Ratio Notation

$3:5$

and read as “three to five”. This means that for every 3 red counters, there are 5 blue counters. The numbers are called the parts of the ratio; they do not state the total number of counters.

Order matters. The ratio of blue counters to red counters is $5:3$, not $3:5$. A useful habit is to write the names above the corresponding parts:

$\begin{array}{cc}\text{red}&\text{blue}\\3&5\end{array}$

A ratio may have more than two parts. For example, $2:3:5$ could compare the amounts of three ingredients, in the order in which they are named.

Equivalent ratios

A group with 3 red counters and 5 blue counters has the same relative composition as one with 6 red counters and 10 blue counters. The second collection is twice as large in each category:

$3:5=6:10$

These are equivalent ratios. Multiplying every part by the same non-zero number preserves the comparison. Dividing every part by the same non-zero number also preserves it:

$12:18\rightarrow 6:9\rightarrow 2:3.$

The same operation must be applied to every part. Adding the same number does not preserve the relationship: $2:3$ is not equivalent to $3:4$.

Reducing a ratio to its simplest form

A ratio is in simplest form when its parts are whole numbers with no common factor greater than 1. To simplify $24:36$, find the highest common factor of 24 and 36. It is 12, so divide both parts by 12:

$24:36=(24\div12):(36\div12)=2:3.$

For a three-part ratio, divide all three parts by their shared HCF. For example, the HCF of 18, 30 and 42 is 6:

$18:30:42=3:5:7.$

If the HCF is not immediately clear, the ratio can be simplified in stages by repeatedly dividing every part by a common factor. The final result is the same.

Put quantities in the same unit first

Quantities must be expressed in the same unit before their numerical values can be compared and simplified. For example, simplify $250\text{ g}:2\text{ kg}$ by first converting $2\text{ kg}$ to $2000\text{ g}$:

$250\text{ g}:2000\text{ g}=250:2000=1:8.$

Once both quantities use the same unit, the units cancel from the ratio. Writing $250:2$ without converting would compare unlike numerical scales and give the wrong relationship.

Ratios containing decimals

A simplest whole-number ratio should not contain decimals. Multiply every part by the same power of 10 to make the parts whole numbers, then simplify. For example,

$1.2:0.9=12:9=4:3.$

Multiplication by 10 does not change the relationship because both parts are multiplied by the same amount.

Ratios containing fractions

To remove fractions, multiply every part by a common multiple of their denominators—usually the lowest common multiple. For example, the denominators in $\frac23:\frac45$ are 3 and 5, whose lowest common multiple is 15:

$\frac23:\frac45=10:12=5:6.$

Here, multiplying both parts by 15 clears the denominators; dividing both resulting parts by 2 then gives the simplest form.

A reliable checking routine

After simplifying, ask three questions:

1. Are the quantities written in the requested order?
2. Were different units converted before simplifying?
3. Are the final parts whole numbers with no common factor greater than 1?

If all three answers are yes, the ratio has been correctly written in simplest form.