Percentages
Proportional Reasoning
Proportional reasoning is the general thinking skill that ties together everything covered in this entire section: recognising when two quantities relate to each other consistently, and using that relationship to find an unknown value. It is less a single technique and more a way of seeing the world – once you can spot a proportional relationship hiding inside a problem, from a recipe to a currency exchange to a scale drawing, you can solve it with confidence.
The most direct tool for proportional reasoning is solving a proportion equation – two equal ratios, like a/b = c/x – using cross-multiplication. This is the same “Rule of Three” method mentioned earlier in this section for direct proportion, generalised to work for any proportional situation you might encounter.
Solving a Proportion
For a/b = c/x, cross-multiply to get a × x = b × c, then solve for x by dividing.
Cross-multiply: 4 × x = 9 × 12 = 108. So x = 108 ÷ 4 = 27.
Applying Proportional Reasoning
Cost per pen = $6 ÷ 3 = $2. For 10 pens: 10 × $2 = $20.
Real-Life Application
- Currency exchange: converting any amount using a known exchange rate.
- Cooking: adjusting any recipe to any number of servings.
- Science: scaling chemical concentrations or dosages proportionally.
Key Takeaways
- Proportional reasoning means recognising and using a consistent relationship between two quantities.
- Cross-multiplication solves any proportion equation a/b = c/x.
- This skill draws together ratios, proportion, rates, and scale – the perfect bridge into the algebraic equations covered next.