Ratios and proportions

A ratio compares, it does not count

"The ratio of red to blue marbles is $4 : 5$" says that for every 4 red marbles there are 5 blue ones. It does not say there are 4 red marbles. There could be 8 and 10, or 40 and 50. A ratio fixes the shape of a group, not its size.

So give the size a name. If red : blue $= 4 : 5$, write

$$\text{red} = 4k, \qquad \text{blue} = 5k$$

for some positive number $k$, the multiplier. Every ratio problem becomes a one-unknown equation once you do this. Any one actual figure, whether a total, a difference or one of the parts, pins down $k$.

Part to part, and part to whole

With red : blue $= 4 : 5$, the whole is $4k + 5k = 9k$. So red marbles are $\frac{4}{9}$ of the total, not $\frac{4}{5}$. The ratio $4 : 5$ is part to part; the fraction $\frac{4}{9}$ is part to whole. Read which one a question gives and which one it asks for. "The ratio of red marbles to all marbles" is $4 : 9$.

Ratios behave like fractions

A ratio $a : b$ can be written $\frac{a}{b}$ and simplified the same way: $18 : 24 = 3 : 4$. Multiplying or dividing both terms by the same number keeps the ratio. Adding the same number to both terms does not: $3 : 4$ becomes $4 : 5$ if you add 1 to each, and $\frac{4}{5} \ne \frac{3}{4}$. Adding the same positive amount to both terms always pulls the ratio towards $1 : 1$.

Three-way ratios through a shared term

If $a : b = 2 : 3$ and $b : c = 4 : 5$, the two ratios mention $b$ with different numbers, 3 and 4. Rescale both so that $b$ gets the same number, their least common multiple, 12:

$$a : b = 8 : 12, \qquad b : c = 12 : 15, \qquad \text{so } a : b : c = 8 : 12 : 15$$

Now $a : c = 8 : 15$. Writing $a : c = 2 : 5$, by simply dropping the middle numbers, is the classic error; it ignores that the two $b$ values were measured on different scales.

Proportions

A proportion says two ratios are equal: $\frac{a}{b} = \frac{c}{d}$. Cross-multiply to solve: $ad = bc$. If 3 kg of flour make 40 loaves, then $x$ kg make 100 loaves where $\frac{3}{40} = \frac{x}{100}$, so $x = 7.5$.

Quantities are directly proportional when their ratio is constant: double one and the other doubles ($y = kx$). They are inversely proportional when their product is constant: double one and the other halves ($xy = k$). Workers and time to finish a fixed job are the standard inverse pair. Ask before you set up the equation: should the answer go up or down?

When the ratio changes

Problems where items are added or removed are ratio problems in two stages. Write the "before" quantities with a multiplier, apply the change to the actual amounts, and set the new ratio equal to the "after" ratio. The multiplier belongs to the original group; never apply the change to the ratio numbers themselves.

Ratios in comparison questions

A ratio on its own never tells you an actual amount. If the only information is "the ratio of $x$ to $y$ is $3 : 4$", then the size of $x$ or of $x + y$ cannot be determined; even which of $x$ and $y$ is larger can flip if negatives are allowed. Ask whether the stem gives a real figure that fixes $k$. If not, look for what the ratio does fix, such as $\frac{x}{y}$ or the fraction of the total.

A quick check

When you have the parts, divide them back out: they should simplify to the given ratio, and they should add up to any given total. The check takes seconds and catches a misread part-to-whole.