Writing Ratios: Notation, Methods and Examples

What is ratio notation?

Ratio notation is the mathematical way to express how much of one quantity there is compared to another. It uses specific formatting to make comparisons clear without writing full sentences.

When describing a relationship, the notation acts as a mathematical shorthand. Instead of saying "there are four red counters for every three blue counters," we write the numerical comparison directly.

Name quantities in the requested order

The sequence of the numbers in a ratio must exactly match the sequence of the items requested. Reversing the numbers describes a completely different relationship.

The order of the values in a ratio is not interchangeable.

If a bowl has $5$ apples and $2$ oranges, the ratio of apples to oranges is $5:2$. Writing $2:5$ would incorrectly mean there are $2$ apples and $5$ oranges. Always read the problem carefully to identify the requested order before writing the numbers.

Write ratios with words, a colon and a fraction

There are three standard ways to write a ratio comparing a value $a$ to a value $b$. All three forms represent the exact same mathematical relationship.

• Word form: $a$ to $b$
• Colon notation: $a:b$
• Fraction form: $\dfrac{a}{b}$

These fractions behave mathematically like division. Fraction form is typically used when comparing a part to a whole, but it is also widely used in algebra for part-to-part comparisons.

Use labels and units correctly

When comparing measurements, the units must be identical before writing the ratio. If one length is given in centimeters and another is given in meters, convert them to the same unit first.

Ratios themselves do not have units. Because a ratio compares quantities of the exact same unit, the units mathematically cancel each other out during the comparison, leaving a pure relative size.

Simplify a written ratio

Ratios are usually written in their simplest form to make the relationship as clear as possible. To simplify a ratio, find the greatest common factor of both numbers and divide each side by that factor.

For an in-depth look at this procedure, explore simplifying ratios.

Worked examples

Here are three examples demonstrating how to write and simplify ratios correctly.

Example 1: Labelled counters

Question: A box contains $6$ red counters and $8$ yellow counters. Write the ratio of red counters to yellow counters in three different ways.

Method:

1. Identify the requested order: red then yellow.
2. Count the red counters ($6$) and yellow counters ($8$).
3. Write the ratio using words, a colon, and a fraction.

Answer: The ratio is $6$ to $8$, $6:8$, or $\dfrac{6}{8}$.

Check: The visual order correctly matches the word order requested.

Example 2: Recipe amounts

Question: A pancake recipe uses $2$ cups of milk and $4$ cups of flour. Write the simplified ratio of milk to flour.

Method:

1. Identify the order: milk to flour.
2. Write the initial ratio: $2:4$.
3. Divide both numbers by their greatest common factor, which is $2$.

Answer: The simplified ratio is $1:2$.

Check: $\dfrac{2}{2} = 1$ and $\dfrac{4}{2} = 2$. The proportion is maintained.

Example 3: Lengths with unit conversion

Question: A string is $40\text{ cm}$ long and a ribbon is $1\text{ m}$ long. What is the simplified ratio of the string's length to the ribbon's length?

Method:

1. Identify the different units: centimeters and meters.
2. Convert meters to centimeters so the units match: $1\text{ m} = 100\text{ cm}$.
3. Write the initial ratio in the requested order: $40:100$.
4. Simplify by dividing both numbers by their greatest common factor, which is $20$.

Answer: The simplified ratio is $2:5$.

Check: $20 \times 2 = 40$ and $20 \times 5 = 100$. The simplified ratio accurately scales back to the original measurements.

Common mistakes

The most common error is assuming the order of a ratio does not matter. Reversing the order describes a completely different situation.

Another frequent mistake is failing to convert units before comparing quantities. If you compare $50\text{ cm}$ directly to $2\text{ m}$ as $50:2$, the mathematical relationship will be completely wrong. Always convert to identical units first.

Frequently asked questions

What is a part-to-part ratio compared to a part-to-whole ratio?

A part-to-part ratio compares two distinct groups, like cars to trucks. A part-to-whole ratio compares one group to the total, like cars to all vehicles. Learn more about part-to-part and part-to-whole ratios.

Can a ratio have three numbers?

Yes. If a recipe calls for $2$ parts sugar, $3$ parts flour, and $1$ part butter, the ratio is written as $2:3:1$. Fraction form is generally not used for ratios with three or more parts.

Why does a ratio have no units?

Because a ratio compares quantities of the exact same unit, the units mathematically cancel each other out. The final ratio represents a pure multiplier rather than a physical measurement.