividing both sides by 2 gives $y = 5x$, so $k = 5$. If instead you're given one matching pair of values, substitute them directly into $k = y/x$.

Worked Example

A factory produces widgets ($W$) in direct proportion to the time ($T$) in minutes. In 15 minutes it produces 120 widgets. Find the equation connecting $W$ and $T$, then use it to predict the output in 2 hours.

1. Find $k$: $k = \dfrac{W}{T} = \dfrac{120}{15} = 8$ widgets per minute.
2. State the equation: $W = 8T$.
3. Convert 2 hours to minutes: $T = 120$.
4. Substitute: $W = 8 \times 120 = 960$ widgets.