Ratio Notation for ESAT Mathematics

Q: How do I simplify a ratio that contains fractions?

To simplify a ratio like $\frac{1}{2} : \frac{3}{4}$, multiply both sides by the lowest common multiple of the denominators. In this case, multiplying both sides by 4 gives $2 : 3$.

Updated July 2026

Ratio notation is a fundamental method for comparing quantities in ESAT Mathematics 1. This topic covers expressing relationships between values, simplifying ratios into integer forms, and ensuring unit consistency. Mastering these skills is essential for solving more complex proportion problems in engineering and science contexts.

Core concept

A ratio expresses the relative sizes of two or more values. Written as $x : y$ or $x$ to $y$ , it indicates how much of one quantity exists in relation to another, where both parts must represent the same units for a direct comparison.

What is ratio notation?

Ratio notation is used to compare the quantities of two or more items. If a bag contains $x$ red sweets and $y$ yellow sweets, then the ratio of red sweets to yellow sweets is expressed as $x : y$ . This relationship can also be written in words as $x$ to $y$ .

In many cases, you may be asked to compare a single part to the total number of parts. Consider the following example:

Worked Example: Expressing a part-to-total ratio

Jana has 63 cm of red ribbon and 59 cm of blue ribbon. What is the ratio of the length of red ribbon to the total length of blue and red ribbon?

First, calculate the total length of the ribbon: $63 + 59 = 122$ cm.
Identify the length of the red ribbon, which is 63 cm.
The ratio of the red ribbon length to the total length is $63 : 122$ .

How to simplify ratios

Ratios are most useful when expressed in their simplest form. Both sides of a ratio can be multiplied or divided by the same positive number without changing the underlying relationship between the quantities. This is similar to the process of simplifying fractions.

Worked Example: Simplifying a ratio containing decimals

A mixture contains 2.5 kg of flour and 1.5 kg of sugar. What is the ratio of flour to sugar, by weight, in the mixture? Give your answer as a ratio of integers.

State the initial ratio: $2.5 : 1.5$ .
To remove the decimals and find an integer ratio, multiply both sides by a common factor. Multiplying both sides by 2 is an efficient choice here: $2.5 \times 2 : 1.5 \times 2$ .
The resulting ratio is $5 : 3$ .

Comparing quantities using ratios

A critical rule when working with ratios is that the units must be the same on both sides. If the quantities being compared are measured in different units, you must convert them to a common unit before writing the ratio.

Worked Example: Handling different units

A mixture contains 2.5 kg of flour and 750 g of sugar. What is the ratio of flour to sugar, by weight, in the mixture? Give your answer as a ratio in its lowest integer terms.

Notice the units are different: kilograms (kg) and grams (g). Convert the flour weight into grams so that both quantities are in the same units: $2.5 \text{ kg} = 2500 \text{ g}$ .
Write the ratio using the common unit: $2500 : 750$ .
Simplify the ratio by dividing both sides by the same number. Dividing both sides by 250 gives: $2500 \div 250 : 750 \div 250$ .
The simplified ratio is $10 : 3$ .

Key takeaways

Ratios can be written in the format $x : y$ or as the phrase $x$ to $y$ .
A ratio is simplified by multiplying or dividing both sides by the same positive constant.
Units must be identical on both sides of a ratio before a valid comparison or simplification can occur.
Ratios should generally be converted into their lowest integer terms for final answers.

Tips

Always read the question carefully to see if it asks for a ratio of one part to another part, or one part to the total. This is a common point of confusion in exam conditions.

Cautions

The most frequent error is forgetting to check the units. Comparing $1 \text{ m}$ to $50 \text{ cm}$ as $1 : 50$ is incorrect: it must be converted to $100 : 50$ and then simplified to $2 : 1$ .

Insight

Ratios are a specific way of looking at multiplicative relationships. While fractions represent a part of a whole, ratios can represent part to part or part to whole, making them highly versatile in engineering fields like material science and structural scaling.

Frequently asked questions

Can ratios contain more than two numbers?

Yes, ratios can compare multiple quantities simultaneously, such as $x : y : z$ . The rules for simplification and unit consistency apply to all parts of the ratio equally.

Do I need to include units in my final ratio answer?

No, ratios are dimensionless quantities. Once you have converted the parts to the same units, the units effectively cancel out, leaving a pure numerical relationship.

How do I simplify a ratio that contains fractions?

To simplify a ratio like $\frac{1}{2} : \frac{3}{4}$ , multiply both sides by the lowest common multiple of the denominators. In this case, multiplying both sides by 4 gives $2 : 3$ .

Ready to test your knowledge?

You've reached the end of this section. Start a practice session to solidify your understanding and master this topic.

Expressing One Quantity As A Fraction Of Another

Dividing Quantities In Ratios