ACARA v9 CONTENT DESCRIPTION “use mathematical modelling to solve practical problems involving ratios and rates, including financial contexts; formulate the problems and interpret and communicate solutions”
Modelling with ratios and rates
Many of the practical problems a student meets are quietly governed by a ratio or by a rate. A ratio shares or mixes quantities in fixed proportions, such as splitting prize money between two people in the ratio 2:3, mixing cordial with water, or scaling a recipe up for more guests. A rate gives an amount for each unit of something else, such as $25 for each hour worked, a price per kilogram at the market, or fuel used per 100 km on a trip. Mathematical modelling means turning the words of the situation into one of these structures and then reasoning with it. The first decision is which kind of structure fits: a ratio splits a whole into parts, while a rate gives an amount per unit. Naming the structure correctly is what makes the rest of the work straightforward.
Two kinds of situation
A ratio splits a whole; a rate gives an amount per unit.
ratios share or mix quantities (2:3), and rates give an amount per unit ($25/h); both can model practical problems.
The modelling cycle
Modelling is not a single calculation but a short cycle of four steps. First you formulate: read the situation and set up the ratio or the rate that describes it. Next you solve: carry out the arithmetic, dividing into parts for a ratio or multiplying the per-unit amount for a rate. Then you interpret: say what the answer actually means back in the situation, in plain words and with the right units or dollar signs. Finally you review: check that the answer is reasonable and that the proportional assumption really holds, for example that the split stays fixed or that the rate does not change. If the review shows the model does not fit, the cycle repeats with an adjusted model. The habit of finishing with a review is what separates a reliable answer from a lucky one.
The modelling cycle
Formulate, solve, interpret, review - then repeat if needed.
modelling follows a cycle: formulate the ratio or rate, solve it, interpret the answer in context, then review whether it is reasonable.
A ratio problem: sharing fairly
Consider sharing $60 between two people in the ratio 2:3. To formulate, add the ratio numbers: 2 + 3 = 5, so the money is divided into 5 equal parts. To solve, find the value of one part by dividing the total by the number of parts: 60 / 5 = $12 per part. The first person receives 2 parts, which is 2 x 12 = $24, and the second receives 3 parts, which is 3 x 12 = $36. The parts add back to the original total, since 24 + 36 = 60, and this built-in check is one of the nicest features of a ratio model: the shares must always rebuild the whole.
A ratio bar for sharing $60 in 2:3
Five equal parts of $12 rebuild the $60 total.
to share $60 in the ratio 2:3, make 5 parts of $12, giving $24 and $36, which add back to $60.
A rate problem: a financial plan
Now take a rate in a financial context. A casual job pays $25 per hour, and you want to know the pay for a 6 hour shift. To formulate, write the relationship as pay = rate x hours, that is pay = 25 x hours. To solve, substitute the 6 hours: 25 x 6 = $150. A rate works by scaling the per-unit amount up to the whole, so each extra hour simply adds another $25. The same style of model prices fruit by the kilogram, charges electricity by the unit, or estimates the fuel for a journey from a constant rate per 100 km.
A rate plan: $25 per hour
Scale the per-hour amount up to the whole shift.
at $25 per hour, 6 hours of work earns 25 x 6 = $150, a rate model in a financial context.
Interpreting and reviewing the answer
Interpreting means stating the answer in words that fit the situation: the two people get $24 and $36, or the shift pays $150. It is worth saying the answer as a full sentence rather than leaving a bare number. Checking comes next: add the shares back to the total, so 24 + 36 = 60, or run a quick unit-rate sanity check, since $150 for 6 hours is $25 an hour as expected. Reviewing then questions the assumptions behind the model. The ratio answer assumes a fixed 2:3 split, and the pay answer assumes a constant $25/h with no overtime. If the rate changed past a certain number of hours, or the split were not fixed, the model would need adjusting, and the cycle would begin again.
Interpret, check, review
State the answer, check it, then question the assumption.
state the answer in context, check it adds back to the total, and review whether the ratio or rate stays fixed.
Why this matters
Ratio and rate models quietly run a great many everyday decisions: splitting a bill, mixing a drink or a batch of concrete, pricing goods by weight, earning wages by the hour, and planning the fuel for a long trip. Having a clear cycle of formulate, solve, interpret and review turns these from guesswork into something dependable, and it gives a student a way to explain and defend an answer. The two slips to watch for are using the wrong number of parts in a ratio, often by forgetting to add the ratio numbers first, and stopping at a number without checking it back against the situation. This is the last Year 8 Measurement unit, and it brings together the ratio and rate ideas from the earlier units into a single way of working.
Quick self-check
1. You share $60 between two people in the ratio 2:3. How many equal parts is that?
2. Sharing $60 in the ratio 2:3, how much is one part worth?
3. So how much do the two people receive when $60 is shared in the ratio 2:3?
4. A job pays $25 per hour. How much is earned for 6 hours?