Expressing a Quantity as a Fraction
This topic covers how to compare two quantities by writing one as a fraction of the other. It is a fundamental skill for setting up problems, especially when quantities are given in different units.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To express quantity 'A' as a fraction of quantity 'B', the correct structure is always A/B.
- The most critical rule is that both quantities must be converted to the same units before creating the fraction.
- It is usually simpler to convert the larger unit into the smaller unit to avoid working with decimals (e.g., convert metres to centimetres).
- If the first quantity is smaller than the second, the fraction will be less than 1.
- If the first quantity is larger than the second, the fraction will be greater than 1 (an improper fraction).
- Always simplify the final fraction to its lowest terms.
Formulae
Fraction = (First Quantity) / (Second Quantity) Use this structure when asked to express the 'First Quantity' as a fraction of the 'Second Quantity'. Ensure units are identical for both.
Definitions
- Expressing one quantity as a fraction of another
- Creating a ratio A/B to show how large quantity A is relative to a reference quantity B. This is a way of comparing their magnitudes.
Worked example
A scientific measurement takes 45 seconds. Express this time as a fraction of a 3-minute interval.
- 1
Identify the two quantities:
45 seconds and 3 minutes.
- 2
Notice the units are different.
Convert them to the same unit, preferably the smaller one (seconds).
- 3
Convert 3 minutes to seconds:
3 minutes × 60 seconds/minute = 180 seconds.
- 4
Set up the fraction using the formula:
(First Quantity) / (Second Quantity) = 45 / 180.
- 5
Simplify the fraction.
Both numbers are divisible by 45.
45/45 = 1 and 180/45 = 4 - 6
The final simplified fraction is 1/4.
Answer: 1/4
Common mistakes
- ×Forgetting to convert units: The most common error is creating a fraction with mixed units (e.g., 45/3 instead of 45/180), leading to a completely wrong answer.
- ×Inverting the fraction: Incorrectly placing the reference quantity in the numerator (e.g., writing B/A instead of A/B). Read the question carefully to see which quantity is being expressed 'as a fraction of' the other.
- ×Errors in simplification: Failing to simplify the fraction completely, or making arithmetic mistakes during simplification.
No-calculator tips
- ✓Convert to the smaller unit: Always convert the larger unit to the smaller one (e.g., hours to minutes, kg to g). This allows you to work with integers, which are easier to manipulate than decimals.
- ✓Simplify in stages: When simplifying a fraction like 180/240, don't try to find the highest common factor at once. Cancel out obvious factors first, like 10 (to get 18/24), then 2 (9/12), then 3 (3/4). This reduces the chance of mental arithmetic errors.