Ratio and Proportions

A ratio is a way of comparing two or more amounts measured in the same units. The ratio $2 : 3$ tells you that for every $2$ of the first thing there are $3$ of the second. The numbers are called the parts and the colon is read as 'to'.

A few important things to keep in mind:

• The order matters. If a recipe uses flour and sugar in the ratio $2 : 3$, that is not the same as $3 : 2$ — swapping them would change the recipe completely.
• Units must agree. To write the ratio of $50$ cm to $1$ m, first convert: $50$ cm to $100$ cm gives $50 : 100$, which simplifies to $1 : 2$.
• A ratio is not the same as a fraction. In the ratio $2 : 3$ the first part is $\tfrac{2}{5}$ of the total (not $\tfrac{2}{3}$), because there are $5$ parts in all.

If a ratio has decimals or fractions in it, scale it up so each part is a whole number. The ratio $0.5 : 1.5$ becomes $1 : 3$ once both parts are doubled. Treating a ratio as a family of equivalent ratios — $1 : 3 = 2 : 6 = 5 : 15$ and so on — is the key idea behind nearly every ratio problem.

A ratio is in its simplest form when its parts share no common factor other than $1$. To simplify a ratio, find the highest common factor (HCF) of all the parts and divide every part by it.

So for $18 : 24$, the HCF of $18$ and $24$ is $6$. Dividing both sides by $6$ gives $18 \div 6 : 24 \div 6 = 3 : 4$.

For a three-part ratio you do the same thing — just include every part. The HCF of $12$, $18$ and $30$ is $6$, so $12 : 18 : 30 = 2 : 3 : 5$.

Some problems ask for a ratio in the form $1 : n$ or $n : 1$. To get the form $1 : n$, divide both parts by the first one. So $4 : 14$ in the form $1 : n$ becomes $1 : 3.5$. The shape is helpful for direct comparisons — for example map scales such as $1 : 50{,}000$.

Sharing in a ratio is a three-step routine. Suppose \120$is shared between Aki and Bea in the ratio$3 : 5$.

1. Add the parts. $3 + 5 = 8$ parts in total.
2. Find the value of one part. \120 \div 8 = $15$ per part.
3. Multiply by each share. Aki gets 3 \times \15 = $45$, Bea gets$5 \times $15 = $75$.

A quick check: \45 + $75 = $120$ — back to the total, as it should be.

The same routine works for three or more parts. If $560$ g of trail mix is made in the ratio $3 : 2 : 5$ of oats, raisins and nuts, the parts total $10$, so one part is $560 \div 10 = 56$ g. The mix then has $168$ g of oats, $112$ g of raisins and $280$ g of nuts.

If you are told the size of one share rather than the total, work backwards: "$3$ parts is \45$, so one part is$$15$, so$5$parts is$$75$, and the total is$$120$."

Two quantities are in direct proportion when one is always the same multiple of the other. If one doubles, the other doubles; if one halves, the other halves. Algebraically you write

$y \propto x,\quad\text{or}\quad y = kx,$

where $k$ is the constant of proportionality.

A classic example: a market sells $4$ apples for \2$. Pricing is in direct proportion, so the unit cost is$$0.50$ per apple. Then:

• $10$ apples cost 10 \times \0.50 = $5$.
• \3$buys$3 \div 0.50 = 6$ apples.

The unitary method — find what one of something costs (or is worth) first — turns every direct-proportion problem into a single multiplication.

A graph of $y$ against $x$ for two quantities in direct proportion is always a straight line through the origin, with gradient $k$. If the line passes through $(0, 0)$ and $(4, 6)$, then $k = \tfrac{6}{4} = 1.5$ and the relationship is $y = 1.5x$.

Two quantities are in inverse proportion when one rises as the other falls, in such a way that their product is constant. Algebraically,

$y \propto \frac{1}{x},\quad\text{or}\quad y = \frac{k}{x},\quad\text{which means}\quad xy = k.$

The everyday picture: if it takes $4$ painters $6$ days to finish a job, then the total work is $4 \times 6 = 24$ painter-days. With more painters working at the same pace the job finishes sooner:

• $8$ painters → $24 \div 8 = 3$ days.
• $3$ painters → $24 \div 3 = 8$ days.

Other inverse-proportion situations: more taps fill a tank faster, more wheels share the road tax, more people share the same pizza for a smaller slice each. The pattern is always more of one $\Rightarrow$ less of the other, with the product staying the same.

If $y = \tfrac{k}{x}$, a graph of $y$ against $x$ is a curve that gets closer to the axes but never touches them — a reciprocal curve.

Ratio and proportion thinking shows up almost every day:

• Cooking and recipes: scaling ingredients up or down for different numbers of people.
• Maps and scale drawings: a map at $1 : 50{,}000$ uses ratio to shrink the real world onto paper.
• Currency and unit conversion: $1$ unit of currency is worth $k$ units of another — direct proportion.
• Science: density, speed, concentration and many physics formulas are proportion in action.

When you face a future problem that involves comparing or scaling, ratio is usually the right tool — and the unitary method is usually the quickest route through.