Expressing Multiplicative Relationships as Ratios and Fractions Revision Notes | GCSE AQA Higher Maths

1. Understanding Multiplicative Relationships

When one quantity is always a fixed multiple of another, we call this a multiplicative relationship. Think about buying apples at £2 per kilogram - if you buy 3kg, you pay £6; if you buy 5kg, you pay £10. The cost is always exactly twice the weight.

These relationships appear everywhere in mathematics and real life. Recipe scaling, currency conversion, speed calculations - they all involve one quantity being a constant multiple of another. The key insight is recognising when this pattern exists.

Once you spot a multiplicative relationship, you can express it in two powerful ways: as a ratio or as a fraction. Both representations give you different tools for solving problems and making comparisons.

2. Expressing Relationships as Ratios

A ratio compares quantities by showing their relative sizes. When you have a multiplicative relationship, the ratio tells you exactly how the quantities relate to each other. If apples cost £2 per kilogram, the ratio of cost to weight is 2:1.

Ratios can be written in different forms but represent the same relationship. The ratio 2:1 is equivalent to 4:2, 6:3, or 10:5. We often simplify ratios to their lowest terms by dividing both parts by their highest common factor.

The beauty of ratios is they work both ways. If the cost to weight ratio is 2:1, then the weight to cost ratio is 1:2. This flexibility helps you tackle problems from different angles.

Worked example: Converting a Multiplicative Relationship to a Ratio

A car travels 120 miles using 8 litres of petrol. Express the relationship between distance and fuel as a ratio.

1. Distance = 120 miles, Fuel = 8 litres — We need to compare these two quantities
2. Distance : Fuel = 120 : 8 — Put the quantities in the order stated in the question
3. HCF of 120 and 8 is 8 — Both numbers divide exactly by 8
4. 120 ÷ 8 : 8 ÷ 8 = 15 : 1 — This gives the simplest form of the ratio

Answer: 15:1 (the car travels 15 miles for every 1 litre of petrol)

3. Expressing Relationships as Fractions

Fractions offer another way to express multiplicative relationships. Instead of comparing quantities side by side like ratios do, fractions show one quantity as a part of another. If 3 eggs need 200g flour, we can write this as the fraction $\frac{200}{3}$ grams of flour per egg.

The fraction form is particularly useful for calculations. When you need to scale up or down, you multiply by the fraction. Need flour for 7 eggs? Calculate $7 \times \frac{200}{3} = \frac{1400}{3} = 466\frac{2}{3}$ grams.

Fractions also reveal rates clearly. The fraction $\frac{120 \text{ miles}}{8 \text{ litres}} = 15$ miles per litre shows the fuel efficiency directly. This rate becomes your multiplier for any calculation.

Worked example: Converting a Ratio to a Fraction

The ratio of boys to girls in a class is 3:5. Express this as fractions of the total class.

1. Total parts = 3 + 5 = 8 — Add all parts of the ratio together
2. Boys = $\frac{3}{8}$ of the class — Boys form 3 parts out of 8 total parts
3. Girls = $\frac{5}{8}$ of the class — Girls form 5 parts out of 8 total parts
4. $\frac{3}{8} + \frac{5}{8} = \frac{8}{8} = 1$ — This confirms our fractions represent the whole class

Answer: Boys: $\frac{3}{8}$, Girls: $\frac{5}{8}$ of the total class

4. Converting Between Ratios and Fractions

Moving between ratio and fraction forms gives you flexibility in problem-solving. Sometimes a ratio makes the relationship clearer; other times a fraction is more useful for calculations. The key is recognising which form suits your current need.

When converting from ratios to fractions, you're often finding what fraction each part represents of the whole. A ratio of 2:3 means the first quantity is $\frac{2}{5}$ of the total and the second is $\frac{3}{5}$ of the total.

Converting from fractions to ratios involves using the numerators directly as ratio parts when fractions share the same denominator. If denominators differ, first convert to a common denominator, then use the numerators.

Worked example: Converting Fractions to a Ratio

In a mixture, water makes up $\frac{2}{7}$ and oil makes up $\frac{5}{7}$. What is the ratio of water to oil?

1. Water: 2 parts, Oil: 5 parts — The numerators tell us the relative amounts
2. Water : Oil = 2 : 5 — Use the numerators as the ratio parts
3. HCF of 2 and 5 is 1 — Since HCF = 1, the ratio is already in simplest form

Answer: 2:5 (water to oil)

5. Problem-Solving with Ratios and Fractions

Real problems often hide multiplicative relationships behind everyday language. 'For every 3 apples, there are 2 oranges' describes a 3:2 ratio. 'The car uses 1 litre per 15 miles' gives you the fraction $\frac{1}{15}$ litres per mile.

The strategy is to identify the two quantities being compared, determine their relationship, then choose whether ratio or fraction form helps you solve the problem more easily. Scaling problems often work better with fractions, while comparison problems suit ratios.

Watch out for the direction of the relationship. 'Miles per litre' is different from 'litres per mile' - they're reciprocals of each other. Always check your answer makes sense in the context of the original problem.

Worked example: Solving a Scaling Problem

A paint mixture uses red and blue paint in the ratio 5:3. If you need 24 litres of blue paint, how much red paint do you need?

1. Red : Blue = 5 : 3 — This means 5 parts red for every 3 parts blue
2. Blue paint needed = 24 litres, ratio part = 3 — We need to scale up from 3 parts to 24 litres
3. Scale factor = 24 ÷ 3 = 8 — Each ratio part represents 8 litres
4. Red paint = 5 × 8 = 40 litres — Multiply the red ratio part by the scale factor

Answer: 40 litres of red paint

6. Exam Techniques and Common Mistakes

Exam questions on multiplicative relationships often test whether you can spot the relationship and choose the right representation. Look for key phrases like 'for every', 'per unit', or 'in the ratio'. These signal multiplicative relationships that you can express as ratios or fractions.

When working with ratios, always simplify to lowest terms unless the question asks otherwise. For fractions, ensure you're clear about what the numerator and denominator represent - is it parts of a whole, or a rate comparing two different quantities?

Context is crucial for checking your answers. If you calculate that a car does 0.1 miles per litre, that should trigger alarm bells - cars typically do 10-15 miles per litre. Always sense-check your final answer against real-world expectations.