ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems involving direct proportion, rates, ratio and scale, including financial contexts; formulate the problems and interpret solutions in terms of the situation; evaluate the model and report methods and findings”
This unit builds on ratio, fractions and percentages, and shares the modelling cycle with the algebra modelling unit. Proportional reasoning is among the most widely used mathematics in everyday life and across the sciences.
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.
Four tools, one cycle
Direct proportion, rates, ratio and scale are four ways of describing a constant relationship; each one feeds the same modelling cycle.
Direct proportion, rates, ratio and scale are four ways of describing a constant relationship between two quantities. Each one feeds the same modelling cycle: formulate the model, solve it, interpret the result, 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.
Direct proportion is a straight line through the origin
A constant of 0.8 fixes the line cost = 0.8 times apples; the points (5, $4) and (12, $9.60) both lie on it.
Five apples for four dollars fixes the constant at eighty cents each, so cost = 0.8 × apples is a straight line through the origin. The points (5, $4) and (12, $9.60) both sit on it, which is exactly what makes the relationship a direct proportion.
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.
A rate compares different units
Distance over time gives 80 km/h; price over mass settles a best buy, where the lower $/kg wins.
A rate carries two different units, like the eighty kilometres per hour from 240 km in 3 h. The same idea settles a best buy: $6.00/kg for the small pack against $5.50/kg for the large one, so the lower rate, the 800 g pack, wins.
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.
Sharing in a ratio
Five equal parts make one part $12, so $60 split 2:3 gives shares of $24 and $36 that add back to $60.
Sharing $60 in the ratio 2:3 means five equal parts. Dividing gives one part = $12, so the shares are two parts ($24) and three parts ($36), which add back to $60.
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.
Scale shrinks reality
Scale is a ratio between drawing and reality: 4 cm at 1:50000 is 2 km, and a 5 m room becomes 5 cm at 1:100.
A scale is a ratio between drawing and reality. At 1:50000, four centimetres stands for 200000 cm, which is 2 km; at 1:100, a five metre room shrinks to a 5 cm line on the plan.
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.
Money as a constant rate
Currency conversion is direct proportion: at 1 AUD = 0.65 USD, 200 AUD maps to 130 USD on a line through the origin.
Currency conversion is a direct proportion: at 1 AUD = 0.65 USD, the graph is a straight line through the origin and 200 AUD maps to 130 USD. It is a constant rate, not the compounding growth of interest.
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.
Quick self-check
1. If 5 apples cost $4, how much do 12 apples cost at the same rate?
2. A car travels 240 km in 3 hours. What is its average speed?
3. Share $60 in the ratio 2:3. What is the larger share?
4. On a map with scale 1:50000, a road is 4 cm long. How long is it in reality?
5. A recipe uses flour and sugar in the ratio 3:1. For 600 g of flour, how much sugar is needed?