Understanding Ratio Notation
Ratios are a fundamental way to compare the relative sizes of two or more quantities without using specific units. This is a core skill that underpins many other ESAT topics, such as scaling diagrams, chemical mixtures, and interpreting proportional relationships.
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
- A ratio written as a: b compares the size of quantity 'a' to quantity 'b'. The order is critical and must match the wording of the question.
- Before forming a ratio, all quantities must be expressed in the same units. For example, compare grams to grams, not kilograms to grams.
- Ratios can be simplified just like fractions by dividing all parts by their highest common factor. The goal is usually to express the ratio using the smallest possible integers.
- If a ratio involves decimals or fractions, you can convert it to an integer ratio by multiplying all parts by a suitable number (e.g., multiply by 10 to clear a decimal, or by a common denominator to clear fractions).
- Distinguish between a part-to-part ratio (e.g., apples to oranges) and a part-to-whole ratio (e.g., apples to total fruit).
Formulae
a : b is equivalent to (k × a) : (k × b) This principle is used to simplify ratios (when k is a fraction, like 1/HCF) or to scale them up (when k is an integer or used to clear decimals/fractions).
Definitions
- Ratio
- A comparison of the relative sizes of two or more quantities, expressed in the form a: b.
- Simplest Form
- A ratio where all its parts are integers and the only common factor between them is 1.
Worked example
A metal alloy is created by mixing 0.6 kg of copper with 150 g of tin. What is the ratio of copper to the total mass of the alloy, expressed in its simplest integer form?
- 1
First, ensure all quantities are in the same unit.
Let's convert kilograms to grams.
1 kg = 1000 g - 2
Convert the mass of copper to grams:
0.6 kg × 1000 g/kg = 600 g.
- 3
The question asks for the ratio of copper to the *total mass*.
Calculate the total mass:
600 g (copper) + 150 g (tin) = 750 g.
- 4
Form the ratio of copper to total mass:
600 :
750.
- 5
Simplify the ratio.
A common factor is 10, so it becomes 60 :
75.
- 6
Both 60 and 75 are divisible by 15.
60 / 15 = 4 and 75 / 15 = 5 - 7
The simplest form of the ratio is 4 :
5.
Answer: 4 : 5
Common mistakes
- ×Failing to convert all parts to a common unit before forming the ratio, for instance, writing 0.6 : 150 instead of 600 : 150.
- ×Providing a part-to-part ratio when a part-to-whole ratio is requested. In the example, this would be giving the copper-to-tin ratio (4 : 1) instead of the copper-to-total ratio (4 : 5).
- ×Incorrectly ordering the ratio. If asked for A to B, writing B : A is a common mistake.
No-calculator tips
- ✓To simplify ratios with large numbers, cancel out common factors in stages. For 600 : 750, first divide by 10 to get 60 : 75, then spot that both end in 0 or 5, so they are divisible by 5, giving 12 : 15. Finally, divide by 3 to get 4 : 5.
- ✓To eliminate decimals, multiply by powers of 10. A ratio of 0.2 : 1.4 can be quickly converted to an integer ratio by multiplying both sides by 10, giving 2 : 14, which simplifies to 1 : 7.