Modelling with Proportion, Rates, Ratio and Scale
Modelling with four related tools
This unit returns to mathematical modelling, the cycle of turning a real problem into mathematics, solving it, and judging how well the answer fits, but now the models are built from four closely related ideas: direct proportion, rates, ratio and scale. Each describes how one quantity relates to another in a constant, predictable way, and each turns up constantly in shopping, cooking, travel, maps and money. As always, the full cycle matters: formulate the model, solve it, interpret the result in context, then evaluate and report.
Direct proportion: a constant multiple
Direct proportion is the simplest relationship. Two quantities are directly proportional when one is always a constant multiple of the other, so doubling one doubles the other and their ratio never changes. If five apples cost four dollars, the cost is proportional to the number of apples, with a constant of eighty cents each. Twelve apples therefore cost nine dollars sixty. The constant multiplier, often called the constant of proportionality, is the heart of the model: find it once and you can scale to any quantity.
Rates: comparing different units
A rate is a special ratio that compares two quantities measured in different units, and it almost always carries those units with it. Speed compares distance with time, so a car travelling 240 kilometres in 3 hours has a speed of 80 kilometres per hour; at that rate it would cover 400 kilometres in 5 hours. Unit pricing is another everyday rate: $4.50 for 3 litres is $1.50 per litre. Rates make comparisons fair, which is exactly how you settle a best-buy question: 500 grams for $3 works out at $6 per kilogram, while 800 grams for $4.40 is only $5.50 per kilogram, so the larger pack is better value.
Ratio: sharing and mixing
Ratio compares quantities of the same kind and is the natural tool for sharing or mixing. To share $60 in the ratio 2 : 3, add the parts to get 5, divide to find one part worth $12, then multiply out to $24 and $36. The same method scales recipes: if flour and sugar are in the ratio 3 : 1 and you use 600 grams of flour, you need 200 grams of sugar. A three-part mixture works identically, splitting a total into shares that add back to the whole.
Scale: maps, plans and models
Scale is ratio applied to lengths, the principle behind every map, plan and model. A scale of 1 : 50 000 means one unit on the map represents 50 000 of the same units in reality. So 4 centimetres on that map is 200 000 centimetres on the ground, which is 2 kilometres. The same idea shrinks the world onto a page or a model: a 1 : 100 floor plan turns a 5 metre room into a tidy 5 centimetre line, and a 1 : 18 model car represents a 4.5 metre vehicle in 25 centimetres.
Money as proportion and rate
Money problems are a rich source of proportion and rate models, and they behave exactly like the examples above. Currency conversion is direct proportion: if one Australian dollar buys 0.65 United States dollars, then 200 Australian dollars converts to 130 United States dollars at that constant rate. It is worth stressing that these are constant-rate relationships, not the compounding growth seen in some financial settings, so a single multiplier does the whole job. Reading a problem to decide which of the four tools fits is the first and most important modelling step.
Evaluate and report the model
Finally, the modelling cycle is only complete when you evaluate and report. A best-buy calculation assumes you actually need the larger quantity and that it will not spoil; a currency rate ignores the fees a real bank would charge; a map scale assumes the ground is flat. Each model is a simplification, trustworthy only within its assumptions, and a careful report states both the answer and those limits. With direct proportion, rates, ratio and scale in your toolkit, and the discipline to interpret and evaluate, a huge range of practical problems becomes straightforward to model and solve.