Define variables and assumptions for a model

A mathematical model is a rule or equation that stands in for a real situation. Defining variables and assumptions means deciding, before any algebra, what each letter stands for, which units it carries and what you are pretending is true. Modelling questions in International Mathematics reward this setup, because the equation is only as useful as the choices behind it.

This is the first lesson in International Mathematics modelling. The later lessons fit, read and test models, and they all start from the definitions you write here.

What goes into a good setup?

Think of it as four short statements.

1. The quantity you want to find. Give it a letter and units, for example “C is the total cost in RM”.
2. The input that you can change or measure. Again give a letter, units and any limits, for example “n is the number of shirts, a whole number with n ≥ 0”.
3. The assumptions. Say what stays fixed or is ignored, for example “the price per shirt stays the same for any order”.
4. The equation linking them. Only now do you write the model.

Notice that the equation comes last. If you write it first, you cannot tell whether a number in it is a rate, a fixed charge or a guess.

Worked example

A class club orders printed T-shirts. The printer charges a design fee of RM60 once, plus RM18 for each shirt. Write a model for the total cost, then use it for an order of 25 shirts.

Step 1, the quantity wanted: let C be the total cost in RM.

Step 2, the input: let n be the number of shirts, a whole number with n ≥ 0.

Step 3, assumptions: the price per shirt stays RM18 for every order size, the design fee is paid once, and there is no delivery charge or tax.

Step 4, the model: C = 60 + 18n.

Step 5, sanity check: with n = 0, C = 60, which is the design fee alone. With n = 1, C = 78, which is the fee plus one shirt. Both make sense.

Step 6, use it: for n = 25, 18 × 25 = 450, then C = 60 + 450 = 510.

The cost of 25 shirts is RM510, under the stated assumptions. Check: 18 × 25 = 18 × 100 ÷ 4 = 450, and 450 + 60 = 510.

The mistake to watch for

A common slip is to swap the fixed charge and the rate.

Mistaken model: C = 18 + 60n

For n = 25 this gives 18 + 1500 = 1518, which is far too high for 25 shirts.

The test that catches it is the n = 0 check. Under this model, ordering nothing costs RM18, but the question says the design fee is RM60.

The number that is added once is the fixed charge (60), and the number that multiplies n is the rate per shirt (18). The correct model is C = 60 + 18n.

Check yourself

Try these on paper, then open each answer.

1. A gym charges a joining fee of RM50 and RM35 for each month of membership. Define the variables, write a model for the total cost and find the cost after 6 months.

2. A plant is 12 cm tall when measured for the first time and grows 2.5 cm each week. Write a model for its height h cm after w weeks, state two assumptions and find the height after 8 weeks.

3. A student writes “let t = the time”. Give two things that are missing from this definition.

Where this leads next

With the setup written, the next step is to choose numbers for the model from data: see fit a simple model to supplied data. When you want mixed questions, try the modelling practice set. The non-calculator working trainer is useful for checking your substitution arithmetic.

Some students can write the equation but leave out the assumptions that earn the explanation marks. Our teachers look for exactly that habit in online one-to-one Mathematics tuition.