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 two hundred and forty kilometres in three hours has a speed of eighty kilometres per hour; at that rate it would cover four hundred kilometres in five hours. Unit pricing is another everyday rate: four dollars fifty for three litres is a dollar fifty per litre. Rates make comparisons fair, which is exactly how you settle a best-buy question: five hundred grams for three dollars works out at six dollars per kilogram, while eight hundred grams for four dollars forty is only five dollars fifty 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 sixty dollars in the ratio two to three, add the parts to get five, divide to find one part worth twelve dollars, then multiply out to twenty-four dollars and thirty-six dollars. The same method scales recipes: if flour and sugar are in the ratio three to one and you use six hundred grams of flour, you need two hundred 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 one to fifty thousand means one unit on the map represents fifty thousand of the same units in reality. So four centimetres on that map is two hundred thousand centimetres on the ground, which is two kilometres. The same idea shrinks the world onto a page or a model: a one to one hundred floor plan turns a five metre room into a tidy five centimetre line, and a one to eighteen model car represents a four and a half metre vehicle in twenty-five 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 nought point six five United States dollars, then two hundred Australian dollars converts to one hundred and thirty 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.