Expressing Quantities as Fractions for the ESAT
Updated July 2026
Understanding how to express one quantity as a fraction of another is essential for the ESAT Mathematics 1 paper. This topic involves comparing two values by forming a ratio and ensuring units are consistent. You will learn to calculate fractions both less than and greater than one accurately.
To express a quantity as a fraction of another quantity , you write the relationship as , ensuring that both values are expressed in the same units before simplifying.
One quantity can be expressed as a fraction of another
In mathematics, we often need to compare two quantities by expressing one as a part of the other. This comparison is written as a fraction. If we want to express a quantity as a fraction of another quantity , the standard form is .
A critical requirement for this process is that both quantities must be in the same units. If the units differ, you must convert them so they match. Generally, it is most efficient to convert the larger unit into the smaller unit to avoid working with decimals prematurely.
Expressing one quantity as a fraction of another: fraction less than 1
When the first quantity is smaller than the second, the resulting fraction will be less than 1. This occurs frequently when calculating proportions of a whole.
Worked Example: Express 200 g as a fraction of 1 kg.
- Identify the units: The quantities are in different units, grams (g) and kilograms (kg).
- Convert to the same units: Change 1 kg to 1000 g.
- Form the fraction: 200 g as a fraction of 1000 g is written as .
- Simplify the fraction: Dividing both the numerator and the denominator by 200 gives .
Expressing one quantity as a fraction of another: fraction greater than 1
It is also possible for the first quantity to be larger than the second. In these cases, the fraction will be an improper fraction (top-heavy), meaning its value is greater than 1. This indicates that the first quantity is a multiple of the second.
Worked Example: Express 1 litre as a fraction of 450 ml.
- Convert to the same units: Start by writing both quantities in millilitres (ml). Since 1 litre is equal to 1000 ml, the comparison is between 1000 ml and 450 ml.
- Form the fraction: 1000 ml as a fraction of 450 ml is .
- Simplify the fraction: Dividing by the common factor 50 gives .
- Convert to a mixed number (optional): .
Key takeaways
- Always convert both quantities to the same unit before forming the fraction.
- The quantity being expressed is the numerator, and the quantity it is being compared to is the denominator.
- Fractions can be less than 1 or greater than 1 depending on the relative sizes of the quantities.
- Simplifying the resulting fraction to its lowest terms is standard practice in ESAT mathematics.
When simplifying large fractions, look for common factors like 10, 50, or 100 first to quickly reduce the numbers before looking for smaller primes.
The most common error is forgetting to convert units. Comparing 200 g to 1 kg by writing will lead to an incorrect answer: always ensure the 1 kg is converted to 1000 g first.
This topic is the foundation for percentages. Once you have a fraction like , multiplying by 100 gives you the percentage (20 percent). Mastery of this allows for easy movement between fractions, decimals, and percentages.
Frequently asked questions
Which unit should I convert to if the quantities have different units?
It is usually best to convert the larger unit to the smaller unit. For example, convert kilograms to grams or litres to millilitres. This prevents the introduction of decimals into your fraction, making simplification easier.
Can a fraction of a quantity be greater than 1?
Yes. If the first quantity is larger than the second quantity , the fraction will be greater than 1. For example, 500 metres as a fraction of 100 metres is .
Do I need to convert improper fractions to mixed numbers?
In many ESAT questions, an improper fraction such as is perfectly acceptable. However, you should be comfortable converting it to a mixed number like if the multiple-choice options require it.